Mixed powerdomains for probability and nondeterminism
LLogical Methods in Computer ScienceVol. 13(1:2)2017, pp. 1–84https://lmcs.episciences.org/ Submitted Mar. 23, 2015Published Jan. 24, 2017
MIXED POWERDOMAINS FOR PROBABILITY ANDNONDETERMINISM
KLAUS KEIMEL a AND GORDON D. PLOTKIN ba Fachbereich Mathematik, Technische Universit¨at Darmstadt b Laboratory for Foundations of Computer Science, School of Informatics, University of Edinburgh
Abstract.
We consider mixed powerdomains combining ordinary nondeterminism andprobabilistic nondeterminism. We characterise them as free algebras for suitable (in)equation-al theories; we establish functional representation theorems; and we show equivalenciesbetween state transformers and appropriately healthy predicate transformers. The extendednonnegative reals serve as ‘truth-values’. As usual with powerdomains, everything comesin three flavours: lower, upper, and order-convex. The powerdomains are suitable convexsets of subprobability valuations, corresponding to resolving nondeterministic choice beforeprobabilistic choice. Algebraically this corresponds to the probabilistic choice operator dis-tributing over the nondeterministic choice operator. (An alternative approach to combiningthe two forms of nondeterminism would be to resolve probabilistic choice first, arriving at adomain-theoretic version of random sets. However, as we also show, the algebraic approachthen runs into difficulties.)Rather than working directly with valuations, we take a domain-theoretic functional-analytic approach, employing domain-theoretic abstract convex sets called Kegelspitzen;these are equivalent to the abstract probabilistic algebras of Graham and Jones, but aremore convenient to work with. So we define power Kegelspitzen, and consider free algebras,functional representations, and predicate transformers. To do so we make use of previouswork on domain-theoretic cones (d-cones), with the bridge between the two of them beingprovided by a free d-cone construction on Kegelspitzen. Introduction
In this paper we investigate mixed powerdomains combining ordinary and probabilisticnondeterminism. These can be defined generally as free algebras over dcpos (directedcomplete posets). The algebraic laws we consider in this regard are for the binary choiceoperators ∪ and + r of ordinary and probabilistic nondeterminism as well as a constant fornontermination (where x + r y expresses a choice of x with probability r versus one of y with Key words and phrases: domain, powerdomain, nondeterminism, probability, functional representation,predicate transformer, d-cone, domain-theoretic functional analysis.
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.23638/LMCS-13(1:2)2017 c (cid:13)
K. Keimel and G. D. Plotkin CC (cid:13) Creative Commons
K. KEIMEL AND G. D. PLOTKIN probability 1 − r ) together with an axiom to the effect that probabilistic choice distributesover ordinary nondeterministic choice, viz.: x + r ( y ∪ z ) = ( x + r y ) ∪ ( x + r z )(see below for axioms for ordinary and probabilistic nondeterminism, or, for example, [16]).We characterise the free algebras as suitable convex sets of subprobability valuations in thecase of domains (continuous dcpos). We do this for all three domain-theoretic notions ofordinary nondeterminism, viz. lower (or Hoare), upper (or Smyth), and convex (or Plotkin),though in the last case we need an additional assumption, that the domains are coherent.We further give suitable notions of predicates and predicate transformers, obtaining adual correspondence between predicate transformers and (mixed) nondeterministic functions(i.e., Kleisli category morphisms). The relevant notions of predicate use R + , the domain ofthe non-negative reals extended with a point at infinity, or its convex powerdomain, as ‘truth-values.’ Our results on predicate transformers are obtained via functional characterisationsof the mixed powerdomains, and are again obtained for all three notions of ordinarynondeterminism. As before these results obtain generally for domains except in the convexcase where, additionally, coherence is again required.In previous joint work with Regina Tix [61, 28] based on Tix’s Ph.D. thesis [60], we carriedout a similar programme for ordinary and so-called ‘extended’ probabilistic computationwhere valuations take values in R + rather than [0 , . By embedding Kegelspitzen ind-cones we are able to make use of our previous results, thereby avoiding a good deal of work.This approach works particularly well when characterising the mixed powerdomains, butless well when considering functional representations. We do obtain strong enough abstractresults for their intended application. However the general results require assumptions(taken from previous work) on the cones in which the Kegelspitzen are embedded, ratherthan natural assumptions directly concerning the Kegelspitzen themselves. One wonders to The German word Kegelspitze means ‘tip of a cone’, suggesting a convex set obtained by cutting off thetop of a cone.
IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 3 what extent one can succeed with more natural assumptions, and, indeed, to what extentone can proceed without making use of d-cones.Several other authors have previously considered the combination of ordinary andprobabilistic nondeterminism, making use of sets of distributions. In a domain-theoreticcontext, the pioneers were the Oxford Programming Research Group [44, 40, 39, 41]. Thatwork was restricted to the case of countable discrete domains, as was that of Ying [69]. Later,Mislove [42] defined mixed powerdomains for all three notions of ordinary nondeterminismover continuous domains required to be coherent in the convex case. Mislove also considerednondeterministic powerdomains over ‘abstract probabilistic algebras’. With the additionof a bottom element, these are the same as the identically named algebras of Graham andJones [14, 18] and equivalent to our Kegelspitzen; however Kegelspitzen seem more in thespirit of domain theory, as we discuss below.Goubault-Larrecq [9, 10, 11, 12] worked at a topological level, considering all three notionsof ordinary nondeterminism combined with various classes of valuations: all valuations,subprobability valuations, and probability valuations, but without explicit consideration ofthe algebraic structures involved. He established functional representation results under quiteweak assumptions on the underlying spaces. When specialised to domains and subprobabilityvaluations, his results correspond to our Corollaries 4.4, 4.7, and 4.10. He worked directlywith the valuation spaces rather than, as we do, making use of abstract structures such ascones and barycentric algebras. In [2, 3] Beaulieu worked algebraically; his results includefree constructions of algebras satisfying the above laws over sets and partial orders, but notdomains.There has been some discussion of other ways to combine nondeterminism with proba-bility. Categorical distributive laws provide a standard means of showing the composition oftwo monads form a third (see [38]). However there is no such law enabling one to composethe monad of ordinary nondeterminism with that of probabilistic nondeterminism — seethe Appendix of [64]. For this reason, Varacca and Winskel [62, 63, 64] reject, or weaken,one of the axioms of (extended) probabilistic nondeterminism, viz. the distributivity law( r + s ) x = rx + sx , and consider certain ‘indexed valuations’ in place of the more usual ones.As shown by Varacca in [63, Chapter 4] this approach applies to domains, where indexedvaluations come in three flavours: Hoare, Smyth, and Plotkin. Categorical distributivity lawsare obtained in several cases, and a freeness result (with the above equational distributivelaw) is given in the case of the combination of Hoare indexed valuations and the Hoarepowerdomain.In [8, 11], and see too [13], Goubault-Larrecq also combined the two types of monads in adifferent order, considering the probabilistic powerdomains over the nondeterministic power-domains; however no algebraic aspects were discussed. This approach is in the spirit of whatis known under the name of random sets in probability theory. The approach can be thoughtof as resolving probabilistic choice before nondeterministic choice; in contrast, approaches K. KEIMEL AND G. D. PLOTKIN employing sets of distributions can rather be thought of as resolving nondeterministic choicefirst.Algebraically, the above distributive law corresponds to resolving nondeterministic choicefirst. The other distributive law x ∪ ( y + r z ) = ( x ∪ y ) + r ( x ∪ z )corresponds to first resolving probabilistic choice. As pointed out in [43] this law leads tosome odd consequences when combined with the other laws, particularly the idempotenceof nondeterministic choice. In Appendix A we show that the equational theory consistingof the laws for nondeterministic and probabilistic choice together with this distributivelaw is equivalent to that of join-distributive bisemilattices [52], i.e., of algebras with twosemilattice operations called join and meet, with join distributing over meet. It thereforehas no quantitative content.An algebraic treatment of this combination of the two forms of nondeterminism wouldtherefore have to weaken some law. One natural possibility is to drop the idempotence ofnondeterministic choice (this was done in a process calculus-oriented context in [68, 4]).There is a natural ‘finite random sets’ functor supporting models of this weaker theory,namely D ω ◦ P + ω where D ω is the finite probability distributions monad, and P + ω is thefinite non-empty sets monad. The barycentric structure is evident, and the nondeterministicchoice operations ∪ X : D ω P + ω ( X ) → D ω P + ω ( X ) are given by:( (cid:88) i = 1 ,...,m α i x i ) ∪ X ( (cid:88) i = 1 ,...,n β j y j ) = (cid:88) i = 1 ,...,mj = 1 ,...,n α i β j ( x i ∪ y j )Unsurprisingly, these finite random set algebras do not provide the free algebras for theweaker theory. More surprisingly, perhaps, they do not provide the free algebras for any equational theory over the relevant signature; this too is shown in Appendix A. As anotherpoint along these lines, we recall a result of Varacca [63, Proposition 3.1.3] that there isno distributive law of P + ω over D ω . Overall, it seems hard to see how there can be anysatisfactory algebraic treatment of the combination of probability and nondeterminism inwhich probabilistic choice is resolved first.In Section 2 we develop the theory of Kegelspitzen, beginning with a notion of (ordered)barycentric algebra. This notion is based on the equational theory of the barycentricoperations rx + (1 − r ) y (for real numbers r between 0 and 1) on convex sets in vectorspaces. There is an extensive relevant literature, which we survey. We need this notionaugmented with a compatible partial order and, in addition, with a distinguished element 0.Finally, specialising to directed complete partial orders and Scott-continuous operations, weintroduce the central notion of Kegelspitzen, axiomatising subprobabilistic powerdomains. Inorder to relate these structures to our previous work on cones, we prove embedding theoremsat the various levels, and then establish the preservation of crucial properties. In the case ofKegelspitzen, the embedding theorem is Theorem 2.35 and the property-preserving results IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 5 are those of Propositions 2.42, 2.43 and 2.44, and Corollary 2.45; the properties preservedinclude continuity and coherence.In Section 3 we define, and give universal algebraic characterisations of, the variousmixed powerdomains, first doing the same for suitable notions of power Kegelspitze. Theuniversal characterisations of the free power Kegelspitzen are given in Theorems 3.4, 3.9,and 3.14 (one for each notion of nondeterminism). The universal characterisations of themixed powerdomains then follow, and are given in Corollaries 3.15, 3.16, and 3.17; the firsttwo hold for any domain, and the third holds for any coherent domain.In Section 4, we consider functional representations. The three functional representationsof power Kegelspitzen are given by Theorems 4.2, 4.6, and 4.9; they largely follow straightfor-wardly from the corresponding results for cones in [28]. The corresponding three functionalrepresentations of the mixed powerdomains over domains are given by Corollaries 4.4, 4.7,and 4.10 and are derived from the corresponding results for Kegelspitzen.In Section 5 we consider predicate transformers for domains, showing the equivalenceof ‘state transformers’, i.e., Kleisli maps, and ‘healthy’ predicate transformers, viz. mapson predicates obeying suitable conditions. The conditions and equivalences follow fromthe functional representation theorems for domains, and are given by Corollaries 5.1, 5.2,and 5.4. There are related general results for Kegelspitzen; these follow from the functionalrepresentation theorems for Kegelspitzen, and are discussed briefly.The results given in Sections 4 and 5 all make use of R + , the d-cone of the non-negativereals augmented by a point at infinity; this is unsurprising given their derivation from thecorresponding results for cones. However, in the context of Kegelspitzen, it is natural tofurther seek results replacing that cone by I , the unit interval Kegelspitze. This is done for themixed powerdomains in Section 6, where both functional and predicate transformer resultsare obtained in all three cases. The functional representation results are Corollaries 6.4, 6.6,and 6.9; the predicate transformer results are Corollaries 6.5, 6.7, and 6.10. With additionalassumptions there are related general results for Kegelspitzen, which we discuss briefly. Terminology . Throughout the paper, we assume familiarity with standard terminologyand notation of domain theory as covered in, say, [7]. In particular, for partially ordered setwe shortly say poset; the abbreviation dcpo stands for ‘directed complete partially orderedset’; ‘bounded directed complete’ means that every upper-bounded directed set has a leastupper bound; the way below relation in a dcpo C is written as (cid:28) C , or simply (cid:28) ; and a domain is a continuous dcpo, that is, a dcpo in which, for every element a , the elementsway-below a form a directed set with supremum a . For any subset X of a poset C we write ↑ C X , or simply ↑ X , for the set of elements above some element of X (and ↓ X is understoodanalogously); similarly, we write (cid:2)(cid:2) C X for the set of elements way-above some element of X .A sub-dcpo is a subset C of a dcpo D that is closed for suprema of directed sets, that is,the supremum of any directed subset of C belongs to C . Sub-dcpos should not be mixed upwith Scott-closed sets which are sub-dcpos and, in addition, lower sets. The intersectionof an arbitrary family of sub-dcpos is a sub-dcpo; thus, for any subset P of a dcpo there K. KEIMEL AND G. D. PLOTKIN is a least sub-dcpo P d containing P . The elements of P d can also be obtained by closingunder directed suprema repeatedly. We will say that P is dense in C , if P d = C . Again,this notion of density is different from dense for the Scott topology.We write R + for the set of nonnegative real numbers with their usual order, and R + = def R + ∪ { + ∞} for the nonnegative real numbers augmented by a top element + ∞ .Finally, I = def [0 ,
1] is the closed and ]0 ,
1[ the open unit interval.
Contents
1. Introduction 12. Kegelspitzen 72.1. Ordered cones and ordered barycentric algebras 72.2. Ordered pointed barycentric algebras 132.3. d-Cones and Kegelspitzen 182.4. Preservation results 262.5. Duality and the subprobabilistic powerdomain 283. Power Kegelspitzen 323.1. Lower power Kegelspitzen 333.2. Upper power Kegelspitzen 373.3. Convex power Kegelspitzen 443.4. Powerdomains combining probabilistic choice and nondeterminism 494. Functional representations 514.1. The lower power Kegelspitze 534.2. The upper power Kegelspitze 554.3. The convex power Kegelspitze 585. Predicate transformers 615.1. The lower case 625.2. The upper case 635.3. The convex case 646. The unit interval 676.1. The lower case 716.2. The upper case 726.3. The convex case 74Acknowledgements 77References 77Appendix A. The other distributive law 80Appendix B. A counterexample 83
IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 7 Kegelspitzen
In this section we introduce the notion of a Kegelspitze, which provides the foundation forour later developments. We begin with its algebraic structure. This is that of an abstractconvex set, which we call a barycentric algebra. We then enrich the structure, first witha compatible partial order, and then with a directed complete partial order. In order tomake use of our previous work [28] on d-cones, we prove embedding theorems of barycentricalgebras in abstract cones at each stage. We conclude by recalling the properties of thesubprobabilistic powerdomain and showing how it fits within our framework.2.1.
Ordered cones and ordered barycentric algebras.
For our work there are twobasic notions abstracted from substructures in real vector spaces, an abstract notion of acone and an abstract notion of a convex set.In a real vector space V , a subset C is understood to be a cone, if x + y ∈ C and r · x ∈ C for all x, y ∈ C and every nonnegative real number r . Generalising, we obtain anabstract notion of a cone: Definition 2.1. An (abstract) cone is a set C together with a commutative associativeaddition ( x, y ) (cid:55)→ x + y : C × C → C that admits a neutral element 0, and a scalarmultiplication x (cid:55)→ r · x : C → C by real numbers r > x, y, z ∈ C and all real numbers r > , s > x + ( y + z ) = ( x + y ) + z ( rs ) · x = r · ( s · x ) x + y = y + x ( r + s ) · x = r · x + s · xx + 0 = x r · ( x + y ) = r · x + r · y · x = x Preserving all the above laws we may extend (and we will tacitly always do so) the scalarmultiplication on a cone to real numbers r ≥ · x = def x ∈ C A map f : C → D between cones is said to be: homogeneous if f ( r · x ) = r · f ( x ) for all r ∈ R + and all x ∈ C , additive if f ( x + y ) = f ( x ) + f ( y ) for all x, y ∈ C , linear if f is homogeneous and additive.In a cone all the equational laws for addition and scalar multiplication that hold invector spaces also hold, except that we restrict scalar multiplication to nonnegative realnumbers and elements x need not have negatives − x . Thus, we may calculate in cones justas we do in vector spaces, except that we have to avoid negatives. As usual, we generallywrite scalar multiplication r · x as rx . K. KEIMEL AND G. D. PLOTKIN
The notion of a barycentric algebra captures the equational properties of convex sets. Asubset A of a real vector space is convex if ra + (1 − r ) b ∈ A for all a, b ∈ A, r ∈ [0 ,
1] (Conv)We may use the same property for defining convexity of subsets in an (abstract) cone. Onevery convex set A we may define for every real number r ∈ [0 ,
1] a binary operation + r , theconvex combination ( a, b ) (cid:55)→ a + r b = def r · a + (1 − r ) · b . Straightforward calculations showthat these operations satisfy the following equational laws: a + b = a (B1) a + r a = a (B2) a + r b = b + − r a (SC)( a + p b ) + r c = a + pr ( b + r − pr − pr c ) provided r < , p < skew commutativity and SA for skew associativity . Definition 2.2. An abstract convex set or barycentric algebra is a set A endowed with abinary operation a + r b for every real number r in the unit interval [0 ,
1] such that the aboveequational laws (B1), (B2), (SC), (SA) hold. A map f : A → B between barycentric algebrasis affine if f ( a + r b ) = f ( a ) + r f ( b ) for all a, b ∈ A and all r ∈ [0 , A are entropic (or commutative ) in the sense that all the operations+ r are affine maps from A × A → A , that is, for all r and s in the unit interval we have theentropic identity ( a + r b ) + s ( c + r d ) = ( a + s c ) + r ( b + s d ) . (E)If c = d this reduces to the distributivity law( a + r b ) + s c = ( a + s c ) + r ( b + s c ) . (D)The entropic identity (E) can be verified by direct calculation. However, calculations inbarycentric algebras are quite tedious as the skew associativity law (SA) is awkward toapply. A simple proof is indicated below after Lemma 2.3.Since barycentric algebras (resp., cones) are equationally defined classes of algebras,there are free barycentric algebras (resp., free cones), and every barycentric algebra (resp.,cone) is the image of a free one under an affine (resp., linear) map.These free objects have a simple description. For any set I we write R ( I ) for the directsum of I copies of R , that is the vector space of all I -tuples x = ( x i ) i ∈ I of real numbers x i such that x i (cid:54) = 0 for finitely many indices i . For i ∈ I we write δ ( i ) for ( δ ij ) j ∈ I , thecanonical basis vector, where δ ij is the Kronecker symbol. Thus, δ maps I to a basis of R ( I ) .Analogously to the the fact that R ( I ) is the free vector space over the set I , we have: Lemma 2.3. (1)
The positive cone R ( I ) + of all x ∈ R ( I ) with nonnegative entries is the free cone over I with unit δ . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 9 (2)
The simplex P I of all x ∈ R ( I ) + such that (cid:80) i x i = 1 is the free barycentric algebra over I with unit δ . The first claim means that, for every map f from I to a cone C , there is a unique linearmap f : R ( I ) + → C such that f = f ◦ δ , namely f ( x ) = (cid:80) i x i f ( i ). Similarly, the second claimtells us that, for every map f from I to a barycentric algebra A , there is a unique affinemap f : P I → A such that f = f ◦ δ .In an equationally definable class of algebras, an equational law holds if and only ifit holds in the free algebras. Thus, for the entropic identity (E) to hold in all barycentricalgebras, it suffices to verify this property for the free barycentric algebras; and an easycalculation shows that (E) holds in any convex subset of a real vector space. The samecalculation can be done after embedding a barycentric algebra in a cone as a convex subset: Standard Construction 2.4.
Let A be a barycentric algebra. On R > × A = { ( r, a ) | < r ∈ R , a ∈ A } define addition and multiplication by scalars t > r, a ) + ( s, b ) = def ( r + s, a + rr + s b ) , t ( r, a ) = def ( tr, a )Adjoin a new element 0: C A = def { } ∪ { ( r, a ) | r > , a ∈ A } = { } ∪ ( R > × A )so that 0 is a neutral element for addition and t · C A thereby becomes a cone, and the map e = ( a (cid:55)→ (1 , a )) : A → C A is an embedding of abarycentric algebra in a cone (i.e., it is affine and injective).We identify elements a ∈ A with the corresponding elements e ( a ) = (1 , a ) ∈ C A , thusidentifying A with the convex subset { } × A of C A . In this way A becomes a base of thecone C A in the sense that A is convex and that every element x = ( r, a ) (cid:54) = 0 in C A can bewritten in the form x = ra , where r and a are uniquely determined by x .We want to add a partial order to our algebraic structure: Definition 2.5. An ordered (abstract) cone is a cone equipped with a partial order ≤ suchthat addition and scalar multiplication are monotone, that is, a ≤ a (cid:48) = ⇒ a + b ≤ a (cid:48) + b , ra ≤ ra (cid:48) . A map f : C → D between ordered cones is said to be: subadditive if f ( a + b ) ≤ f ( a ) + f ( b ) for all a, b ∈ C , superadditive if f ( a + b ) ≥ f ( a ) + f ( b ) for all a, b ∈ C , sublinear if f is homogeneous and subadditive, superlinear if f is homogeneous and superadditive.The linear maps are those that are sublinear and superlinear. Definition 2.6. An ordered barycentric algebra is a barycentric algebra with a partial order ≤ such that the barycentric operations + r are monotone, that is, a ≤ a (cid:48) = ⇒ a + r b ≤ a (cid:48) + r b for 0 ≤ r ≤
1. A map f : A → B between ordered barycentric algebras is said to be: convex if f ( a + r b ) ≤ f ( a ) + r f ( b ) for 0 ≤ r ≤ a, b ∈ C , concave if f ( a + r b ) ≥ f ( a ) + r f ( b ) for 0 ≤ r ≤ a, b ∈ C .The affine maps are those that are both convex and concave.Every barycentric algebra can be understood to be ordered by the discrete order a ≤ b iff a = b and, if A and B both are discretely ordered, the convex maps between them arethe affine ones.Convex subsets of ordered vector spaces and ordered cones are ordered barycentricalgebras with respect to the induced order.In complete analogy to Standard Construction 2.4, we can construct embeddings ofordered barycentric algebras in ordered cones, by which we mean monotone affine maps whichare also order embeddings (a monotone map between partial orders is an order embedding ifit reflects the partial order). Standard Construction 2.7.
For an ordered barycentric algebra A we use the embeddingof A in the abstract cone C A as in Standard Construction 2.4 and we extend the order on A by defining an order ≤ on C A by 0 ≤ r, a ) ≤ ( s, b ) ⇐⇒ r = s and a ≤ b in A With this order, C A becomes an ordered (abstract) cone and the affine map e = a (cid:55)→ (1 , a )from A to C A is an affine order embedding.The following surprising lemma will be used in later sections: Lemma 2.8.
Let a, b, c be elements of an ordered barycentric algebra A . If a + p c ≤ b + p c holds for some p with < p < , then this holds for all such p .Proof. We may view A as a convex subset of an ordered cone C according to 2.7. Supposenow that a, b, c are elements of A and that the inequality a + p c ≤ b + p c holds for some0 < p < a + q c ≤ b + q c for q = p p : By the above hypothesis we have theinequality pa + (1 − p ) c ≤ pb + (1 − p ) c . We use this inequality twice for establishing this firstclaim: qa +(1 − q ) c = p (2 pa +(1 − p ) c ) = p ( pa + pa +(1 − p ) c ) ≤ p ( pa + pb +(1 − p ) c ) ≤ p ( pb + pb + (1 − p ) c ) = p (2 pb + (1 − p ) c ) = qb + (1 − q ) c .Secondly, we define recursively p = p and p n +1 = p n p n . Our first claim allows toconclude a + p n b ≤ b + p n c for all n . As the p n form an increasing sequence converging to 1,for every q <
1, there is an n such that q < p n . Thus the following third claim finishes theproof of our lemma. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 11
Claim: If a + p c ≤ b + p c holds for some p , then it also holds for all q ≤ p . Indeed, a + q c = qa +(1 − q ) c = qp ( pa +(1 − p ) c )+ p − qp c ≤ qp ( pb +(1 − p ) c )+ p − qp c = qb +(1 − q ) c = b + q c . Remarks 2.9. ( Historical Notes and References ) (1) The axiomatization of convexsets arising in vector spaces over the reals has a long history. The first axiomatization seemsto be due to M. H. Stone [56]. Independently, H. Kneser [32] gave a similar axiomatizationmotivated by von Neumann’s and Morgenstern’s work on game theory [46]. Stone’s andKneser’s results are not restricted to the reals; they axiomatise convex sets embeddable invector spaces over linearly ordered skew fields. Such an axiomatization cannot be equational.For a barycentric algebra to be embeddable in a vector space one has to add a cancellationaxiom: a + r c = b + r c = ⇒ a = b (for 0 < r <
1) (C1)Similarly, one can show that an ordered barycentric algebra is embeddable in an orderedvector space if, and only if, it satisfies the order cancellation axiom: a + r c ≤ b + r c = ⇒ a ≤ b (for 0 < r <
1) (OC1)(2) Several authors independently developed the equational theory of convex sets and thecorresponding notion of an abstract convex set (while ignoring each other). Our extensionto ordered structures seems to be new. We now try to give as complete as possible anaccount of these developments. The reader should be aware that we always stay within theequational theory of convex sets in real vector spaces, and we do not go into generalities,such as abstract convexities in the sense of, for example, van de Vel [65].(3) W. Neumann [45] seems to be the first to have looked at the equational theory ofconvex sets. He remarked that barycentric algebras may be very different from convex setsin vector spaces. Indeed ∨ -semilattices become examples of barycentric algebras if we define a + r b = def a ∨ b for 0 < r <
1. Neumann [45] noticed that the semilattices form the onlyproper nontrivial equationally definable subclass of the class of all barycentric algebras. Everybarycentric algebra has a greatest homomorphic image which is a semilattice. This semilatticeis significant; indeed, for a convex subset of a real vector space this greatest homomorphicsemilattice image is the (semi-)lattice of its faces. W. Neumann also characterised the freebarycentric algebras, a characterisation that we reproduced in Lemma 2.3.The equational axioms (B1), (B2), (SC), (SA) that we use in our definition of barycentricalgebras are due to ˇSwirszcz [58] and have been reproduced by Romanowska and Smith[53, Section 5.8]; the same axioms have also been used by Graham [14] and by Jones andPlotkin [19, 18] when they introduced the notion of an abstract probabilistic powerdomain .Romanowska and Smith introduced the term barycentric algebra for an abstract convex set;their monograph, cited above, is an exhaustive source on barycentric algebras and relatedstructures.(4) The notion of an abstract cone has emerged under the name of quasilinear space in interval mathematics in works of O. Mayer [37]; one may also consult papers by W.
Schirotzek, in particular [54], where ordered quasilinear spaces appear, the closed intervalsin the reals with the Egli-Milner order forming a prime example. The embedding of abarycentric algebra as a convex subset in the abstract cone C A is due to J. Flood [6]. Thesurprising lemma 2.8 is due to W. Neumann [45] in the unordered case. The possibilityof embedding a barycentric algebra in a cone makes calculations much easier as alreadyremarked by J. Flood [6]. The proof of Lemma 2.8 illustrates this advantage when comparedwith Neumann’s original proof in the unordered case.Let us note in passing that every ∨ -semilattice with a zero becomes an abstract cone bydefining a + b = a ∨ b and ra = a for r >
0, but ra = 0 for r = 0. The class of ∨ -semilatticeswith 0 is the only proper nontrivial equationally definable subclass of the class of all abstractcones.(5) Another early axiomatisation of abstract convex sets is due to Gudder [15]. Heuses as operations convex combinations of two and three elements and axiomatises theseoperations. Without introducing the notion of an abstract cone he gives the construction ofthe embedding of an abstract convex set into a cone viewed as a convex set.(6) Equivalent to the equational one, there is another approach to abstract convex setsinitiated by T. ˇSwirszcz [58] who characterises them as the Eilenberg-Moore algebras ofthe monad P of probability distributions with finite support over the category of sets. Infunctional analysis this approach has been rediscovered by G. Rod´e [51] and developedfurther by H. K¨onig [35] without any background in category theory. The approach hasbeen pursued further in a series of papers by Pumpl¨un, R¨ohrl, Kemper and others (see e.g.[48, 49, 50, 30, 67]). We summarise it as follows:For all natural numbers m >
0, the set P m of all probability measures q = ( q , . . . , q m )on an m -element set is a compact convex set and, for every finite set q , . . . , q n ∈ P m , theconvex combination (cid:80) ni =1 p i q i is again an element of P m for every p = ( p , . . . , p n ) ∈ P n .The sum (cid:80) ni =1 p i q i is the barycenter of masses p , . . . , p n placed at points q , . . . , q n . Theextreme points of P n are the Dirac measures δ i , i = 1 , . . . , n , given by the Kronecker symbol δ ki . A convex space is a nonempty set X together with a family of mappings p X : X n → X for p ∈ P n , n = 1 , , . . . , satisfying for every x = ( x , . . . , x m ) ∈ X m the identities δ Xi ( x ) = x i , p X ( q X ( x ) , . . . , q Xn ( x )) = (cid:0) n (cid:88) i =1 p i q i (cid:1) X ( x )for all p ∈ P n and q , . . . , q n ∈ P m . Using the formal notation (cid:80) ni =1 p i x i = p X ( x ) the twoequations take the form n (cid:88) i =1 δ ik x i = x k (A1)where δ ik is the Kronecker symbol, and n (cid:88) i =1 p i (cid:0) m (cid:88) k =1 q ik x k (cid:1) = m (cid:88) k =1 (cid:0) n (cid:88) i =1 p i q ik (cid:1) x k (A2) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 13 for p = ( p , . . . , p n ) ∈ P n , and q i = ( q i , . . . , q im ) ∈ P m , i = 1 , . . . , n .It should be added that the main interest of the authors using the latter approach wasnot in convex but in superconvex spaces. For those one replaces the sets P n of probabilitymeasures on finite sets by the set P I of discrete probability measures on an infinite countableset I , using the same defining identities as above replacing finite by infinite sums.2.2. Ordered pointed barycentric algebras.
Convex sets containing 0 in vector spacesand in (abstract) cones are pointed barycentric algebras in the following sense:
Definition 2.10. A pointed barycentric algebra is a barycentric algebra A with a distin-guished element usually written 0. A map f : A → B between pointed barycentric algebrasis or linear if it is affine and preserves the distinguished element: f (0) = 0.If A is a convex set containing 0 in a vector space or in an abstract cone, we have ra ∈ A for all a ∈ A and all r ∈ [0 , r ∈ [0 , r ∈ [0 ,
1] for an arbitrary pointed barycentricalgebra A . For an element a ∈ A and r ∈ [0 ,
1] define r · a = def a + r · a = 0 = r · , · a = a, ( rs ) · a = r · ( s · a ) , r · ( a + s b ) = r · a + s r · b as is straightforward to check (the last property follows from the distributive law (D)). Everylinear map f : A → B of pointed barycentric algebras is homogeneous in the sense that f ( r · a ) = r · f ( a ) for all a ∈ A and 0 ≤ r ≤
1. Indeed, we have: f ( r · a ) = f ( a + r
0) = f ( a ) + r f (0) = f ( a ) + r r · f ( a ).Every cone is a pointed barycentric algebra. Thus, for a map f between cones we havetwo notions of homogeneity, one where r ranges over all nonnegative real numbers, andanother where r only ranges over the unit interval. But the two notions are equivalent forcones; indeed, if f ( ra ) = rf ( a ) for all r in the unit interval, then f ( a ) = f ( r − ra ) = r − f ( ra )for all r ≥
1, whence rf ( a ) = f ( ra ) for all r ≥
1, too. This shows that our terminology oflinearity for maps between cones and pointed barycentric algebras, respectively, is consistent.As for barycentric algebras (see 2.3), there is a simple description for the free pointedbarycentric algebra over a set I : Lemma 2.11.
The convex set S I = { x = ( x i ) i ∈ I ∈ R ( I ) + | (cid:88) i x i ≤ } ⊆ R ( I ) of all finitely supported subprobability distributions on I with (0) i ∈ I as distinguished elementis the free pointed barycentric algebra over I with unit δ . Proof.
Let A be a pointed barycentric algebra with distinguished element 0 and f : I → A an arbitrary function. We add a new element i to I and extend f to this new element by f ( i ) = 0. There is a unique affine map g from the free barycentric algebra P I ∪{ i } to A such that g ◦ δ = f . We now compose g with the affine bijection from S I to P I ∪{ i } thatmaps the subprobability distribution x = ( x i ) i ∈ I on I to the probability distribution y on I ∪ { i } with y i = x i for i ∈ I , and y i = 1 − (cid:80) i ∈ I x i . In this way we obtain an affine map f : S I → A such that f = f ◦ δ and f (0) = 0. Clearly, f is unique with these properties.We now add a partial order: Definition 2.12. An ordered pointed barycentric algebra is an ordered barycentric algebrawith a distinguished element 0. A map f : A → B of ordered pointed barycentric algebras is sublinear (resp., superlinear ) if it is homogeneous and convex (resp., concave).The linear maps are those that are both sublinear and superlinear.For general reasons, there is a free ordered cone Cone ( A ) over any ordered pointedbarycentric algebra A . We need a concrete construction of this free object. The idea is tostretch the multiplication by scalars from those in the unit interval to all nonnegative reals: Standard Construction 2.13.
Let A be an ordered pointed barycentric algebra. Sinceevery pointed barycentric algebra is the image of a free one under a linear map, by Lemma2.11 there is a linear surjection f : S I → A , where S I is the pointed barycentric algebra offinitely supported subprobability measures on some set I .For every x ∈ R ( I ) + there is a greatest real number 0 < r ≤ rx ∈ S I , and forevery real number s with 0 < s ≤ r we also have sx ∈ S I . We define a relation (cid:45) on R ( I ) + : x (cid:45) x (cid:48) if f ( rx ) ≤ f ( rx (cid:48) ) for some r, < r ≤ , such that rx ∈ S I and rx (cid:48) ∈ S I . If this holds for some 0 < r ≤
1, then it also holds for all s with 0 < s ≤ r , since f ishomogeneous. Lemma 2.14.
The relation (cid:45) is a preorder on the cone R ( I ) + compatible with the coneoperations, that is, x (cid:45) x (cid:48) implies x + y (cid:45) x (cid:48) + y and rx (cid:45) rx (cid:48) for every y in the cone andevery nonnegative real number r .Proof. Clearly, the relation (cid:45) is reflexive and transitive. For compatibility, consider elements x (cid:45) x (cid:48) in S I . There is an r with 0 < r ≤ rx, rx (cid:48) ∈ S I and f ( rx ) ≤ f ( rx (cid:48) ). Given y ,choose s such that 0 < s ≤ r and s ( x + y ) , s ( x (cid:48) + y ) ∈ S I . Using that f is linear on S I we obtain f ( s ( x + y )) = f ( s ( x + y )) = f ( sx + sy ) = f ( sx ) + f ( sy ) ≤ f ( sx (cid:48) ) + f ( sy ) = f ( s ( x (cid:48) + y )),that is x + y (cid:45) x (cid:48) + y . The property that tx (cid:45) tx (cid:48) for t ∈ R + is straightforward. Corollary 2.15.
The relation x ∼ x (cid:48) if x (cid:45) x (cid:48) and x (cid:48) (cid:45) x is a congruence relation on thecone R ( I ) + . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 15
We write
Cone ( A ) for R ( I ) + / ∼ , the quotient cone ordered by ρ ( x ) ≤ ρ ( y ) if x (cid:45) y , where ρ : R ( I ) + → R ( I ) + / ∼ is the (linear monotone) quotient map. Restricted to S I , the quotientmap factors through f . Indeed, if x, y are elements of S I such that f ( x ) ≤ f ( y ), then x (cid:45) y by the definition of (cid:45) . Thus, there is a unique monotone linear map u : A → Cone ( A ) suchthat ρ | S I = u ◦ f .We note two important properties of this map: Lemma 2.16. (1) If u ( a ) ≤ u ( b ) , for a, b ∈ A , then ra ≤ rb for some < r ≤ . (2) Every b ∈ Cone ( A ) has the form ru ( a ) for some a ∈ A and r ≥ .Proof. (1) Suppose that u ( f ( x )) ≤ u ( f ( y )), for x, y ∈ S I . Then ρ ( x ) ≤ ρ ( y ). So x (cid:45) y and wehave f ( rx ) ≤ f ( ry ) for some 0 < r ≤
1, which is to say that rf ( x ) ≤ rf ( y ) for some0 < r ≤ x ∈ S I such that ρ ( x ) = b ∈ Cone ( A ). There is a y ∈ S I and an r ≥ x = ry . Set a = def f ( y ). Then: ru ( a ) = ru ( f ( y )) = rρ ( y ) = ρ ( ry ) = ρ ( x ) = b .The two properties of the map u : A → Cone ( A ) picked out in this lemma yield acharacterisation of when a monotone linear map from a pointed ordered barycentric algebrato an ordered cone is universal as expressed by the freeness property in the followingproposition: Proposition 2.17.
Let u : A → C be a monotone linear map from a pointed orderedbarycentric algebra to an ordered cone C . Then C is the free ordered cone over A with unit u (in the sense that every monotone homogeneous map h from A into an ordered cone D hasa unique monotone homogeneous extension (cid:101) h : C → D along u ) if, and only if, the followingtwo properties hold of u : (1) If u ( a ) ≤ u ( b ) , for a, b ∈ A , then ra ≤ rb for some < r ≤ . (2) Every x ∈ C has the form ru ( a ) for some a ∈ A and r ≥ .For such universal maps u , the extension (cid:101) h is sublinear, superlinear, linear, respectively if,and only if, h is. Moreover, (cid:101) h ≤ (cid:101) g if and only if h ≤ g .Proof. Suppose the two properties hold. For uniqueness of the extension use the secondassumption and note that, for such an (cid:101) h , we have: (cid:101) h ( ru ( a )) = r (cid:101) h ( u ( a )) = rh ( a )We then use this property to define (cid:101) h ; to show (cid:101) h well-defined and monotone we chose a, a (cid:48) ∈ A and r, r (cid:48) ≥ ru ( a ) ≤ r (cid:48) u ( a (cid:48) ) then rh ( a ) ≤ r (cid:48) h ( a (cid:48) ). For if ru ( a ) ≤ r (cid:48) u ( a (cid:48) ) then u ( rs − a ) ≤ u ( r (cid:48) s − a (cid:48) ), where s = def max( r, r (cid:48) ); so, by the first assumption, trs − a ≤ tr (cid:48) s − a (cid:48) for some 0 < t ≤
1. So, in turn, we have trs − h ( a ) = h ( trs − a ) ≤ h ( tr (cid:48) s − a (cid:48) ) = tr (cid:48) s − h ( a ), and so rh ( a ) ≤ r (cid:48) h ( a ) as required. It is straightforward that (cid:101) h ishomogeneous. That the two properties are necessary follows from Lemma 2.16, which provides anexample of a map u : A → Cone ( A ) possessing them. Suppose indeed that u (cid:48) : A → C has the universal property. Then there are mutually inverse monotone homogeneous maps (cid:101) u : C → Cone ( A ) and (cid:101) u (cid:48) : Cone ( A ) → C such that u = (cid:101) u ◦ u (cid:48) and u (cid:48) = (cid:101) u (cid:48) ◦ u . For property1, take a ≤ b in A such that u (cid:48) ( a ) ≤ u (cid:48) ( b ). Then u ( a ) = (cid:101) u ( u (cid:48) ( a )) ≤ (cid:101) u ( u (cid:48) ( b )) = u ( b ), hence ra ≤ rb for some 0 < r ≤
1. For property 2, take x ∈ C . Then (cid:101) u ( x ) = ru ( a ) for some a ∈ A and r ≥
1. Applying (cid:101) u (cid:48) yields x = (cid:101) u (cid:48) ( ru ( a )) = r (cid:101) u (cid:48) ( u ( a )) = ru (cid:48) ( a ).Given such a map u : A → C , suppose h is sublinear. Then so is (cid:101) h . To see this choose x, x (cid:48) ∈ C . By the first property x = ru ( a ) and x (cid:48) = r (cid:48) u ( a (cid:48) ) for some a, a (cid:48) ∈ A and r, r (cid:48) ≥ s = def r + r (cid:48) and calculate: (cid:101) h ( x + y ) = (cid:101) h ( ru ( a ) + r (cid:48) u ( a (cid:48) )) = (cid:101) h ( su ( a + r/s a (cid:48) )) = sh ( a + r/s a (cid:48) ) ≤ s ( h ( a ) + r/s h ( a (cid:48) ))(as h is sublinear) = rh ( a ) + r (cid:48) h ( a (cid:48) ) = (cid:101) h ( x ) + (cid:101) h ( y ). Theconverse is evident as u is linear. The assertion for superlinearity is proved similarly, andthen the assertion for linearity follows.Finally we show that (cid:101) h ≤ (cid:101) g if h ≤ g (the converse is obvious). Assuming h ≤ g , for any a ∈ A and r ≥ (cid:101) h ( ru ( a )) = rh ( a ) ≤ rg ( a ) = (cid:101) g ( ru ( a )).Together with Lemma 2.16 we now have: Theorem 2.18.
For any ordered pointed barycentric algebra A , Cone ( A ) is the free orderedcone over A with unit u in the following strong sense: Every monotone homogeneousmap h from A into an ordered cone D has a unique monotone homogeneous extension (cid:101) h : Cone ( A ) → D along u . The extension (cid:101) h is sublinear, superlinear, linear, respectively if,and only if, h is. Moreover, (cid:101) h ≤ (cid:101) g if and only if h ≤ g . It would be nice, if the monotone linear map u from A into the universal cone would bean order embedding. But this is not always the case. We give an example of an orderedpointed barycentric algebra that cannot be embedded in any ordered cone at all. Indeed, inan ordered cone, if for some 0 < r < rx ≤ ry , then x ≤ y by multiplying with thescalar r − . This need not be true in an ordered pointed barycentric algebra: Example 2.19.
We consider the unit interval and replace the element 1 by two elements1 , . On each of the sets [0 , ∪ i , i = 1 , , we take convex combinations as usual inthe unit interval, and the set { , } is considered as a join-semilattice with x + r y = 1 whenever x (cid:54) = y and 0 < r <
1. In this way we obtain a pointed barycentric algebra with 0as distinguished element. Clearly, rx = r = ry whenever 0 < r < x, y ∈ { , } .We therefore pay attention to the order cancellation law: rx ≤ ry = ⇒ x ≤ y (for 0 < r <
1) (OC2)and its specialised form rx = ry = ⇒ x = y (for 0 < r <
1) (C2)These laws are particular instances of the cancellation laws (OC1) and (C1). For example(C2) can be rewritten as x + r y + r ⇒ x = y . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 17
In view of the properties of scalar multiplication, the map x (cid:55)→ rx of A into itself islinear and monotone. The axiom (OC2) is equivalent to the statement that this map is alsoan order embedding whenever 0 < r ≤
1, which implies that the image rA is an orderedpointed barycentric algebra isomorphic to A .Property (OC2) is less restrictive than it seems. Indeed, using Lemma 2.8 for the case c = 0 we obtain: Remark 2.20.
If in an ordered pointed barycentric algebra ra ≤ rb for some 0 < r < r .We now answer the question which ordered pointed barycentric algebras can be embeddedinto ordered cones, where by embedding we mean a linear order embedding: Proposition 2.21.
For an ordered pointed barycentric algebra A satisfying (OC2) , theuniversal map u : A → Cone ( A ) is an embedding. An ordered pointed barycentric algebra canbe embedded into an ordered cone if, and only if, it satisfies (OC2) .Proof. Consider u : A → Cone ( A ), and, given a, a (cid:48) ∈ A , suppose that u ( a ) ≤ u ( a (cid:48) ). Then byLemma 2.16.1, ra ≤ ra (cid:48) for some 0 < r ≤
1, and so a ≤ a (cid:48) , if A satisfies (OC2). Thus u isan order embedding. Thus, if A satisfies (OC2), it can be embedded in an ordered cone.Conversely, if an ordered pointed barycentric algebra can be embedded in an ordered cone,it satisfies (OC2), since (OC2) is satisfied in every ordered cone.We can also identify which embeddings are universal: Corollary 2.22.
An embedding u : A → C of an ordered pointed barycentric algebra A inan ordered cone C is universal if, and only if, every x ∈ C has the form r · u ( a ) , for some a ∈ A and r ≥ . Under the assumption (OC2), we may identify an ordered pointed barycentric A withits image in Cone ( A ) under the embedding u by the previous proposition and we will do sowithout mentioning in the sequel. With this identification for every c ∈ Cone ( A ), there is an0 < r ≤ rc ∈ A ; we will frequently use this fact.It would be desirable to embed an ordered pointed barycentric algebra in an orderedcone as a lower set. If an ordered pointed barycentric algebra A is embedded in an orderedcone C as a lower set, then one has for all a, b ∈ A : a ≤ rb = ⇒ ∃ a (cid:48) ∈ A. a = ra (cid:48) (for 0 < r <
1) (OC3)Indeed, if a ≤ rb for a, b ∈ A and 0 < r <
1, then a (cid:48) = def r a ≤ b in C . Then a (cid:48) ∈ A , as A is alower set in C , and a = ra (cid:48) . Property (OC3) is not satisfied for all ordered pointed barycentricalgebras. As an example, let A be the convex hull of the points (0 , , (1 , , (0 , , (2 ,
2) in R + . Then A is a pointed barycentric algebra embedded in the ordered cone R + , but A is nota lower set. As it does not satisfy Property (OC3), A cannot be embedded in any orderedcone as a lower set. We also have a converse: Lemma 2.23.
An ordered pointed barycentric algebra A satisfying order cancellation (OC2) is embedded in the ordered cone Cone ( A ) as a lower set if, and only if, it satisfies Property (OC3) .Proof. By Proposition 2.21 we can embed A into the ordered cone Cone ( A ). To show that A is embedded as a lower set, suppose that x is an element of Cone ( A ) such that x ≤ b forsome b ∈ A . Then rx ∈ A for some r with 0 < r < rx ≤ rb . By Property (OC3) thereis an x (cid:48) ∈ A such that rx (cid:48) = rx which implies x = x (cid:48) ∈ A by multiplying by r .In the presence of the order cancellation property (OC2) one has a (cid:48) ≤ b for the element a (cid:48) whose existence is postulated in (OC3). Remark 2.24. ( Historical Notes and References ) As far as we know, pointed barycen-tric algebras have not attracted much attention. They are identical to the finitely positivelyconvex spaces in the sense of Wickenh¨auser, Pumpl¨un, R¨ohrl, and Kemper [67, 48, 49, 50, 30].To define them, one uses the same setting and the identities used for convex spaces (seehistorical remark 2.9), but replaces the convex sets P n of probability measures on n -elementsets by the pointed convex sets S n = { ( q , . . . , q n ) ∈ [0 , n | n (cid:88) i =1 q i ≤ } of subprobability measures on n -element sets. As before, the main interest of these authorswas directed towards the positively superconvex spaces and their applications in functionalanalysis, where the S n are replaced by the set S of subprobability measures on an infinitecountable set. Our standard construction 2.13 for constructing the free ordered cone over anordered pointed barycentric algebra is simpler than Pumpl¨un’s construction of a free coneover an (unordered) positively superconvex space (see [48, Definition 4.17 ff.]).In the same way as join-semilattices can be considered to be barycentric algebras,join-semilattices with a distinguished element can be considered to be pointed barycentricalgebras.Recently, convex and positively convex spaces were taken up by A. Sokolova and H.Woracek [55]. These authors are particularly interested in finitely generated barycentric andpointed barycentric algebras, that is, homomorphic images of polyhedra and pointed polyhe-dra in finite dimensional vector spaces, and they prove that finitely generated barycentricand pointed barycentric algebras, respectively, are finitely presented.2.3. d-Cones and Kegelspitzen. We now endow partially ordered sets with their Scotttopology. In particular, the sets R + , R + , and the unit interval [0 ,
1] are endowed with theirusual order and the corresponding Scott topology. Maps are restricted to Scott-continuousones.Thus, for an ordered cone C it is natural to ask for addition ( a, b ) (cid:55)→ a + b : C × C → C and scalar multiplication ( r, a ) (cid:55)→ ra : R + × C → C to be Scott-continuous (in both IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 19 arguments). Note that the continuity of scalar multiplication in the first argument impliesthat 0 is the least element of C ; indeed, Scott-continuity with respect to scalars implies that r (cid:55)→ ra : R + → C is monotone, whence 0 ≤ · a ≤ · a = a . Definition 2.25.
An ordered cone in which addition ( a, b ) (cid:55)→ a + b : C × C → C and scalarmultiplication ( r, a ) (cid:55)→ ra : R + × C → C are Scott-continuous (in both arguments) will becalled an s-cone . If in addition the order is directed complete (resp., bounded directedcomplete), we say that C is a d-cone (resp., a b-cone ).We are heading towards a similar connection between the algebraic and the orderstructure on ordered pointed barycentric algebras. For this we have to restrict the scalars r to the unit interval: Definition 2.26. A Kegelspitze is a pointed barycentric algebra K equipped with a directedcomplete partial order such that, for every r in the unit interval, convex combination( a, b ) (cid:55)→ a + r b : C × C → C and scalar multiplication ( r, a ) (cid:55)→ ra : [0 , × C → C areScott-continuous in both arguments. (An alternative name would be pointed barycentricd-algebra .)We only need to require scalar multiplication to be continuous in its first argument inthe definition, since a (cid:55)→ ra = a + r Lemma 2.27.
Every Kegelspitze satisfies the order cancellation property (OC2) : If ra ≤ rb for some < r < , then a ≤ b .Proof. Indeed, if ra ≤ rb for some 0 < r <
1, the order theoretical version of Neumann’slemma 2.8 implies that ra ≤ rb for all r < a ≤ b by the Scott continuity ofthe map r (cid:55)→ ra : [0 , → K .We would like to embed every Kegelspitze K into a d-cone, where embeddings ofKegelspitzen in d-cones are Scott-continuous linear maps which are order embeddings. Weproceed in two steps. In a first step we use the embedding of K (considered as a pointedbarycentric algebra) in the ordered cone Cone ( K ) according to Standard Construction 2.13.By Proposition 2.21 this Standard Construction yields indeed a linear order embedding u of K in Cone ( K ), since K satisfies (OC2) by Lemma 2.27. By the following lemma, thisembedding is Scott-continuous: Lemma 2.28.
Let u be a homogeneous order embedding of a Kegelspitze K in an s-cone C in such a way that for every element y ∈ C there is an r, < r < , such that ry ∈ u ( K ) .Then u is Scott-continuous. Proof.
Let ( x i ) i be a directed family in K and x its supremum in K . Since u is monotone,clearly u ( x i ) ≤ u ( x ). In order to show that u ( x ) is the supremum of the u ( x i ) in C , considerany upper bound y ∈ C of the u ( x i ). Choose an r, < r < , such that ry ∈ u ( K )and y (cid:48) ∈ K with u ( y (cid:48) ) = ry . Then ry = u ( y (cid:48) ) is an upper bound of the directed family ru ( x i ) = u ( rx i ) (using homogeneity of u ). Since u is an order embedding, y (cid:48) is an upperbound of the rx i in K . By the Scott continuity of scalar multiplication, rx is the leastupper bound of the rx i in K , whence rx ≤ y (cid:48) . Using homogeneity and monotonicity of u ,we deduce ru ( x ) = u ( rx ) ≤ u ( y (cid:48) ) = ry which implies u ( x ) ≤ y .We now want to show that scalar multiplication and addition on Cone ( K ) are Scott-continuous. This is no problem for scalar multiplication:We first recall that a (cid:55)→ ra : Cone ( K ) → Cone ( K ) is Scott-continuous for every r > r − .We now verify that r (cid:55)→ ra : R + → Cone ( K ) is Scott-continuous. Suppose indeed that r i is an increasing family in R + with r = sup i r i . Choose an s, < s < , such that sr ≤ sa ∈ K . We then use the continuity of r (cid:55)→ ra : [0 , → K to obtain sup i ( sr i )( sa ) =(sup i sr i )( sa ) = ( sr )( sa ). Now in Cone ( K ) we have s sup i r i a = sup i ( sr i )( sa ) = srsa = s ( ra ) which implies sup i r i a = ra .We now turn to addition. To prove that a (cid:55)→ a + b : Cone ( K ) → Cone ( K ) is Scott-continuous for every fixed b ∈ Cone ( K ), we have to show: If a i is a directed system in Cone ( K ) which has a sup a = sup i a i then the family a i + b has a sup and a + b = sup i ( a i + b ).For this we choose an s, < s < , such that sa ∈ K and sb ∈ K . Because sa i ≤ sa , wewould like to conclude that sa i ∈ K , since then the Scott continuity of convex combinationin K implies that sup i ( ( sa i ) + ( sb )) = sup i ( sa i ) + ( sb ) = ( sa ) + ( sb ), whence in Cone ( K ) we have sup i ( a i + b ) = 2 s − sup i ( ( sa i ) + ( sb )) = 2 s − ( ( sa ) + ( sb ) = a + b asdesired.Thus, we would like to use that K is a lower set in Cone ( K ). By Lemma 2.23, this isequivalent to the requirement that K satisfies Property (OC3). As we often use this propertywe make a definition: Definition 2.29.
A Kegelspitze is said to be full if it satisfies Property (OC3).We now can state:
Proposition 2.30.
For a full Kegelspitze K , the free cone Cone ( K ) over K according toStandard Construction 2.13 is a b-cone and K is Scott-continuously embedded in Cone ( K ) as a Scott-closed convex set.The b-cone Cone ( K ) is the free b-cone over K w.r.t. Scott-continuous homogeneousmaps as, for every such map f from K into a b-cone D , the unique homogeneous extension (cid:101) f : Cone ( K ) → D is Scott-continuous. Moreover, (cid:101) f is sublinear, superlinear, or linear, if,and only if, f is. Further, f ≤ g if and only if (cid:101) f ≤ (cid:101) g . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 21
Proof.
In the presence of (OC3), Lemma 2.23 shows that we have a linear order embeddingof K in Cone ( K ) as a lower set and convexity is evident. The embedding is Scott-continuousby Lemma 2.28 and so K is embedded as a Scott-closed set.The arguments preceding the statement of the proposition show that addition and scalarmultiplication are Scott-continuous on Cone ( K ). We even have a b-cone: Indeed, if ( x i ) i is a directed family in Cone ( K ) bounded above by some x , choose an r, < r < , suchthat rx ∈ K . Using that K is a lower set, ( rx i ) i is a directed family in K and so has a sup y = sup i rx i in K . We conclude that the ( x i ) i have a sup namely r − y = sup i x i .Now let f : K → D be a homogeneous function into a b-cone D . By Theorem 2.18 it hasa unique homogeneous extension (cid:101) f : Cone ( K ) → D . If f is Scott-continuous, then (cid:101) f is Scott-continuous, too: Indeed, let ( x i ) i be a bounded directed family in Cone ( K ) with x = sup i x i .Choose any r > rx ∈ K . Since K is a lower set in Cone ( K ) we also have rx i ∈ K and rx = sup i rx i in K . By the continuity of f on K we have f ( rx ) = sup i f ( rx i ), whence (cid:101) f ( x ) = r − (cid:101) f ( rx ) = r − f ( rx ) = r − sup i f ( rx i ) = r − sup i (cid:101) f ( rx i ) = sup i (cid:101) f ( x i ).The remaining claims follow directly from Theorem 2.18.In a second step we use a completion procedure following Zhang and Fan [70], Keimeland Lawson [26, 27] and Jung, Moshier, and Vickers [21], in order to embed the b-cone Cone ( K ) in a d-cone. Standard Construction 2.31. A universal (or free ) dcpo-completion of a poset P consistsof a dcpo P and a Scott-continuous map ξ : P → P enjoying the universal property thatevery Scott-continuous map f from P to a dcpo Q has a unique Scott-continuous extension f : P → Q satisfying f = f ◦ ξ . A universal dcpo-completion, if it exists, is evidently uniqueup to a canonical isomorphism.Let us extract relevant information about universal dcpo-completions of a poset P fromthe literature:(1) Every poset P has a universal dcpo-completion. One may, for example [70, Theorem 1],take the least sub-dcpo P of the dcpo of all nonempty Scott-closed subsets of P (orderedby inclusion) containing the principal ideals ↓ x with x ∈ P, and ξ = def ( x (cid:55)→ ↓ x ) : P → P as canonical map.(2) Let ξ : P → D be a topological embedding (for the respective Scott topologies) of a poset P into a dcpo D . Then the least sub-dcpo P of D containing the image ξ ( P ) togetherwith the corestriction ξ : P → P is a universal dcpo-completion and the Scott topology of P is the subspace topology induced by the Scott topology on D . (See [26, Theorem 7.4]).(3) A function ξ : P → D of a poset P into a dcpo D is a universal dcpo-completion if, andonly if,(i) ξ is a topological embedding (for the Scott topologies) and In the literature [70, 26, 27] the term dcpo-completion is used instead of universal dcpo-completion . Forour purposes we prefer the the latter terminology, since there are Scott-continuous order embeddings ofposets into dcpos which are dense for directed suprema but not universal, for example the embedding of R + into the dcpo obtained by adding a top element. (ii) the image ξ ( P ) is dense in D (This follows from the previous item, since any two universal dcpo-completions areisomorphic.)(4) Let P , . . . , P n , Q be posets and let P , . . . , P n , Q be universal dcpo-completions thereof.Then every Scott-continuous function f : P ×· · ·× P n → Q has a unique Scott-continuousextension f : P × · · · × P n → Q . Further, if g is another such function, then f ≤ g if,and only if, f ≤ g . (Here the products are understood to have the product order. Thus,the claim follows from [27, Proposition 5.6], since functions defined on finite productsare Scott-continuous if, and only if, they are separately Scott-continuous in each of theirarguments.)(5) The universal dcpo-completion of a finite direct product of posets is the direct product ofthe universal dcpo-completions of its factors. More precisely, if ξ i : P i → P i ( i = 1 , . . . , n ) are universal dcpo-completions, then so is ξ = def ξ × · · · × ξ n : P × · · · × P n → P × · · · × P n (This follows directly from the previous item.)In the characterisation 2.31(3) of universal dcpo-completions above, the first condition —being a topological embedding — is the critical one. As we now show, it holds automaticallyin many situations.To begin with, we remark that, for a Scott-closed subset C of a poset P , the canonicalembedding of C into P is topological, that is, the intrinsic Scott topology on C is thesubspace topology induced by the Scott topology on P .But even on a lower subset P of a dcpo Q , the intrinsic Scott topology of P may bestrictly finer than the subspace topology induced by the Scott topology of Q . A simpleexample for this phenomenon is given by P = R + × R + with the coordinatewise order and Q = P (cid:62) , the dcpo obtained by attaching to P a top element. Here [0 , × R + is Scott-closedin P , but the Scott closure of this subset in Q is all of Q .Following [36] we say that a dcpo P is meet continuous if for any x ∈ P and any directedset D ⊆ P with x ≤ (cid:87) ↑ D , x is in the Scott closure of ↓ x ∩ ↓ D . All domains are meetcontinuous as are all dcpos with a Scott-continuous meet operation. Lemma 2.32.
The canonical embedding of a lower subset P of a meet continuous dcpo Q is a topological embedding (for the respective Scott topologies).Proof. One checks that every directed sup in P is also a directed sup in Q . Consequentlythe inclusion of P in Q is Scott-continuous. So, for every subset V of Q that is Scott-openin Q , V ∩ P is open for the Scott topology on P .In the other direction we have to show: Let U be a subset of P which is open for theScott topology on P . Then there is a Scott-open subset V of Q such that U = V ∩ P . Since ↑ U ∩ P = U , it suffices to show that ↑ U is Scott-open in Q , where ↑ U is the upper set in Q generated by U . For this take any directed subset D in Q with supremum d in ↑ U . Wehave to show that there is a c ∈ D with c ∈ ↑ U . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 23
To this end, as d in ↑ U , there is an x ∈ U with x ≤ d . By meet continuity, x is in theScott closure w.r.t. Q of ↓ x ∩ ↓ D . Following the remark above on the Scott topologies ofclosed subsets of partial orders, we see that the intrinsic Scott topology on ↓ x agrees withboth the subspace topology induced by the Scott topology on Q , and the subspace topologyinduced by the Scott topology on P . As the P -closure of the set ↓ x ∩ ↓ D intersects withthe P -open set U , the set ↓ x ∩ ↓ D itself intersects U . For an element x (cid:48) in the intersection,one has x (cid:48) ∈ U and x (cid:48) ≤ c for some c ∈ D , whence c ∈ ↑ U , and we see that this c has therequired properties.The previous lemma yields a sufficient condition for the universality of dcpo-completions: Corollary 2.33.
The canonical embedding of a lower subset P of a meet continuous dcpo Q is a universal dcpo-completion if, and only if, P is dense in Q . We now consider the universal dcpo-completion of an s-cone.
Proposition 2.34.
Let C be an s-cone and C a universal dcpo-completion. Then additionand scalar multiplication on C extend uniquely to Scott-continuous operations on the dcpo-completion C which thus becomes a d-cone. The unique Scott-continuous extension f : C → D of a Scott-continuous function f from C to a d-cone D is homogeneous, sublinear, superlinear,or linear, respectively, if f is. Moreover, f ≤ g if and only if f ≤ g .Proof. By property 2.31(4) of universal dcpo-completions, addition and scalar multiplicationon C extend uniquely to Scott-continuous operations on C . As a consequence of [27,Proposition 8.1], the extended operations obey the same equational laws as in C , that is, C is a d-cone.Now let D be any d-cone and f : C → D a Scott-continuous map. By the universalproperty, f has a unique Scott-continuous extension f : C → D . If f is subadditive, let usshow that f is subadditive, too. For this we consider the two maps g : ( a, b ) (cid:55)→ f ( a + b ) and h : ( a, b ) (cid:55)→ f ( a ) + f ( b ) from C × C to D . Clearly, ( a, b ) (cid:55)→ f ( a + b ) and ( a, b ) (cid:55)→ f ( a ) + f ( b )are Scott-continuous extensions of g and h to C × C . Since C × C is the dcpo-completionof C × C by property 2.31(4) of dcpo-completions, these are the unique Scott-continuousextensions g and h , respectively. The subadditivity of f is equivalent to the statement that g ≤ h which, by the last part of property 2.31(4) of dcpo-completions, is equivalent to g ≤ h which again is equivalent to the subadditivity of f . The argument for superadditivity issimilar and for homogeneity it is even simpler.We now consider a full Kegelspitze K and apply the completion procedure above to Cone ( K ) which, by Proposition 2.30, is the universal b-cone over K ; we write d - Cone ( K )for its universal dcpo-completion Cone ( K ). We know that K is embedded in Cone ( K )as a Scott-closed convex set. Since Cone ( K ) is bounded directed complete, it is a lowerset in its dcpo-completion. Thus K is a lower set in Cone ( K ), too. Further, since K is embedded in Cone ( K ) and since universal dcpo-completions preserve existing directed sups, K is a sub-dcpo of Cone ( K ). So K is Scott-closed in the universal dcpo-completion Cone ( K ) = d - Cone ( K ).Using Propositions 2.30 and 2.34 we then obtain: Theorem 2.35.
Let K be a full Kegelspitze. The universal dcpo-completion d - Cone ( K ) of the b-cone Cone ( K ) according to the Standard Construction 2.13 is a d-cone and K isembedded in this d-cone as a Scott-closed convex set.The embedding of K into d - Cone ( K ) is universal in the sense that every Scott-continuoushomogeneous map f from K into a d-cone D has a unique Scott-continuous homogeneousextension f : d - Cone ( K ) → D . Moreover, the extension f is sublinear, superlinear, or linearif, and only if, f is. Moreover, f ≤ g if, and only if, f ≤ g . Below, we wish to identify some naturally occurring embeddings as universal. Tothat end, we begin with a proposition characterising universal embeddings. The standardfactorisation of a Scott-continuous linear order embedding K e −→ C of a Kegelspitze in ad-cone is K u −→ B ξ −→ C where B is the sub-cone of C with carrier { re ( a ) | r > , a ∈ K } and the induced order, u isthe co-restriction of e , and ξ is the inclusion. Lemma 2.36.
Let K u −→ B ξ −→ C be the standard factorisation of a Scott-continuous linearorder embedding K e −→ C of a full Kegelspitze K in a d-cone. Then B is the free b-cone over K ,with unit u , with respect to Scott continuous homogeneous maps, and ξ is a Scott-continuouslinear map.Proof. Clearly K u −→ B is an embedding of an ordered pointed barycentric algebra in anordered cone. Using Corollary 2.22 and Proposition 2.30, we then see that B is the free b-coneover K , with unit u , with respect to both monotone homogeneous maps and Scott-continuoushomogeneous maps.The map ξ is a monotone homogeneous map from B into C extending e along u (i.e., ξ ◦ u = e ). As e is Scott-continuous and B is the free b-cone over K , with unit u , with respectto both monotone and Scott-continuous homogeneous maps, ξ is in fact Scott-continuous. Itis evidently linear. Proposition 2.37.
Let K u −→ B ξ −→ C be the standard factorisation of an embedding K e −→ C of a full Kegelspitze K in a d-cone. Then e is universal for Scott-continuous homogeneousmaps if, and only if, C is the universal dcpo-completion over B , with unit ξ .Proof. In one direction, suppose that C is the universal dcpo-completion over B , with unit ξ .By Lemma 2.36 we also have that B is the free b-cone over K , with unit u . It then followsfrom Proposition 2.34 that C is the free d-cone over B , with unit ξ . So C is the free d-coneover K , with unit ξ ◦ u = e , as required. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 25
In the other direction assume that e is universal, and consider the standard constructionof the free d-cone over K : K u s −→ Cone ( K ) ξ s −→ d - Cone ( K )As Cone ( K ) is the free b-cone over K with unit u s and, by Lemma 2.36, B is the freeb-cone over K with unit u , there is a homogeneous dcpo-isomorphism α : Cone ( K ) ∼ = B suchthat α ◦ u s = u . Next, as d - Cone ( K ) is the free d-cone over Cone ( K ) with unit ξ s , and as α and ξ are Scott-continuous homogeneous maps (the latter by Lemma 2.36), there is aScott-continuous homogeneous map β : d - Cone ( K ) → C such that β ◦ ξ s = ξ ◦ α . We thenhave β ◦ ( ξ s ◦ u s ) = e . But, as d - Cone ( K ) is the free d-cone over K with unit ξ s ◦ u s and, byassumption, C is the free d-cone over K with unit e , it follows that β is a dcpo-isomorphism.Putting all this together, we see that we have dcpo-isomorphisms α : Cone ( K ) ∼ = B and β : d - Cone ( K ) ∼ = C such that β ◦ ξ s = α ◦ ξ . Then, as Cone ( K ) ξ s −→ d - Cone ( K ) is adcpo-completion, it follows that B ξ −→ C is a dcpo-completion, as required.As a first application of the proposition, we next give a sufficient condition for anembedding of a Kegelspitze in a d-cone to be universal in the meet continuous case. Thecriterion applies in particular to continuous d-cones as indicated just before the statementof Lemma 2.32. Proposition 2.38.
Let e : K → C be a Scott-continuous linear order embedding of a fullKegelspitze K in a meet continuous d-cone. Suppose that: (1) e ( K ) is a lower subset of C , and (2) B = def { re ( a ) | a ∈ K, r > } is dense in C .Then e is universal.Proof. Endowing B with the ordered cone structure induced by C , we obtain K u −→ B ξ −→ C ,the standard factorisation of e . We may suppose, w.l.o.g., that u , and so e , is an inclusion.As e ( K ) is a lower subset of C , so is B . Then, by Proposition 2.37, Corollary 2.33 and theassumption that B is dense in C , we obtain the desired result.As another application of Proposition 2.37, we check that, given two universal embeddings K i e i −→ C i ( i = 1 ,
2) of full Kegelspitzen in d-cones, their product K × K e × e −−−−→ C × C isitself a universal embedding (we use the evident definitions of the products of Kegelspitzenand of d-cones). Let K i u i −→ B i ξ i −→ C i be the standard factorisation of e i . Then one checksthat the standard factorisation of e × e is K × K u × u −−−−→ B × B ξ × ξ −−−→ C × C (we usethe evident definition of the product of ordered cones). By Proposition 2.37, the B i ξ i −→ C i are universal dcpo-completions, and so, by 2.31(4), B × B ξ × ξ −−−→ C × C is also a universaldcpo-completion. Using Proposition 2.37 again, we see that e × e is universal. Remark 2.39. ( Historical Notes and References ) The abstract probabilistic algebras of Graham and Jones [14, 18] are barycentric algebras on a dcpo P with a bottom element 0 such that the map + : [0 , × P = ( r, ( x, y )) (cid:55)→ x + r y is continuous, taking the Hausdorfftopology on [0 , P , and the product topology on [0 , × P .Let us add that Jones [18, Section 4.2] proves that an abstract probabilistic algebra canequivalently be defined to be a dcpo P together with Scott-continuous maps S n × P n → P , n ∈ N , informally written (cid:80) ni =1 q i x i , satisfying the equations (A1) and (A2) in remark 2.9,where S n is the domain of subprobability measures on an n -element set as in remark 2.24with the Scott topology. Jones explicitly adds the requirement that the operations (cid:80) ni =1 q i x i are commutative in the sense that they are invariant under any permutation of the indices,a property that other authors (see the remarks 2.9 and 2.24) silently hide in the suggestivenotation of a sum.The notion of an abstract probabilistic algebra P of C. Jones is equivalent to our notionof a Kegelspitze. Indeed, one can show that a barycentric algebra over a dcpo is an algebrain this sense if, and only if, it is a Kegelspitze in our sense, so the two notions are equivalent.However, with our definition one can directly use domain-theoretic methods, for example forcompleting Kegelspitzen to d-cones.2.4. Preservation results.
We now turn to the question as to which additional propertiesof a Kegelspitze K are inherited by the universal d-cone d - Cone ( K ) over K . First we considercontinuity. We define a Kegelspitze K to be continuous if it is continuous as a dcpo.We will use the following standard lemma: Lemma 2.40.
Let P be a Scott-closed subset of a continuous poset Q . Then P is acontinuous poset and the way-below relation (cid:28) P on P is the restriction of the way-belowrelation (cid:28) Q on Q .Proof. Let x, y be elements of P . Clearly x (cid:28) Q y implies x (cid:28) P y . Thus, if Q is continuous,the same holds for P . Conversely, let x (cid:28) P y . Since Q is continuous, the elements z (cid:28) Q y form a directed set with supremum y . Since this directed set is in P , there is some z (cid:28) Q y such that x ≤ z , whence x (cid:28) Q y .In any d-cone, x (cid:55)→ rx is an order isomorphism for r >
0, whence x (cid:28) y ⇐⇒ rx (cid:28) ry for every r >
0. This statement is not true for Kegelspitzen, in general.For elements x, y in a Kegelspitze K and 0 < r < rx (cid:28) ry implies x (cid:28) y . Indeed if y ≤ sup i x i , then ry ≤ r sup i x i = sup i rx i , whence rx ≤ rx i for some i which implies x ≤ x i by Lemma 2.27. But x (cid:28) y does not imply rx (cid:28) ry in general, asthe counterexample in Appendix B shows. Fortunately, this difficulty disappears when werequire property (OC3): Lemma 2.41.
In a continuous full Kegelspitze K we have x (cid:28) y if and only if rx (cid:28) ry for every r with < r < .Proof. Let K be a continuous Kegelspitze and 0 < r <
1. By Property (OC2), the map x (cid:55)→ rx is an order isomorphism from K onto rK and rK is a sub-dcpo of K . Moreover, IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 27 if x and y are elements of K such that x (cid:28) y in K , then rx (cid:28) ry in rK . Property (OC3)implies that rK is a lower set, hence, a Scott-closed subset of K . By Lemma 2.40, theway-below relation of the dcpo rK is the restriction of the way-below relation on K . Thus rx (cid:28) ry in K . Proposition 2.42.
For a full Kegelspitze K , the cone d - Cone ( K ) is continuous if and onlyif K is a continuous Kegelspitze. If this is the case, then every element x ∈ d - Cone ( K ) isthe supremum of a directed family of elements a i in Cone ( K ) with a i (cid:28) x .Proof. If the d-cone d - Cone ( K ) is continuous, then the Scott-closed subset K is continuousby Lemma 2.40. Suppose conversely that K is a continuous Kegelspitze satisfying (OC3).Then K may be considered to be a Scott-closed convex subset of the ordered cone Cone ( K ).For x, y in Cone ( K ), we have x (cid:28) Cone ( K ) y iff rx, ry ∈ K and rx (cid:28) K ry for some r >
0. Indeed, suppose that x (cid:28) Cone ( K ) y . Let r > rx, ry ∈ K and let u i ∈ K be a directed family with ry ≤ sup i u i . Then y ≤ r − sup i u i = sup i r − u i whence x ≤ r − u i for some i which implies that rx ≤ u i . Conversely, suppose that rx (cid:28) K ry andconsider a directed family v i ∈ Cone ( K ) such that y ≤ sup i v i . There is an s > s sup i v i ∈ K . Moreover, we may choose s < r and s <
1. By Lemma 2.41, we know that sx = sr rx (cid:28) K sr ry = sy . Since sy ≤ s sup i v i = sup i sv i we conclude that that sx ≤ sv i forsome i , whence x ≤ v i by Lemma 2.27. We conclude that Cone ( K ) is a continuous b-cone.For the continuous b-cone Cone ( K ), the universal dcpo-completion d - Cone ( K ) agreeswith the round ideal completion which is a continuous d-cone by [27, Corollary 8.3]. Theelements of a round ideal constitute a directed family of elements way below the elementdefined by the round ideal.Recall that we say that the way-below relation on an ordered cone is additive if a (cid:28) b and a (cid:48) (cid:28) b (cid:48) = ⇒ a + a (cid:48) (cid:28) b + b (cid:48) Similarly, in a Kegelspitze we say that convex combinations preserve the way-below-relationif a (cid:28) b and a (cid:48) (cid:28) b (cid:48) = ⇒ a + r a (cid:48) (cid:28) b + r b (cid:48) Proposition 2.43.
Let K be a continuous full Kegelspitze. Then d - Cone ( K ) has an additiveway-below relation if and only if convex combinations preserve the way-below-relation in K .Proof. Suppose first that convex combinations preserve the way-below relation on K . Choose a (cid:28) a (cid:48) and b (cid:28) b (cid:48) in d - Cone ( K ). Interpolate elements a (cid:28) a (cid:48)(cid:48) (cid:28) a (cid:48) and b (cid:28) b (cid:48)(cid:48) (cid:28) b (cid:48) . Then a (cid:48)(cid:48) and b (cid:48)(cid:48) belong to Cone ( K ), since, by Proposition 2.42, a (cid:48) and b (cid:48) are suprema of directedfamilies in Cone ( K ), and since Cone ( K ) is a lower set in d - Cone ( K ). Thus we can find an r with 0 < r < ra (cid:48)(cid:48) ∈ K and rb (cid:48)(cid:48) ∈ K . Since ra (cid:28) ra (cid:48)(cid:48) and rb (cid:28) rb (cid:48)(cid:48) in K , we usethe property that convex combinations preserve the way-below relation to conclude that ra + rb (cid:28) ra (cid:48)(cid:48) + rb (cid:48)(cid:48) . Multiplying by 2 r yields a + b (cid:28) a (cid:48)(cid:48) + b (cid:48)(cid:48) ≤ a (cid:48) + b (cid:48) . Thus theway-below relation is additive on d - Cone ( K ). The converse is straightforward. Another noteworthy property is coherence. Recall that a dcpo is called coherent if theintersection of any two Scott-compact saturated subsets is Scott-compact.
Proposition 2.44.
Let K be a continuous full Kegelspitze. Then the d-cone d - Cone ( K ) iscoherent if, and only if, K is coherent.Proof. Clearly, if d - Cone ( K ) is coherent, then the Scott-closed subset K is coherent, too.For the converse recall that a continuous poset is said to have property M with respectto a basis B if, for any x , x , y , y ∈ B with y (cid:28) x and y (cid:28) x there is a finite set F ⊂ B such that ↑ x ∩ ↑ x ⊆ ↑ F ⊆ ↑ y ∩ ↑ y . By [7, Proposition III-5.12] the following areequivalent for a continuous dcpo P :(1) P is coherent.(2) P satisfies M for every basis B .(3) P satisfies M for some basis B .Now let K be a continuous full Kegelspitze which is coherent. Thus, K satisfies property Mwith B = K . We conclude that Cone ( K ) satisfies property M with B = Cone ( K ). Supposeindeed that x , x , y , y are elements of Cone ( K ) such that x (cid:28) y and x (cid:28) y . We canfind an r with 0 < r < ry ∈ K and ry ∈ K . As rx (cid:28) ry and rx (cid:28) ry hold in K , by the coherence of K we can find a finite subset F of K such that ↑ rx ∩ ↑ rx ⊆ ↑ F ⊆ ↑ ry ∩ ↑ ry . We then have ↑ x ∩ ↑ x ⊆ ↑ r − F ⊆ ↑ y ∩ ↑ y .Since Cone ( K ) is a basis of d - Cone ( K ), we conclude that d - Cone ( K ) is coherent.By [7, Corollary II-5.13] we conclude: Corollary 2.45.
Let K be a continuous full Kegelspitze. Then d - Cone ( K ) is Lawsoncompact if, and only if, K is Lawson compact. Duality and the subprobabilistic powerdomain.
We next give a brief introductionto duality for d-cones and Kegelspitzen, together with our main examples, function spacesand probabilistic powerdomains. For any dcpos P and Q we write Q P for the dcpo of allScott-continuous maps f : P → Q with the pointwise order, and note that it is a continuouslattice whenever P is a domain and Q is a continuous lattice (see, e.g., [7]). Example 2.46. (d-Cone function spaces) Let P be a dcpo. For every d-cone ( C, + , , · ),the dcpo C P is a d-cone when equipped with the pointwise sum and scalar multiplication:( f + g )( x ) = def f ( x ) + g ( x ) , ( r · f )( x ) = def r · f ( x )We write L P for the d-cone R P + of all Scott-continuous functions f : P → R + . We will usethe following properties of the d-cone L P later on:(a) If P is a domain then L P is a continuous lattice, hence a continuous d-cone.(b) For any domain P , the way-below relation of L P is additive, if, and only if, P iscoherent (Tix [61, Propositions 2.28 and 2.29]). IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 29
Example 2.47. (Kegelspitze function spaces) For every Kegelspitze ( K, + r ,
0) the dcpo K P becomes a Kegelspitze with the pointwise barycentric operations:( f + r g )( x ) = def f ( x ) + r g ( x )and with the constant function with value 0 as distinguished element.Taking K to be the unit interval I = [0 , L ≤ ( P ) = def I P .It is a Scott-closed convex subset of L P forming a sub-Kegelspitze of L P (considered as aKegelspitze), and therefore full. Further, L P is the universal d-cone over the Kegelspitze L ≤ P , that is L P ∼ = d - Cone ( L ≤ P ). The inclusion L ≤ ( P ) ⊆ L ( P ) is the universal embed-ding, as follows from Proposition 2.38, noting that L P has a continuous meet, and every f ∈ L P is the sup of the sequence of bounded functions p n ∧ f , where p n is the projectionsending R + to [0 , n ]. Finally, L ≤ P is a domain if P is, and its way-below relation is inheritedfrom that of L P and is preserved by the barycentric operations if, and only if, P is coherent.For any two d-cones C and D , the Scott-continuous linear functions f : C → D form asub-d-cone L lin ( C, D ) of the function space D C . Similarly, for any Kegelspitze K and d-cone D , the set L lin ( K, D ) of Scott-continuous linear functions is a sub-d-cone of D K . Note thatif K is C , regarded as a Kegelspitze, then L lin ( C, D ) and L lin ( K, D ) are identical.Duality will play an important rˆole: every d-cone C and Kegelspitze K has a duald-cone, viz. C ∗ = def L lin ( C, R + ) and K ∗ = def L lin ( K, R + ), respectively.By Theorem 2.35, if K is full then every Scott-continuous linear functional f : K → R + has a unique Scott-continuous linear extension (cid:101) f : C → R + , where C = d - Cone ( K ), and so f (cid:55)→ (cid:101) f is a natural d-cone isomorphism between K ∗ and C ∗ . We will use this isomorphismfreely.Duality leads to the weak Scott topology on a d-cone C , which, as the name implies,is coarser than the Scott topology. It is the coarsest topology on C for which the Scott-continuous linear functionals on C remain continuous. The sets U f = def { x ∈ C | f ( x ) > } , f ∈ C ∗ form a subbasis of this topology.We need the notion of a reflexive cone. For any d-cone C we have a canonical map ev C from C into the bidual C ∗∗ . It assigns the evaluation map f (cid:55)→ f ( x ) to every x ∈ C and isScott-continuous and linear. We say that C is reflexive if ev C is an isomorphism of d-cones.If C is a reflexive d-cone, then its dual C ∗ is also reflexive, and its dual is (isomorphic to) C .We will need, as we did in [28], the notion of a convenient d-cone. This is a continuousreflexive d-cone C whose weak Scott topology agrees with its Scott topology, and whosedual C ∗ is continuous and has an additive way-below relation. The valuation powerdomainconstruction, which we consider next, exemplifies these strong requirements. Example 2.48. (Probabilistic powerdomains) Probability measures on dcpos are modelledby valuations, which assign a probability to Scott-open subsets in a continuous way. Thispermits the development of a satisfactory theory of integration of Scott-continuous functions.
For large classes of dcpos, Scott-continuous valuations correspond to regular Borel probabilitymeasures and so it does not make much difference whether one works with Scott-continuousvaluations or probability measures. For the sake of applications in semantics it is usefulnot to restrict to probabilities, where the whole space has probability 1, but to admitsubprobabilities, where the whole space has probability of at most 1. For theoreticalpurposes, on the other hand, it is more convenient to extend the notion of probability to thatof a measure, where the measure of the whole space can be any nonnegative real number, oreven + ∞ .A valuation on a dcpo P is a map µ defined on the complete lattice O P of Scott-opensubsets of P taking nonnegative real values, including + ∞ , which (replacing finite additivity)is strict and modular: µ ( ∅ ) = 0 µ ( U ∪ V ) + µ ( U ∩ V ) = µ ( U ) + µ ( V ) for all U, V ∈ O P The point (or
Dirac ) valuations δ x ( x ∈ P ) are given by: δ x ( U ) = (cid:40) x ∈ U )0 ( x / ∈ U )The simple valuations are the finite linear combinations of point valuations. The set V P ofall Scott-continuous valuations µ : O P → R + forms a sub-d-cone of the function space R O P + ,with the pointwise partial order and with pointwise addition and scalar multiplication:( µ + ν )( U ) = def µ ( U ) + ν ( U ) , ( r · µ )( U ) = def r · µ ( U )We call it the valuation powerdomain over P (it is also known as the extended probabilisticpowerdomain ). The Scott-continuous valuations µ such that µ ( P ) ≤ subproba-bility valuations ; they form a Scott-closed convex subset V ≤ P of the d-cone V P , and hencea full sub-Kegelspitze. We call it the subprobabilistic powerdomain over P . We will use thefollowing results later on:(a) Let P be a domain. The valuation powerdomain V P is a continuous d-cone with anadditive way-below relation and (hence) the subprobabilistic powerdomain V ≤ P isa continuous Kegelspitze in which the barycentric operations preserve its way-belowrelation, which is inherited from V P (Jones [18, Theorem 5.2 and Corollary 5.4], Kirch[31], Tix [59] and see [7, Theorem IV-9.16]).(b) Let P be a coherent domain. Then the powerdomains V P and V ≤ P are also coherent(Jung and Tix [22], Jung [20]). In each case the relevant simple valuations form a basis.(c) For any dcpo P there is a canonical Scott-continuous map δ P : P → V ≤ P ⊆ V P whichassigns to any x ∈ P the Dirac measure δ P ( x ) which has value 1 for all Scott-openneighbourhoods U of x and value 0 otherwise. Then, if P is a domain, V P is thefree d-cone over P (Kirch [31], and see [7, Theorem IV-9.24]), and V ≤ P is the freeKegelspitze over P (Jones [18, Theorem 5.9]). That is, for every d-cone C and every IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 31
Scott-continuous function f : P → C , there is a unique Scott-continuous linear map f : V P → C such that the following diagram commutes: P f V Pδ P (cid:63) f (cid:45) C (cid:45) and for every Kegelspitze K and every Scott-continuous map f : P → K there is aunique Scott-continuous linear map f : V ≤ P → K such that the following diagramcommutes: P f V ≤ Pδ P (cid:63) f (cid:45) K (cid:45) (d) Let P be a domain. Then the valuation powerdomain V P is the universal d-cone overthe Kegelspitze V ≤ P , that is: V P ∼ = d - Cone ( V ≤ P ) (the inclusion V ≤ P ⊆ V P is theuniversal arrow). This is because V P and V ≤ P are, respectively, the free d-cone andthe free Kegelspitze over P .(e) For any dcpo P , the valuation powerdomain V P is the dual of L P up to isomorphism,and, if P is a domain, the d-cone L P is the dual of V P , which implies that both V P and L P are reflexive (Kirch [31, Satz 8.1 and Lemma 8.2], and Tix [59, Theorem 4.16]).The proof is based on the appropriate notion of an integral of a Scott-continuousfunction f : P → R + with respect to a Scott-continuous valuation µ . Among the variousways to define this integral, the most elegant is via the Choquet integral: (cid:90) f dµ = def (cid:90) + ∞ µ (cid:0) f − (cid:0) ] r, + ∞ ] (cid:1)(cid:1) dr The Choquet integral should be read as the generalised Riemann integral of thenonnegative monotone-decreasing function r ∈ [0 , + ∞ [ (cid:55)→ µ (cid:0) f − (cid:0) ] r, + ∞ ] (cid:1)(cid:1) , which maytake infinite values. The integral was originally defined as a Lebesgue integral byJones [18]; Tix later proved the Choquet definition equivalent [59].The isomorphism between V P and ( L P ) ∗ , the dual of L P , is the map µ (cid:55)→ λf. (cid:82) f dµ ,establishing a kind of Riesz representation theorem. And if P is a domain, theisomorphism between L P and the dual ( V P ) ∗ is the map f (cid:55)→ λµ. (cid:82) f dµ .(f) For a domain P , the Scott topology and the weak Scott topology agree on both L P and V P (Tix [59, Satz 4.10]). Summarising the properties reported in Examples 2.46 and 2.48, we can say that thevaluation powerdomain V P over a continuous coherent domain P is a convenient d-cone.3. Power Kegelspitzen
We now construct three kinds of power Kegelspitzen, proceeding analogously to the con-structions of the three types of powercone in [61]. Under various assumptions on a givenKegelspitze K , we will construct its lower, upper , and convex power Kegelspitzen , H K , S K ,and P K . Power Kegelspitzen and powercones have a choice operation, so we begin bydefining the three kinds of Kegelspitzen and cones enriched with a choice operation thatthereby arise.A Kegelspitze semilattice is a Kegelspitze equipped with a Scott-continuous semilatticeoperation ∪ over which convex combinations distribute, that is, for all x, y, z ∈ K and r ∈ [0 ,
1] we have: x + r ( y ∪ z ) = ( x + r y ) ∪ ( x + r z )It is a Kegelspitze join-semilattice if ∪ is the binary supremum operation (equivalently, if x ≤ x ∪ y always holds). It is a Kegelspitze meet-semilattice if ∪ is the binary infimum operation(equivalently, if x ∪ y ≤ x always holds). A morphism of Kegelspitze semilattices is a morphismof Kegelspitzen which also preserves the semilattice operation. Using the distributivityaxiom it is straightforward to show that the semilattice operation is homogeneous and thatthe barycentric operations are ⊆ -monotone (a function between two semilattices is said to be ⊆ -monotone if it preserves the partial order ⊆ naturally associated to semilattices). Further,the following convexity identity holds x ∪ ( x + r y ) ∪ y = x ∪ y (CI)This can be proved beginning with the equation x ∪ y = ( x ∪ y ) + r ( x ∪ y ), then expandingout the right-hand side using the distributivity of + r over ∪ , and then using the inclusion ⊆ associated with the semilattice operation.There is an analogous notion of d-cone semilattice . This is a d-cone C equipped with aScott-continuous semilattice operation ∪ over which the cone operations distribute, i.e., forall x, y, z ∈ C and r ∈ R + we have: x + ( y ∪ z ) = ( x + y ) ∪ ( x + z ) r · ( x ∪ y ) = r · x ∪ r · y Such a cone is a d-cone join-semilattice ( d-cone meet-semilattice ) if ∪ is the binary supremumoperation (respectively, the binary infimum operation). Every d-cone semilattice (d-conejoin-semilattice, d-cone meet-semilattice) can be regarded as a Kegelspitze semilattice(respectively, Kegelspitze join-semilattice, Kegelspitze meet-semilattice). A morphism ofd-cone semilattices is a morphism of d-cones which also preserves the semilattice operation.Much as before, distributivity implies that ∪ is homogeneous, and that the d-cone operationsare ⊆ -monotone. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 33
The lower, upper, and convex power Kegelspitzen will be, respectively, Kegelspitzejoin-semilattices, meet-semilattices, and semilattices. Possibly under further assumptions,they will be the free such Kegelspitze semilattices on K . The analogous result in the lowercase for powercones was proved in [61]. Freeness results were also proved in the other twocases, but of a different character, having weaker assumptions and weaker conclusions.In order to verify the properties of the various power Kegelspitzen, we will embed theKegelspitzen in d-cones, and then use the embeddings to transfer results from [61] aboutpowercones to the power Kegelspitzen.Another way to proceed is to view the power Kegelspitzen as retracts of the powerdomainsof the domains underlying the Kegelspitzen. One can then transfer results about thepowerdomains (such as the preservation of continuity) to the Kegelspitzen. Similar usesof retracts already occur in [61, 9, 10, 11, 12]. We have not explored this option, but it iscertainly possible to strengthen some of our results in this way. However, the results wepresent are sufficient for their intended application in Section 3.4 to mixed powerdomainscombining probabilistic choice and nondeterminism.3.1. Lower power Kegelspitzen.
We first investigate the convex lower (or
Hoare ) powerKegelspitze . We need some closure properties of convex sets: Lemma 3.1.
Let K be a Kegelspitze. Then: (1) Any directed union of convex subsets of K is convex. (2) The Scott closure of a convex subset of K is convex. (3) If X and Y are convex subsets of K then so is X + r Y = def { x + r y | x ∈ X, y ∈ Y } Proof.
The first statement is immediate. For the second, note that the Scott closure of a setis obtained by repeating the operations of downwards closure and taking directed unionstransfinitely many times. As each of these operations can be seen to preserve convexity, andas taking directed unions does too, it follows that the Scott closure of a convex set is convex.Finally the third statement follows using the entropic law (E).The lower power Kegelspitze H K , of a given full Kegelspitze ( K, + r , K , ordered by subset, with zero { } and with convex combination operators+ rH given by: X + rH Y = def X + r Y for r ∈ [0 , X ⊆ K , X is the closure of X in the Scott topology). Thatthese operators are well-defined follows from Lemma 3.1.As we said above, in order to verify the properties of H K we will make use of theembedding of K into a d-cone C and the properties of the lower powercone (or Hoarepowercone) of C . So let us begin by reviewing the definition and properties of the lower powercone H C of a d-cone ( C, + , , · ) [61, Section 4.1]. As a partial order, it is the collectionof all nonempty Scott-closed convex subsets of C ordered by inclusion ⊆ . It has arbitrarysuprema, given by: (cid:95) i ∈ I X i = conv (cid:91) i ∈ I X i with directed suprema given by: (cid:95) ↑ i ∈ I X i = (cid:91) ↑ i ∈ I X i Addition and scalar multiplication are lifted from C to H C as follows: X + H Y = def X + Y r · H X = def r · X where X + Y = { x + y | x ∈ X, y ∈ Y } , and r · X = { r · x | x ∈ X } . Convex combinationsare then given by r · H X + H (1 − r ) · H Y = X + r Y . Further, the following is proved in [61,Section 4.1]: Theorem 3.2.
Let ( C, + , , · ) be a d-cone. Then ( H C, + H , , · H ) is also a d-cone, and,equipped with binary suprema, it forms a d-cone join-semilattice.If C is continuous, then so is H C . The non-empty finitely generated convex Scott-closedsets conv F , where F is a finite, non-empty subset of C , form a basis for H C ; further, forany X, Y ∈ H C , X (cid:28) H C Y if, and only if, X ⊆ conv F and F ⊆ (cid:3)(cid:3) Y , for some such F . If,in addition, the way-below relation of C is additive, so is that of H C . We can now show:
Theorem 3.3.
Let ( K, + r , be a full Kegelspitze. Then ( H K, + rH , { } ) is a full Kegelspitze.It has arbitrary suprema, given by: (cid:95) i ∈ I X i = conv (cid:91) i ∈ I X i with directed suprema given by: (cid:95) ↑ i ∈ I X i = (cid:91) ↑ i ∈ I X i and, equipped with binary suprema, it forms a Kegelspitze join-semilattice.If, further, K is a continuous Kegelspitze, then so is H K . The non-empty finitelygenerated convex Scott-closed sets conv F , where F is a finite, non-empty subset of K , forma basis for H K ; further, for any X, Y ∈ H K , X (cid:28) H K Y if, and only if, X ⊆ conv F and F ⊆ (cid:3)(cid:3) Y , for some such F . If, in addition, the way-below relation of K is closed underconvex combinations, so is that of H K .Proof. Using Theorem 2.35 we can regard K as a Scott-closed convex subset of the d-cone C = def d - Cone ( K ), with its partial order and algebraic structure inherited from that of C .It is then immediate that H K embeds as a sub-partial order of H C . Further, as K isScott-closed and convex, one easily shows, using the above formulas for the suprema andconvex combinations of H C , that H K is a Scott-closed convex subset of H C , bounded above IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 35 by K . Therefore H K has all sups and a Kegelspitze structure, and they are inherited from H C . Explicitly, arbitrary sups and and directed sups are given by the claimed formulas, asthe corresponding formulas hold for H C (we may, equivalently, take Scott closure of subsetsof K with respect to C or K ). The inherited convex combinations and zero agree with thoseof H C . Convex combinations distribute over the semilattice operation (here binary sups), as+ and r · − do. So, equipped with binary suprema, H K is a Kegelspitze join-semilattice. Itautomatically satisfies Property (OC3) as it is embedded in the d-cone H C as a Scott-closedsubset.Next, suppose that K is a continuous Kegelspitze. Then, by Proposition 2.42, C iscontinuous, and so, by Theorem 3.2, H C is too. As H K is a Scott-closed subset of H C ,it follows that H K is continuous with way-below relation the restriction of that of H C to H K . That the non-empty finitely-generated Scott-closed sets form a basis of H K , andthe characterisation of the way-below relation of H K then follow from the correspondingparts of Theorem 3.2. Further, as r · − preserves the way-below relation of H C ( r · − isan order-isomorphism of cones), it also preserves the restriction to H K , and so H K iscontinuous as a Kegelspitze.Finally, suppose additionally that K ’s way-below relation is closed under convex com-binations. Then, by Proposition 2.43, C has an additive way-below relation. We then seefrom [61] that the d-cone operations of H C preserve its way-below relation, and so thatconvex combinations preserve the way-below relation of H K (it is the restriction of that of H C as H K is a closed subset of H C ).We now show that H K is the free Kegelspitze join-semilattice over any full Kegelspitze K , with unit the evident Kegelspitze morphism η H : K → H K , where η H ( x ) = ↓ x . Theorem 3.4.
Let K be a full Kegelspitze. Then the map η H is universal. That is, forevery Kegelspitze join-semilattice L and Kegelspitze morphism f : K → L there is a uniqueKegelspitze semilattice morphism f † : H K → L such that the following diagram commutes: K f H Kη H (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) L f ( X ) Proof.
To show uniqueness, choose an X ∈ H K . It can be written as a non-empty sup, viz. (cid:87) x ∈ X η H ( x ). Then, noting that continuous binary join morphisms preserve all non-emptysups (for such sups are directed sups of finite non-empty sups, and finite non-empty sups are iterated binary ones) we calculate: f † ( X ) = f † ( (cid:87) x ∈ X η H ( x ))= (cid:87) x ∈ X f † ( η H ( x ))= (cid:87) x ∈ X f ( x )To show existence we therefore set f † ( X ) = (cid:95) x ∈ X f ( x )and verify that it makes the diagram commute and is both a Kegelspitze and a semilatticemap. The first of these requirements holds as we calculate: f † ( η H ( x )) = f † ( { y | y ≤ x } ) = (cid:95) y ≤ x f ( y ) = f ( x )For the second we need to show that f † is strict and continuous and preserves convexcombinations. Strictness is a consequence of the diagram commuting, as both η H and f arestrict. For continuity we first show that (cid:95) f ( A ) = (cid:95) f ( A )for any A ⊆ K . Choose A ⊆ K . We evidently have (cid:87) f ( A ) ≤ (cid:87) f ( A ), and it remains toprove the converse inequality (cid:87) f ( A ) ≤ (cid:87) f ( A ). By the continuity of f , we have f ( A ) ⊆ f ( A )whence (cid:87) f ( A ) ≤ (cid:87) f ( A ). So we only have to show that x ≤ (cid:87) f ( A ) for any x ∈ f ( A ).This follows from the fact that the Scott closure of a set is obtained by transfinitely manyrepetitions of the operations of downwards closure and taking directed sups.We can now calculate: f † ( (cid:87) ↑ i ∈ I X i ) = f † ( (cid:83) ↑ i ∈ I X i )= (cid:87) f ( (cid:83) ↑ i ∈ I X i )= (cid:87) f ( (cid:83) ↑ i ∈ I X i )= (cid:87) ↑ i ∈ I (cid:87) x ∈ X i f ( x )= (cid:87) ↑ i ∈ I f † ( X i )For convex combinations we calculate: f † ( X + rH Y ) = f † ( X + r Y )= f † ( X + r Y )= (cid:87) z ∈ X + r Y f ( z )= (cid:87) x ∈ X, y ∈ Y f ( x + r y )= (cid:87) x ∈ X, y ∈ Y f ( x ) + r f ( y )= ( (cid:87) x ∈ X f ( x )) + r ( (cid:87) y ∈ Y f ( y ))= f † ( X ) + r f † ( Y ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 37
The second equation holds as f is Scott-continuous. The sixth equation holds as convexcombinations in L distribute over arbitrary non-empty sups, as we may see by analysingsuch sups as before, i.e., as directed sups of iterated binary ones.Finally, we need to show that binary sups are also preserved, and so calculate: f † ( X ∨ Y ) = f † (conv( X ∪ Y ))= f † (conv( X ∪ Y ))= (cid:87) x ∈ X, y ∈ Y (cid:87) ≤ r ≤ f ( x + r y )= (cid:87) x ∈ X, y ∈ Y (cid:87) We next investigate the convex upper (or Smyth ) powerKegelspitze S K , of a given continuous full Kegelspitze ( K, + r , K , ordered by reverseinclusion ⊇ , with zero K and with convex combination operators + rS given by: X + rS Y = def ↑ ( X + r Y )for r ∈ [0 , X + r Y is Scott-compact, as it is the image under + r of X × Y , which is compact in the product topology on K × K , which latter is the same as the Scott topology, as K is continuous. Then note thatthe upper closure of a Scott-compact (convex) set is Scott-compact (respectively convex).The upper power Kegelspitze has binary infima, which make it a Kegelspitze meet-semilattice. They are given by: X ∧ Y = ↑ conv( X ∪ Y )In order to verify the properties of S K , we follow our general methodology, using theembedding of K into a d-cone C and the properties of the upper powercone (or Smythpowercone) of C . Let us begin by recalling the definition and properties of the upperpowercone S C of a continuous d-cone ( C, + , , · ) [61, Section 4.2]. It consists of all nonemptyScott-compact convex saturated subsets ordered by reverse inclusion ⊇ . It has directedsuprema given by intersection: (cid:95) ↑ i ∈ I X i = (cid:92) ↓ i ∈ I X i and binary infima given by: X ∧ Y = ↑ conv( X ∪ Y ) Addition and scalar multiplication are lifted from C to S C as follows: X + S Y = def ↑ ( X + Y ) r · S X = def ↑ ( r · X )Note that r · S X = ↑{ } = C if r = 0 and r · S X = r · X if r > 0. Convex combinations aregiven by r · S X + S (1 − r ) · S Y = ↑ ( r · X + (1 − r ) · Y ). Further, the following is provedin [61, Section 4.2]: Theorem 3.5. Let ( C, + , , · ) be a continuous d-cone. Then ( S C, + S , C, · S ) is a continuousd-cone and, equipped with binary infima, it forms a d-cone meet-semilattice.The non-empty finitely generated convex saturated Scott-compact sets ↑ conv F , where F is a finite, non-empty subset of C , form a basis for S C ; further, for any X, Y ∈ S C , X (cid:28) S C Y if, and only if, X ⊇ ↑ conv F and (cid:2)(cid:2) F ⊇ Y , for some such F . If the way-belowrelation of C is additive, so is that of S C . For the proof of the next theorem we recall that a (monotone) retraction pair betweentwo partial orders P and Q is a pair of monotone maps P e −→ Q r −→ P such that r ◦ e = id P ; it is a (monotone) closure pair if, additionally, e ◦ r ≥ id Q . We cannow show: Theorem 3.6. Let ( K, + r , be a continuous full Kegelspitze. Then ( S K, + rS , K ) is acontinuous Kegelspitze meet-semilattice. Directed suprema are given by intersection: (cid:95) ↑ i ∈ I X i = (cid:92) ↓ i ∈ I X i and binary infima are given by: X ∧ Y = ↑ conv( X ∪ Y ) The non-empty finitely generated convex saturated Scott-compact sets ↑ conv F , where F is afinite, non-empty subset of K , form a basis for S K ; further, for any X, Y ∈ S K , X (cid:28) S K Y if, and only if, X ⊇ ↑ conv F and (cid:2)(cid:2) F ⊇ Y , for some such F . The way-below relation (cid:28) S K ispreserved by r · S K − , and if the way-below relation of K is closed under convex combinations,so is that of S K . If K is coherent then S K is a bounded-complete domain, hence coherenttoo.Proof. Using Theorem 2.35, we can regard K as a Scott-closed convex subset of the d-cone C = def d - Cone ( K ), with its partial order and algebraic structure inherited from that of C .By Proposition 2.42, C is continuous, and so, by Lemma 2.40, the way-below relation of K is also inherited from that of C .To relate S K to S C we first define a Scott-closed convex subset L of S C and then showthat S K is a closure of L (with partial ordering inherited from S C ). This enables us totransport structure from S C to S K via L . We take L to be the collection of elements of S C intersecting K . It is evidently a lower set (for ≤ = ⊇ ). If X i is a directed subset of L IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 39 then X i ∩ K is a ⊆ -filtered collection of non-empty saturated Scott-compact subsets of K and so has a non-empty intersection by a consequence of the Hofmann-Mislove Theorem [7,Corollary II-1.22]. This shows that (cid:87) S C X i is in L . The convexity of L follows from that of K . We order L by reverse inclusion ⊇ , i.e., as a sub-partial order of S C . As L is a Scott-closed subset of S C , it is a continuous sub-dcpo of S C , with way-below relation inheritedfrom that of S C , and with basis B ∩ L , for any basis B of S C . Further, as L is a convexScott-closed subset of S C , it inherits a Kegelspitze structure from S C , with zero C , andwith convex combinations given by X + rL Y = r · S X + S (1 − r ) · Y = ↑ ( r · X + (1 − r ) · Y ).Binary infima are also inherited by L , and as these distribute over + C and · C , L forms aKegelspitze meet-semilattice.We now define a monotone closure pair: S K e −→ L c −→ S K by setting e ( X ) = def ↑ C X and c ( Y ) = def Y ∩ K .Since ( e, c ) is a monotone closure pair, the existence of directed suprema in L impliesthe existence of directed suprema in S K and c preserves these directed suprema, that is, c is Scott-continuous. The following shows that the supremum of a directed collection X i in S K is calculated as expected: (cid:95) ↑ i X i = c ( (cid:95) ↑ i e ( X i )) = ( (cid:92) ↓ i ↑ C X i ) ∩ K = (cid:92) ↓ i ( ↑ C X i ) ∩ K = (cid:92) ↓ i X i For the continuity of e , we have to show for any directed collection X i in S K that: ↑ C ( (cid:92) i X i ) ⊇ ( (cid:92) i ↑ C X i )(the other direction holds as e is monotone). Suppose, for the sake of contradiction,that there is a y ∈ C , with y in every ↑ C X i , but not in ↑ C ( (cid:84) i X i ). We then have that (cid:84) i X i ⊆ { x ∈ C | x (cid:54)≤ y } ∩ K . As the latter set is Scott-open in K , and as K is well-filtered(this follows from the Hofmann-Mislove theorem, see [7, Theorem II-1.21]), there is an i such that X i ⊆ { x ∈ C | x (cid:54)≤ y } ∩ K , which contradicts our assumption. Thus ( c, e ) is aScott-continuous closure pair and we can conclude from the continuity of L that S K is acontinuous dcpo.Turning to the characterisation of the way-below relation on S K , choose X, Y in S K .By general properties of retractions we know that X (cid:28) S K Y holds iff there is a U ∈ L such that X ≤ c ( U ) and U (cid:28) L ↑ C Y (equivalently e ( X ) ≤ U and U (cid:28) L ↑ C Y , as ( e, c ) is anadjoint pair). As the way-below relation on L is the restriction of that on S C , Theorem 3.5tells us that U (cid:28) L ↑ C Y holds iff there is a non-empty finite subset F of C such that U ⊇ ↑ C conv F and (cid:2)(cid:2) C F ⊇ ↑ C Y . Putting these together, we have that X (cid:28) S K Y holdsiff there is a non-empty finite subset F of C such that ↑ C X ⊇ ↑ C conv F and (cid:2)(cid:2) C F ⊇ ↑ C Y (equivalently, ↑ C X ⊇ conv F and (cid:2)(cid:2) C F ⊇ Y ). As Y ⊆ K and (cid:2)(cid:2) C F ⊇ Y , and K is a Scott-closed subset of C , F ∩ K is non-empty. Sowe have that X (cid:28) S K Y holds if, and only if, there is a non-empty finite subset F (cid:48) of K suchthat ↑ C X ⊇ conv F (cid:48) and (cid:2)(cid:2) C F (cid:48) ⊇ Y (in one direction, given F , set F (cid:48) = F ∩ K ; in the otherdirection, given F (cid:48) , take F = F (cid:48) ). For such an F (cid:48) we have ↑ C X ⊇ conv F (cid:48) iff X ⊇ conv F (cid:48) (as K is a convex subset of C ) and (cid:2)(cid:2) C F (cid:48) ⊇ Y iff (cid:2)(cid:2) K F (cid:48) ⊇ Y , and we have established thedesired characterisation of (cid:28) S K .Using this characterisation, and the fact that K is continuous, it follows immediatelythat (cid:28) S K is preserved by r · S K − . It also follows immediately that the non-empty finitelygenerated convex saturated sets form a basis of S K .Next, a calculation now shows that e preserves the convex combination operation: e ( X )+ rS e ( Y ) = ↑ C ( r · ( ↑ C X ) + (1 − r ) · ( ↑ C Y ))= ↑ C ( ↑ C r · X + ↑ C (1 − r ) · Y )= ↑ C ( r · X + (1 − r ) · Y )= ↑ C ↑ K ( r · X + (1 − r ) · Y )= e ( X + rK Y )It follows that the convex combination operation on S K can be defined in terms of thaton L as we have: X + rS Y = ce ( X + rS Y ) = c ( e ( X )+ rL e ( Y )). So, as e, c, + rL are allScott-continuous, so is + r S K .As ( c, e ) is a closure pair and L has binary infima, so does S K and e preserves them.For any X, Y ∈ S K we can then calculate: X ∧ Y = c ( e ( X ) ∧ e ( Y ))= ( ↑ C conv( ↑ C X ∪ ↑ C Y )) ∩ K = ( ↑ C conv ↑ C ( X ∪ Y )) ∩ K = ( ↑ C conv( X ∪ Y )) ∩ K = ↑ K conv( X ∪ Y )showing that binary meets are given as required. We have also seen that ∧ S K can be definedas a composition of e, c, ∧ L , and so is Scott-continuous.As e is an order-mono (i.e., it reflects the partial order) and as it preserves convexcombinations and binary meets, any inequations between these operations holding in L alsohold in S K . So (also using the fact that both convex combinations and binary meets aremonotone) S K is an ordered barycentric algebra, binary meets form a meet-semilattice, andconvex combinations distribute over binary meets. Therefore, as convex combinations andbinary meets are both Scott-continuous, and as r (cid:55)→ ra is Scott-continuous, we see that S K is a Kegelspitze meet semilattice with zero K . We also know that S K is continuous and soit is a continuous Kegelspitze.Next, we show that convex combinations in S K preserve its way-below relation, as-suming the same is true of K . Suppose that X (cid:28) S K Y and X (cid:48) (cid:28) S K Y (cid:48) . Using thecharacterisation of (cid:28) S K we see that there are finite, non-empty F, F (cid:48) ⊆ K such that X ⊇ ↑ conv F , (cid:2)(cid:2) F ⊇ Y , X (cid:48) ⊇ ↑ conv F (cid:48) , and (cid:2)(cid:2) F (cid:48) ⊇ Y (cid:48) , and then that we need only show that IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 41 X + rS X (cid:48) ⊇ ↑ conv( F + r F (cid:48) ) and (cid:2)(cid:2) ( F + r F (cid:48) ) ⊇ Y + rS Y (cid:48) , i.e., that ↑ ( X + r X (cid:48) ) ⊇ ↑ conv( F + r F (cid:48) )and (cid:2)(cid:2) ( F + r F (cid:48) ) ⊇ ↑ ( Y + r Y (cid:48) ). The first of these requirements holds as we have: ↑ ( X + r X (cid:48) ) ⊇ ↑ ( ↑ conv F + r ↑ conv F (cid:48) ) ⊇ ↑ (conv F + r conv F (cid:48) ) ⊇ ↑ conv( F + r F (cid:48) )(with the last inclusion holding because of the entropic law). The second holds as convexcombinations in K preserve (cid:28) K .Finally if K is also (Scott-)coherent, i.e., if the intersection of two Scott-compactsaturated sets is again such, then S K is a bounded complete dcpo. So, as it is alsocontinuous, it is coherent (see [7, Proposition III-5.12]).We note that the proof also establishes that the map u = X (cid:55)→ ↑ X : S K → S ( d - Cone ( K ))is a d-cone meet semilattice embedding (that is, it is a d-cone semilattice morphism thatreflects the partial order).Theorem 3.6 does not assert that S K satisfies Property (OC3), but only that r · S K − preserves (cid:28) S K . Indeed, S K need not satisfy Property (OC3): Fact 3.7. The upper power Kegelspitze of the subprobabilistic powerdomain V ≤ { , } ofthe two-element discrete partial order does not satisfy Property (OC3). Proof. For notational convenience we replace V ≤ { , } by the isomorphic Kegelspitzeobtained from S = { ( r, s ) ∈ [0 , | r + s ≤ } by ordering it coordinatewise and equippingit with the evident convex combination operators.Set Y ∈ S S to be { (1 , } , and take any 0 < r < X (cid:48) ∈ S S to be the saturatedconvex closure of { (0 , , ( r, } , which is: { ( r (1 − s ) , s ) | s ∈ [0 , , r ≤ r ≤ } Clearly X (cid:48) ⊇ ↑ ( r · Y ), i.e., X (cid:48) ≤ r · S Y . Suppose, for the sake of contradiction, that X (cid:48) = r · S X = ↑ ( r · X ) for some X ∈ S S . Then, as (0 , ∈ X (cid:48) = ↑ r · X we have (0 , ≥ r · x for some x ∈ X . As r · x ∈ r · S X = X (cid:48) , r · x has the form ( r (1 − s ) , s ) for some s ∈ [0 , r ≤ r ≤ 1. Then as (0 , ≥ r · x = ( r (1 − s ) , s ), and so 0 ≥ r (1 − s ), we see that s = 1and so that r · x = (0 , r < S . However it is not a problem for this paper as iteratingthe upper mixed powerdomain does not involve iterating S .We next show that S K is the free Kegelspitze meet-semilattice over any Kegelspitze K satisfying suitable assumptions. The unit η S : K → S K is the evident Kegelspitze morphism η S ( x ) = def ↑ K x . Lemma 3.8. Let K be a continuous full Kegelspitze in which convex combinations preservethe way-below relation. Suppose that F, G are non-empty subsets of K such that (cid:2)(cid:2) G ⊇ F .Then ↑ conv G (cid:28) S K ↑ conv F . Proof. As convex combinations preserve (cid:28) K and (cid:2)(cid:2) G ⊇ F , we have (cid:2)(cid:2) conv G ⊇ ↑ conv F .Then, using the compactness of ↑ conv F , we see that there is a non-empty finite subset H of conv G such that (cid:2)(cid:2) H ⊇ ↑ conv F . The conclusion follows by the characterisation of (cid:28) S K given in Theorem 3.6. Theorem 3.9. Let K be a continuous full Kegelspitze in which convex combinations preservethe way-below relation. Then the map η S is universal. That is, for every Kegelspitze meet-semilattice L and Kegelspitze morphism f : K → L there is a unique Kegelspitze semilatticemorphism f † : S K → L such that the following diagram commutes: K f S Kη S (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) ↑ { (cid:94) f ( F ) | F ⊆ fin K, F (cid:54) = ∅ , ↑ conv F (cid:28) S K X } Proof. Using the basis of (cid:28) S K given in Theorem 3.6, for any X ∈ S K we have X = (cid:95) ↑ { (cid:94) η S ( F ) | F ⊆ fin K, F (cid:54) = ∅ , ↑ conv F (cid:28) S K X } where we make use of the easily proved fact that for any finite non-empty subset F of K ,we have: ↑ conv F = (cid:86) b ∈ F η S ( b ).It then follows for any Kegelspitze semilattice morphism f † which makes the diagramcommute that f † ( X ) = (cid:95) ↑ { (cid:94) f ( F ) | F ⊆ fin K, F (cid:54) = ∅ , ↑ conv F (cid:28) S K X } establishing uniqueness.For existence we define f † by means of this formula and verify that it makes the diagramcommute and is both a Kegelspitze and a semilattice map. For continuity, it is evident that f † is monotone, and so it suffices to show that for any directed set X i , i ∈ I , in S K and anyfinite, non-empty F ⊆ K with ↑ conv F (cid:28) S K (cid:87) i X i we have: (cid:94) f ( F ) ≤ (cid:95) i { (cid:94) f ( G ) | G ⊆ fin K, G (cid:54) = ∅ , ↑ conv G (cid:28) S K X i } This holds as if ↑ conv F (cid:28) S K (cid:87) i X i then ↑ conv F (cid:28) S K X i , for some i .Next, it is helpful to prove that f † ( (cid:94) η S ( F )) = (cid:94) f ( F ) ( ∗ )for any finite non-empty set F , that is, that: (cid:95) ↑ { (cid:94) f ( G ) | G ⊆ fin K, G (cid:54) = ∅ , ↑ conv G (cid:28) S K ↑ conv F } = (cid:94) f ( F ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 43 To show the left-hand side is ≤ the right-hand side, suppose we have a non-empty finitesubset G of K such that ↑ conv G (cid:28) S K ↑ conv F . Then, by the characterisation of (cid:28) S K given in Theorem 3.6, there is a finite non-empty H ⊆ K such that ↑ conv G ⊇ ↑ conv H and (cid:2)(cid:2) H ⊇ ↑ conv F So, for any a ∈ F , there is a b ∈ H such that b (cid:28) a , and so a c ∈ conv G such that c ≤ a . Let G be the set of such c ’s. We then have: (cid:94) f ( G ) = (cid:94) f ( G ) ∧ (cid:94) f ( G ) ≤ (cid:94) f ( G ) ≤ (cid:94) f ( F )where the equality follows from the convexity identity (CI). This shows the left-hand side is ≤ the right-hand side.Conversely, given F = { a , . . . , a n } , with n > 0, choose b (cid:28) a , . . . , b n (cid:28) a n and take G = { b , . . . , b n } . By Lemma 3.8 we have ↑ conv G (cid:28) S K ↑ conv F . So the left-hand side is ≥ (cid:86) f ( G ), and so ≥ (cid:86) f ( F ), as G consists of an arbitrary choice of an elements way-beloweach element of F .Taking F to be a singleton in ( ∗ ), we see that, as required, the diagram commutes; this,in turn, implies that f † is strict, as η S and f are. As regards preservation of the semilatticeoperation ∧ , as every element is a directed supremum of non-empty finite infima of elementsof the form η S ( b )and as ∧ is Scott-continuous, we need only verify it for such non-emptyfinite infima, and that follows immediately from ( ∗ ).We finally show that f † preserves convex combinations. Since f † is Scott-continuousand every element of S K is a directed supremum of meets of the form (cid:86) b ∈ F η S ( b ) ( F ⊆ K non-empty and finite), it suffices to show that f † preserves convex combinations of suchfinite meets. To that end, given F, G ⊆ K non-empty and finite, we calculate: f † ( (cid:86) b ∈ F η S ( b )+ r S K (cid:86) c ∈ G η S ( c )) = f † ( (cid:86) b ∈ F,c ∈ G ( η S ( b ) + r S K η S ( c )))= f † ( (cid:86) b ∈ F,c ∈ G η S ( b + r c ))= (cid:86) b ∈ F,c ∈ G f ( b + r c )= (cid:86) b ∈ F,c ∈ G ( f ( b ) + r f ( c ))= (cid:86) b ∈ F f ( b ) + r (cid:86) c ∈ G f ( c )= f † ( (cid:86) b ∈ F η S ( b )) + r f † ( (cid:86) c ∈ G η S ( c ))where the third and sixth equalities follow from ( ∗ ), and the first and fifth follow fromdistributivity.This result contrasts with the corresponding universality result for upper powerconesin [61]. There the assumptions are weaker, but so are the conclusions: there is no assumptionof preservation of the way-below relation, but the universality relates only to continuousd-cone semilattices, not to all of them. Further the proof methods for the two theoremsare different. It would be interesting to know if the assumption made in Theorem 3.9 thatconvex combinations preserve the way-below relation is needed. Convex power Kegelspitzen. We next investigate the convex (or Plotkin ) powerKegelspitze P K , of a given continuous and coherent (so Lawson compact) full Kegelspitze.Note that we have to suppose not only continuity but also coherence in order to provethe desired results. First we need some definitions from [61]. Nonempty Lawson-compactorder-convex subsets of a Lawson-compact domain are called lenses . Both Scott-closed setsand saturated Scott-compact sets are lenses, as they are both Lawson-compact, and everylens X can be written as the intersection of a non-empty Scott-closed convex set and anon-empty Scott-compact saturated convex one, as we have: X = X ∩ ↑ X . We also have X = ↓ X for any lens X . If a lens X of a continuous Lawson-compact Kegelspitze is alsoconvex, then so are ↓ X and ↑ X . The Egli-Milner ordering is defined on order-convex subsetsof a partial order ≤ by: X ≤ EM Y ≡ def ∀ x ∈ X. ∃ y ∈ Y. x ≤ y ∧ ∀ y ∈ Y. ∃ x ∈ X. x ≤ y which can equivalently be written as X ≤ EM Y ≡ ↓ X ⊆ ↓ Y ∧ ↑ Y ⊆ ↑ X We define P K to be the collection of convex lenses of K ordered by the Egli-Milnerordering, with zero { } and with convex combination operators + rP given by: X + rP Y = def ( ↓ X + rH ↓ Y ) ∩ ( ↑ X + rS ↑ Y )for r ∈ [0 , X + rP Y = X + r Y ∩ ↑ ( X + r Y ). The convex power Kegelspitzeis a Kegelspitze semilattice when equipped with the semilattice operator ∪ P defined by: X ∪ P Y = def ( ↓ X ∨ H K ↓ Y ) ∩ ( ↑ X ∧ S K ↑ Y )Using the explicit definitions of the semilattice operations for the lower and upper powerKegelspitzen one sees that X ∪ P Y = conv( X ∪ Y ) ∩ ↑ conv( X ∪ Y ); note too that ↓ ( X ∪ Y ) = ↓ X ∨ H K ↓ Y and ↑ ( X ∪ P Y ) = ↑ X ∨ S K ↑ Y .In order to verify the properties of P K we proceed as before, via embeddings into cones.Let us begin by recalling the definition and properties of the convex powercone P C of acontinuous Lawson-compact d-cone ( C, + , , · ) [61, Section 4.3]. This is the collection of allconvex lenses of C partially ordered by the Egli-Milner ordering. It has directed supremagiven by: (cid:95) ↑ i ∈ I X i = ( (cid:95) ↑ i ∈ I ↓ X i ) ∩ ( (cid:95) ↓ i ∈ I ↑ X i )where, on the right, we take directed suprema in H C and S C , respectively. More explicitly,we have: (cid:95) ↑ i ∈ I X i = ( (cid:91) ↑ i ∈ I ↓ X i ) ∩ ( (cid:92) ↓ i ∈ I ↑ X i )Addition and scalar multiplication are lifted from C to P C as follows: X + P Y = def ( ↓ X + H ↓ Y ) ∩ ( ↑ X + S ↑ Y ) r · P X = def ( r · H ↓ X ) ∩ ( r · S ↑ X ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 45 Using the explicit definitions of addition and scalar multiplication in the lower and upperpowercones, these definitions simplify to: X + P Y = X + Y ∩ ↑ ( X + Y ) r · P X = r · X Convex combinations are given by: r · P X + P (1 − r ) · P Y = r · X + (1 − r ) · Y ∩ ↑ ( r · X + (1 − r ) · Y )There is also a Scott-continuous semilattice operation. It is defined by: X ∪ P Y = def ( ↓ X ∨ H C ↓ Y ) ∩ ( ↑ X ∨ S C ↑ Y )which simplifies to X ∪ P Y = conv( X ∪ Y ) ∩ ↑ conv( X ∪ Y ). Further, the following is provedin [61, Section 4.3]: Theorem 3.10. Let ( C, + , , · ) be a continuous coherent d-cone. Then ( P C, + P , { } , · P ) isalso a continuous coherent d-cone, and, equipped with the semilattice operation ∪ P , it formsa d-cone semilattice.The finitely generated convex lenses k C ( F ) = def conv F ∩ ↑ conv F , where F is a finite,non-empty subset of C , form a basis for P C , and, for any X, Y ∈ P C , we have X (cid:28) P C Y if, and only if, X ≤ EM k C ( F ) and F ⊆ (cid:3)(cid:3) Y and (cid:2)(cid:2) F ⊇ Y (i.e., F (cid:28) EM Y ) for some such F .If the way-below relation of C is additive, so is that of P C . We can now show: Theorem 3.11. Let ( K, + r , be a continuous coherent full Kegelspitze. Then ( P K, + rP , { } ) is also a continuous coherent full Kegelspitze and, equipped with the Scott-continuous semilattice operation ∪ P , it forms a Kegelspitze semilattice. Directed supremaare given by: (cid:95) ↑ i ∈ I X i = ( (cid:91) ↑ i ∈ I ↓ X i ) ∩ ( (cid:92) ↓ i ∈ I ↑ X i ) The finitely generated convex lenses k K ( F ) = def conv F ∩ ↑ conv F , where F is a finite,nonempty subset of K , form a basis for P K , and, for any X, Y ∈ P K , we have X (cid:28) P K Y if, and only if, X ≤ EM k K ( F ) and F ⊆ (cid:3)(cid:3) Y and (cid:2)(cid:2) F ⊇ Y (i.e., F (cid:28) EM Y ) for some such F .If, in addition, the way-below relation of K is closed under convex combinations, so is thatof P K .Proof. Applying Theorem 2.35 we can regard K as a Scott-closed convex subset of thed-cone C = def d - Cone ( K ), with its partial order and algebraic structure inherited from thatof C . Applying Propositions 2.42 and 2.44, we see that C is continuous and coherent. Itfollows that K is a sub-dcpo of C , that its way-below relation is inherited from that of C ,that a subset of K is Scott-compact in the topology of K if, and only if, it is Scott-compactin C , and that a subset of K is a lens of K if, and only if, it is a lens of C .We therefore see that P K is a subset of P C . It also evidently inherits its partial orderfrom that of P C . We next show that P K is a Scott-closed convex subset of K . To see it is a lower set, suppose X ≤ EM Y ∈ P K . Then X ⊆ ↓ C X ⊆ ↓ C Y ⊆ K (the last as Y ⊆ K and K is a lower set), and so X ∈ P K . For closure under directed suprema, suppose X i is adirected subset of P K . Then (cid:95) ↑ i ∈ I X i = Y ∩ Z , where Y = def ( (cid:91) ↑ i ∈ I ↓ C X i ) and Z = def (cid:92) ↓ i ∈ I ↑ C X i with the closure being taken in C . As X i ⊆ K , Y is in fact a Scott-closed subset of K .Therefore the directed supremum is a subset of K and so a lens of K , as required, and wehave shown that P K is a Scott-closed subset of P C . It follows in particular that P K is asub-dcpo of P C . Noting that we can write Y equivalently taking the lower closure and thetopological closure in K , and that we can write Z ∩ K as (cid:84) i ∈ I ↑ K X i we further see thatdirected suprema in P K are given as claimed.For convex closure, recall that convex combinations are given by r · X + (1 − r ) · Y ∩ ↑ C ( r · X + (1 − r ) · Y )where the closure is taken in C . Taking X, Y ∈ P K we see that r · X + (1 − r ) · Y is asubset of K , as K is a convex subset of C . So the closure can equivalently be taken in K and we find that the convex combination is a subset of K and so, as required, a lens in K .Intersecting ↑ C ( r · X + (1 − r ) · Y ) with K , we note that we can write the convex combinationequivalently as r · X + (1 − r ) · Y ∩ ↑ K ( r · X + (1 − r ) · Y ) with the closure taken in K . Thusthe convex combination operators of P K are the same as those inherited from P C .As P K is a Scott-closed convex subset of P C , it inherits a continuous coherent Kegel-spitze structure satisfying Property (OC3) from P C , with way-below relation the restrictionof that of P C , and with basis B ∩ P K , where B is any basis of P C . As the zero of P K isevidently that of P C , we see from the above that the partial order and algebraic structuredefined on P K is that inherited from P C , and so P K is indeed a continuous coherentKegelspitze satisfying Property (OC3).One checks that the operation ∪ defined on P K is the restriction of ∪ P to P K . It istherefore, as claimed a Scott-continuous semilattice operation. Further as + P and r · P − both distribute over ∪ P , we see that, equipped with ∪ , P K is, as claimed, a Kegelspitzesemilattice.The finitely generated convex lenses k C ( F ) = conv F ∩ ↑ C conv F that are in P K forma basis of P K . As then F ⊆ k C ( F ) ⊆ P K , the closure can be equivalently be taken in K ,and we see that k C ( F ) = k K ( F ). So P K has a basis as claimed. The characterisation of (cid:28) P K can then be read off from the characterisation of (cid:28) P C , as (cid:28) P K is the restriction of (cid:28) P C to P K .If (cid:28) K is closed under convex combinations, we can assume by Proposition 2.43 that (cid:28) C is closed under sums. Then (cid:28) P C is also closed under sums, and so, too, under convexcombinations. As P K inherits convex combinations and its way-below relation from P C ,we see that (cid:28) P K is closed under convex combinations, concluding the proof. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 47 We next show that P K is the free Kegelspitze semilattice over any Kegelspitze K satisfying suitable assumptions. The unit η P : K → P K is the evident Kegelspitze morphism η P ( x ) = def { x } . We first need two lemmas. Lemma 3.12. Let K be a continuous coherent full Kegelspitze. Suppose that H , G , F arenon-empty finite subsets of K such that H (cid:28) EM G and k K ( G ) (cid:28) P K k K ( F ) . Then there arefinite sets H ⊆ conv H and F ⊆ conv F such that H ∪ H ≤ EM F ∪ F .Proof. By the characterisation of (cid:28) P K given in Theorem 3.11, there is a non-empty finite I ⊆ fin K such that k K ( G ) ≤ EM k K ( I ) and I (cid:28) EM k K ( F ).We first show: ∀ c ∈ H. ∃ a ∈ conv F. c ≤ a ( ∗ )Choosing c ∈ H , as H (cid:28) EM G we find a b ∈ G with c (cid:28) b . Then, as k K ( G ) ≤ EM k K ( I )we find an i (cid:48) in the closed set conv I such that b ≤ i (cid:48) . So, as c (cid:28) i (cid:48) ∈ conv I , there is an i ∈ conv I such that c (cid:28) i , and so c ≤ i . As convex combinations are monotone, it is thenenough to show that every i (cid:48)(cid:48) ∈ I is below an element of conv F . This follows as, since I (cid:28) EM k K ( F ), every such i (cid:48)(cid:48) is way-below an element of the closed set conv F .We next show: ∀ i ∈ I. ∃ c ∈ conv H. c ≤ i ( ∗∗ )Choosing i ∈ I , as k K ( G ) ≤ EM k K ( I ) there is a b ∈ conv G with b ≤ i . As H (cid:28) EM G andconvex combinations are monotone, we then find the required c ∈ conv H .We now build a finite set H of elements of conv H by picking one below each elementof I , as guaranteed by ( ∗∗ ). We then have: ∀ c ∈ H ∪ H . ∃ a ∈ conv F. c ≤ a ( ∗∗∗ )Indeed, for c ∈ H , the conclusion is given by ( ∗ ), and, for c ∈ H , we use that c ≤ i for some i ∈ I and that every element of I is below an element of conv F , as in the argument proving( ∗ ). We next build a finite set F of elements of conv F by picking one above each elementof H ∪ H , as guaranteed by ( ∗∗∗ ).We claim that H ∪ H ≤ EM F ∪ F , as required. This follows as, on the one hand,by ( ∗∗∗ ), every element of H ∪ H is below an element of F , and, on the other hand, as I (cid:28) EM k K ( F ), every element of conv F (and so of F ∪ F ) is above an element of I and so,by the choice of H , above an element of H . Lemma 3.13. Let K be a continuous coherent full Kegelspitze in which convex combinationspreserve the way-below relation, Suppose that F , G are finite non-empty subsets of K suchthat G (cid:28) EM F . Then k K ( G ) (cid:28) P K k K ( F ) .Proof. By the characterisation of (cid:28) P K given in Theorem 3.11 it suffices to find a finite set H such that k K ( G ) ≤ EM k K ( H ), H ⊆ (cid:3)(cid:3) k K ( F ) and (cid:2)(cid:2) H ⊇ k K ( F ).As convex combinations preserve (cid:28) K and G (cid:28) EM F , we have conv G (cid:28) EM conv F .Then, using the compactness of conv F , we see that there is a non-empty finite subset G (cid:48) of conv G such that (cid:2)(cid:2) G (cid:48) ⊇ conv F . Let H = G ∪ G (cid:48) . Since G ⊆ H ⊆ conv G , we have k K ( G ) = k K ( H ). We also have H ⊆ (cid:3)(cid:3) conv F ⊆ (cid:3)(cid:3) k K ( F ), whence H ⊆ (cid:3)(cid:3) k K ( F ). Finally, (cid:2)(cid:2) H ⊇ (cid:2)(cid:2) G (cid:48) ⊇ conv F , and so we have (cid:2)(cid:2) H ⊇ ↑ conv F ⊇ k K ( F ). Theorem 3.14. Let K be a continuous coherent full Kegelspitze and in which convexcombinations preserve the way-below relation. Then the map η P is universal. That is, forevery Kegelspitze semilattice L and Kegelspitze morphism f : K → L there is a uniqueKegelspitze semilattice morphism f † : P K → L such that the following diagram commutes: K f P Kη P (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) ↑ { (cid:91) L f ( F ) | F ⊆ fin K, F (cid:54) = ∅ , F (cid:28) EM X } Proof. For any non-empty finite set F ⊆ K we have k K ( F ) = (cid:91) P η P ( F )as ↓ (cid:91) P η P ( F ) = (cid:87) H K {↓ η P ( b ) | b ∈ F } = (cid:87) H K {↓ b | b ∈ F } = conv F = ↓ k K ( F ), and(proved similarly) ↑ (cid:91) P η P ( F ) = ↑ k K ( F ).Using this and the basis given in Theorem 3.11, for any X ∈ P K we then have: X = (cid:95) ↑ { (cid:91) P η P ( F ) | F ⊆ fin K, F (cid:54) = ∅ , k K ( F ) (cid:28) P K X } It follows that f † ( X ) = (cid:95) ↑ { (cid:91) L f ( F ) | F ⊆ fin K, F (cid:54) = ∅ , k K ( F ) (cid:28) P K X } establishing uniqueness.For existence we define f † by means of this formula and then verify that it makes thediagram commute and is both a Kegelspitze and a semilattice map. It is clearly continuous.Next, as in the proof of Theorem 3.9 , it helpful to prove that, for any non-empty F ⊆ fin K ,we have: f † ( (cid:91) P η P ( F )) = (cid:91) L f ( F ) ( ∗ )that is, that: (cid:95) ↑ { (cid:91) L f ( G ) | G ⊆ fin K, G (cid:54) = ∅ , k K ( G ) (cid:28) P K k K ( F ) } = (cid:91) L f ( F )To show that the left-hand side is ≤ (cid:83) L f ( F ), suppose given G = { b , . . . , b n } ⊆ K , with n > 0, such that k K ( G ) (cid:28) P K k ( F ). Choose c (cid:28) b , . . . , c n (cid:28) b n and set H = { c , . . . , c n } .By Lemma 3.12, there are finite sets H ⊆ conv H and F ⊆ conv F such that H ∪ H ≤ EM F ∪ F . We then have: (cid:91) L f ( H ) = (cid:91) L f ( H ) ∪ L (cid:91) L f ( H ) ≤ (cid:91) L f ( F ) ∪ L (cid:91) L f ( F ) = (cid:91) L f ( F ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 49 where the two equalities follow using the fact that L satisfies the convexity identity (CI)several times. So (cid:83) L f ( G ) ≤ (cid:83) L ( F ) as H consists of an arbitrary choice of elementsway-below each element of G .Conversely, supposing F = { a , . . . , a n } , with n > 0, choose b (cid:28) a , . . . , b n (cid:28) a n andtake G = { b , . . . , b n } . By Lemma 3.13 we have k K ( G ) (cid:28) P K k K ( F ). So the left-hand sideis ≥ (cid:83) L f ( G ), and so ≥ (cid:83) L f ( F ), as G consists of an arbitrary choice of elements way-beloweach element of F .Given ( ∗ ), the rest of the proof follows exactly as did that of Theorem 3.9.Similarly to the case of upper semilattices, this universality result contrasts with thecorresponding universality result for convex powercones in [61]. As before, it would beinteresting to know if the preservation assumption made here is needed.3.4. Powerdomains combining probabilistic choice and nondeterminism. Power-domains combining probabilistic choice and nondeterminism exist on arbitrary dcpos forgeneral reasons. That is, there is always a free Kegelspitze semilattice over any dcpo, andthe same is true for Kegelspitze join- and meet-semilattices. This is because each of thesekinds of structure can be axiomatised by inequations over a signature of finitary operations,possibly (Scott-)continuously parameterised by an auxiliary dcpo, and free algebras overdcpos satisfying such inequations always exist (this can be shown using the General AdjointFunctor Theorem, and see [17]). These various free semilattices over a dcpo are automaticallycontinuous if the dcpo is, as follows from [57] (but not from the less general results on freealgebras in [1], which do not apply when there is parameterisation).Free Kegelspitze semilattices are given by the inequational theory with: a binary opera-tion symbol + r , for each r ∈ [0 , · r , continuously parameterisedby r , ranging over the dcpo [0 , ∪ ; and a constant 0. Theequations consist of: equations for a Kegelspitze, by which we mean the barycentric algebraequations for + r , as given in Section 2 and the equation · r ( x ) = x + r 0; equations assertingthat ∪ is associative, commutative, and idempotent; and the equation x + r ( y ∪ z ) = ( x + r y ) ∪ ( x + r z )saying that + r distributes over ∪ in its second argument, for any r ∈ [0 , 1] (and so alsoin its first one). For Kegelspitze join-semilattices one adds the inequation x ≤ x ∪ y ; formeet-semilattices one instead adds the inequation x ∪ y ≤ x .While we do not know any general characterisation of these various free constructions,by making use of our previous results we can characterise them for domains (assumedalso coherent in the convex case). From the discussion in Section 2.5 we know that thesubprobabilistic power domain V ≤ P over a dcpo is a full Kegelspitze; that, in case P is adomain, it is a continuous Kegelspitze with convex combinations preserving the way-belowrelation; and that, in case P is also coherent, then so is V ≤ P . We further know that, if P is a domain, then the subprobabilistic powerdomain V ≤ P is the free Kegelspitze over P , withunit x (cid:55)→ δ x , where δ x is the Dirac distribution, with mass 1 at x (given a Scott-continuous f : P → K , we write f : V ≤ P → K for its extension to a Kegelspitze map).Therefore we can form the three power Kegelspitzen HV ≤ P , SV ≤ P , and PV ≤ P ,assuming that P is a domain (and a coherent one, in the convex case); it is immediate fromthe above remarks on the subprobabilistic powerdomain and Theorems 3.4, 3.9, and 3.14that these yield, respectively, the free Kegelspitze join-semilattice, the free Kegelspitzemeet-semilattice, and the free Kegelspitze semilattice over a given domain. We record theseresults as corollaries. Corollary 3.15. Let P be a domain. Then the map η HV = def ↓ δ x : P → HV ≤ P isuniversal. That is, for every Kegelspitze join-semilattice L and Scott-continuous map f : P → L there is a unique Kegelspitze semilattice morphism f † : HV ≤ P → L such thatthe following diagram commutes: P f HV ≤ Pη HV (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) f ( X ) Corollary 3.16. Let P be a domain. Then the map η SV = def ↑ δ x : P → SV ≤ P is universal.That is, for every Kegelspitze meet-semilattice L and Scott-continuous map f : P → L thereis a unique Kegelspitze semilattice morphism f † : SV ≤ P → L such that the followingdiagram commutes: P f SV ≤ Pη SV (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) ↑ { (cid:94) f ( F ) | F ⊆ fin V ≤ P, F (cid:54) = ∅ , (cid:2)(cid:2) F ⊇ X } Corollary 3.17. Let P be a coherent domain. Then the map η P V = def { δ x } : P → PV ≤ P is universal. That is, for every Kegelspitze semilattice L and Scott-continuous map f : P → L there is a unique Kegelspitze semilattice morphism f † : PV ≤ P → L such that the following IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 51 diagram commutes: P f PV ≤ Pη P V (cid:63) f † (cid:45) L (cid:45) The morphism is given by: f † ( X ) = (cid:95) ↑ { (cid:91) L f ( F ) | F ⊆ fin V ≤ P, F (cid:54) = ∅ , (cid:2)(cid:2) F ⊇ X } Functional representations In [28, Sections 4 and 6], the various powercones over a d-cone were represented by functionals.We will use those results to obtain similar functional representations of the correspondingpower Kegelspitzen, and then deduce corresponding functional representations for mixedpowerdomains.Some context may help. For a functional representation of a monad T one chooses atest space O , say, and represents an object T ( X ) by a suitable collection of functionals, withdomain a space of ‘test functions’ from X to O and range O . One general such method is towork in a symmetric monoidal closed category, when one has available the ‘continuation’ or‘double-dualisation’ monad [[ X, O ] , O ] (writing [ X, Y ] for the function space). Assuming thatthe monad T is strong, there is then a 1-1 correspondence between T -algebras α : T ( O ) → O and morphisms T → [[ − , O ] , O ] of strong monads [29, 33, 34]. If there are sufficiently manytest functions, this morphism will be a monomorphism, and, perhaps with further restrictionson the functionals, it may corestrict to an isomorphism; one may also have to restrict tocertain objects X .In our case, we would work with the category of Kegelspitzen and continuous linearmaps, when [ K, L ] would be the Kegelspitze formed from such maps with the pointwiseorder and algebraic structure, and the extended reals R + provide a natural test space. As wewill see below, there is a natural choice of functionals for all three of our power-Kegelspitzen,but in no case are such functionals generally linear: for example in the Hoare case they arerather sublinear. When, later, we apply our results to obtain functional representationsof mixed powerdomain monads, we are working in the cartesian-closed category of dcpos,the above general framework does apply and our functional representations are, in fact,submonads of the relevant continuation monads (but, for non-essential reasons, with someminor differences in the convex case).Throughout this section, we generally work with full Kegelspitzen, that is, those satisfyingProperty (OC3). We consider such Kegelspitzen K to be embedded in their universal d-cones C = d - Cone ( K ) as Scott-closed convex sets (Theorem 2.35) and we recall that the universald-cones are continuous whenever the Kegelspitzen are (Proposition 2.42). From Section 2.5 we recall that the subprobabilistic powerdomain V ≤ P of a dcpo P is a full Kegelspitze, that d - Cone ( V ≤ P ) ∼ = V P , the valuation powerdomain of P , and that, in case P is a domain, V ≤ P is continuous.We will make use of norms on d-cones C , taking them to be Scott-continuous sublinearfunctionals || - || : C → R + such that || x || > x (cid:54) = 0; normed d-cones are then d-conesequipped with a norm. A map f : C → D from one normed d-cone C to another D is nonexpansive if || f ( x ) || ≤ || x || for all x ∈ C ; it is a morphism of normed d-cones if it isnonexpansive and a morphism of d-cones (i.e., Scott-continuous and linear).Various function space d-cones will be involved in our development. As well as thoseconsidered in Section 2.5 we note that, for any d-cones C and D , the subsets L sub ( C, D ),and L sup ( C, D ) of D C of, respectively, the sublinear, and superlinear functions form sub-d-cones of D C . Regarding d-cone semilattices, if D is a d-cone semilattice (respectively,join-semilattice, meet-semilattce) then, with the pointwise structure, so is D P , for any dcpo P . Further, for any d-cone C , if D is a join-semilattice (meet-semilattice) then, as is easilychecked, L sub ( C, D ) (respectively, L sup ( C, D )) is a sub-d-cone join-semilattice (respectively,sub-d-cone meet-semilattice) of D C .Supposing additionally the cones C and D to be normed, the collection L ≤ ( C, D ) of allScott-continuous sublinear nonexpansive functions, with D a d-cone join-semilattice, formsa Kegelspitze join-semilattice; indeed it is a sub-Kegelspitze join-semilattice of L sub ( C, D ),regarding the latter as a Kegelspitze join-semilattice.A trivial example of a normed cone is R + with norm the identity function: || x || = x A less trivial example is provided by the dual cone K ∗ of a Kegelspitze K equipped withthe sup norm, defined by: || f || ∗ K = sup x ∈ K f ( x )where the index K indicates the dependency of this norm on the d-cone K ∗ on the Kegelspitze K . Notice that || f || ∗ K = + ∞ if there is an x ∈ K such that f ( x ) = + ∞ .If K satisfies Property (OC3) then we can define a norm on C ∗ , where C = def d - Cone ( K )by: || f || ∗ K = def || f (cid:22) K || ∗ K = sup x ∈ K f ( x )where the index now indicates the dependency of this norm on C ∗ on K . With this norm,the d-cone isomorphism between K ∗ and C ∗ given in Section 2.5, Example 2.47 becomes anisomorphism of normed d-cones.Recall that, for any element x ∈ C = d - Cone ( K ), the evaluation map ev C ( x ) : C ∗ → R + sends f to f ( x ). We note that ev C ( x ) ≤ || - || ∗ K if x ∈ K , with the converse holding if K iscontinuous. For, if x ∈ K then we have ev C ( x ) ≤ || - || ∗ K , since f ( x ) ≤ sup x ∈ K f ( x ) = || f || ∗ K ,for f ∈ C ∗ . And if x (cid:54)∈ K , then using the Strict Separation Theorem [61, Theorem 3.8], weobtain an f ∈ C ∗ such that f ( y ) ≤ y ∈ K but f ( x ) > 1, whence || f || ∗ K = sup y ∈ K f ( y ) ≤ < f ( x ) = ev C ( x )( f ), and so ev C ( x ) (cid:54)≤ || - || ∗ K . Thus, if the d-cone C = d - Cone ( K ) is IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 53 continuous and reflexive, the Scott-continuous linear functionals ϕ ≤ || - || ∗ K on C ∗ are givenby evaluations at points x ∈ K .4.1. The lower power Kegelspitze. We regard R + as a d-cone join-semilattice with thesemilattice operation r ∨ s = max( r, s ) (as such it is isomorphic to H R + , now regarding R + as a d-cone). We take an arbitrary d-cone C with its lower powercone H C and its dual C ∗ .Consider the map Λ C : H C → R C ∗ + where:Λ C ( X )( f ) = def sup x ∈ X f ( x )Fixing f ∈ C ∗ , we obtain the map Λ C ( − )( f ) : H C → R + , which is the unique d-conejoin-semilattice morphism extending f along the canonical embedding η : C → H C by [28,Proposition 3.2].Fixing X ∈ H C , we obtain the functionalΛ C ( X ) : C ∗ → R + where: Λ C ( X )( f ) = sup x ∈ X f ( x )As the pointwise supremum of the Scott-continuous linear functionals ev C ( x ) ( x ∈ X ),Λ C ( X ) is Scott-continuous and sublinear. In this way we obtain a d-cone join-semilatticemorphism Λ C : H C −→ L sub ( C, R + )which represents the lower convex powercone by the Scott-continuous sublinear functionalson the dual cone C ∗ . Theorem 4.1 ([28, Proposition 6.1 and Theorem 6.2]) . Let C be a d-cone. Then we have ad-cone join-semilattice morphism Λ C : H C → L sub ( C, R + ) , where: Λ C ( X ) = sup x ∈ X f ( x ) If, in addition, C is continuous then Λ C is an order embedding; if, further, C is reflexivewith a continuous dual then it is an isomorphism. To apply the above considerations to the universal d-cone C = d - Cone ( K ) over K wenow consider a full Kegelspitze K . The power Kegelspitze H K is a Scott-closed convex join-subsemilattice of the powercone H C . The functionals Λ C ( X ) representing Scott-closed convexsubsets X of K are the sublinear functionals Λ C ( X ) dominated by the norm || - || ∗ K = Λ C ( K ),and all of them if K is continuous. For certainly if X ⊆ K then Λ C ( X ) ≤ Λ C ( K ) = || - || ∗ K ,and, assuming the converse, for any x ∈ X we have ev C ( x ) ≤ Λ C ( X ) ≤ || - || ∗ K , and so x ∈ K by the above discussion (assuming K continuous).Recalling that L ≤ ( K ∗ , R + ) is the collection of nonexpansive functionals in L sub ( K ∗ , R + ),we therefore have a Kegelspitze join-semilattice morphismΛ K : H K → L ≤ ( K ∗ , R + ) viz. the composition H K Λ C (cid:22) H K −−−−−→ L ≤ ( C ∗ , R + ) ∼ = L ≤ ( K ∗ , R + )of the restriction of Λ C to H K with the isomorphism L ≤ ( C ∗ , R + ) ∼ = L ≤ ( K ∗ , R + ) arisingfrom the normed d-cone isomorphism between K ∗ and C ∗ . Theorem 4.1 then yields thedesired functional representation theorem, adapting its hypotheses to Kegelspitzen: Theorem 4.2. Let K be a full Kegelspitze. Then we have a Kegelspitze join-semilatticemorphism Λ K : H K → L ≤ ( K ∗ , R + ) . It is given by: Λ K ( X )( f ) = def sup x ∈ X f ( x ) If K is continuous then Λ K is an order embedding. If, further, the dual cone K ∗ is continuousand the universal d-cone d - Cone ( K ) is reflexive, then Λ K is an isomorphism. With the aid of this theorem we can obtain a corresponding result for the lower mixedpowerdomain. For any dcpo P , making use of Section 2.5, we see that the predicate extensionand restriction mapsEXT P = def f (cid:55)→ f : L P → ( V ≤ P ) ∗ and RES P = def f (cid:55)→ f ◦ δ : ( V ≤ P ) ∗ → L P are d-cone morphisms, and mutually inverse isomorphisms if P is a domain.Next, for any dcpo P , we equip the d-cone L P with the sup norm, i.e., the one definedby: || f || ∞ = sup x ∈ P f ( x ). Lemma 4.3. Let P be a dcpo. Then the extension map EXT P : L P → ( V ≤ P ) ∗ preservesthe norm. The restriction map RES P : ( V ≤ P ) ∗ → L P is nonexpansive and preserves thenorm if P is a domain. So EXT P and RES P are normed d-cone morphisms, and mutuallyinverse isomorphisms if P is a domain.Proof. We wish first to show that || f || ∞ = || f || ∗ ( V ≤ P ) for a given f ∈ L P (where f ( µ ) = (cid:82) f dµ ).In one direction, we have || f || ∞ ≤ || f || ∗ ( V ≤ P ) as, for any x ∈ P , f ( x ) = (cid:82) f dδ x = f ( δ x ). Inthe other direction it suffices to show that f ( µ ) ≤ || f || ∞ for all µ ∈ V ≤ P . This holds as wehave: f ( µ ) = (cid:82) f dµ ≤ (cid:82) ( x (cid:55)→ || f || ∞ ) dµ = µ ( P ) || f || ∞ ≤ || f || ∞ . Next, the restriction map isevidently nonexpansive. If P is continuous it preserves the norm as then it is right inverseto the extension map, and that preserves the norm.We will make use of the mapping:Φ P : R ( V ≤ P ) ∗ + → R L P + where P is a dcpo and Φ P ( F ) = F ◦ EXT P . It is a d-cone morphism, and preserves pointwisejoins and meets. If P is a domain it is an isomorphism, with inverse Φ rP = def F (cid:55)→ F ◦ RES P . Corollary 4.4. Let P be a dcpo. Then we have a Kegelspitze join-semilattice morphism: Λ P : HV ≤ P −→ L ≤ ( L P, R + ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 55 It is given by: Λ P ( X )( f ) = def sup µ ∈ X (cid:90) f dµ If P is a domain then Λ P is an isomorphism.Proof. We first check that both Φ P and Φ rP preserve sublinearity and nonexpansiveness, thelatter by Lemma 4.3. So Φ P cuts down to a morphism L ≤ (( V ≤ P ) ∗ , R + ) → L ≤ ( L P, R + )of Kegelspitze join-semilattices that is an isomorphism if P is a domain.Next, as discussed in Section 2.5, V ≤ P is a full Kegelspitze, and, if P is a domain, then V ≤ P is continuous and d - Cone ( V ≤ P ) ∼ = V P ; further, if P is continuous then ( V P ) ∗ (whichis isomorphic to ( V ≤ P ) ∗ ) is continuous (being isomorphic to L P ), and V P is reflexive. Soif P is a domain then V ≤ P satisfies all the other various hypotheses of Theorem 4.2.An easy calculation then displays Λ P as the following composition of Kegelspitzejoin-semilattice morphisms that are isomorphisms if P is a domain: HV ≤ P Λ K −−→ L ≤ (( V ≤ P ) ∗ , R + ) Φ P −−→ L ≤ ( L P, R + )We remark that nonexpansiveness has a simple formulation for monotone homogeneousfunctionals F : L P → R + , viz. that F ( P ) ≤ P is the constant function on P with value 1. The condition is evidently is aspecial case of nonexpansiveness, as || P || ∞ = 1. Conversely, for any g ∈ L P , noting that g ≤ || g || ∞ P , we have: || F ( g ) || ≤ || F ( || g || ∞ P ) || = || g || ∞ || F ( P ) || ≤ || g || ∞ .4.2. The upper power Kegelspitze. We regard R + as a d-cone meet-semilattice withthe semilattice operation r ∧ s = min( r, s ) (as such it is isomorphic to S R + , now regarding R + as a d-cone). Take a continuous d-cone C with its upper powercone S C and its dual C ∗ .Consider the map Λ C : S C → R C ∗ + where:Λ C ( X )( f ) = def inf x ∈ X f ( x )Fixing f ∈ C ∗ , we obtain the map Λ C ( − )( f ) : S C → R + , which is the unique Scott-continuous linear meet-semilattice homomorphism extending f along the canonical embedding η : C → S C which maps x to ↑ x (this follows from [28, Proposition 3.5], using the aboveisomorphism).Fixing X ∈ S C , we obtain the functionalΛ C ( X ) : C ∗ → R + where: Λ C ( X )( f ) = inf x ∈ X f ( x ) As the pointwise infimum of linear functionals, Λ C ( X ) is superlinear. It is also Scott-continuous. Indeed, for a Scott-compact set X , the image f ( X ) is Scott-compact in R + , hencehas a smallest element min f ( X ) = inf x ∈ X f ( x ) = Λ C ( X )( f ); thus ↑ Λ C ( X )( f ) = S ( f )( X );since f (cid:55)→ S ( f ) is Scott-continuous, f (cid:55)→ S ( f )( X ) : C ∗ → S ( R + ) is Scott-continuous, too;composing with the isomorphism S ( R + ) ∼ = R + yields the Scott continuity of Λ C ( X ).In this way we obtain a d-cone meet-semilattice morphismΛ C : S C −→ L sup ( C ∗ , R + )representing the upper powercone by the Scott-continuous superlinear functionals on thedual cone C ∗ . We need the quite strong hypothesis of a convenient d-cone (see Section 2.5)to obtain the analogue of Theorem 4.1: Theorem 4.5 ([28, Proposition 6.4 and Theorem 6.5]) . Suppose that C is a continuousd-cone. Then we have a d-cone meet-semilattice morphism Λ C : S C → L sup ( C ∗ , R + ) , whichis an order embedding, where: Λ C ( X )( f ) = inf x ∈ X f ( x ) Further, if C is convenient then Λ C is an isomorphism. We now consider a continuous full Kegelspitze K . The universal d-cone C = d - Cone ( K )is also then continuous and we apply the above considerations to it. The upper powerKegelspitze S K consists of all nonempty Scott-compact saturated convex subsets X of K .As discussed in Section 3.2 the map u : S K → S C , where u ( X ) = ↑ X is a d-cone meet-semilattice morphism which is an order-embedding. The functions Λ C ( u ( X )) : C ∗ → R + ( X ∈ S K ) are Scott-continuous and superlinear. We want to characterise the Scott-continuous and superlinear functionals F on C ∗ that represent the elements of S K in thisway. It turns out that, unlike the case of the lower power Kegelspitze, being nonexpansive isnot sufficient. We notice that, for any X ∈ S K , the representing functional F : C ∗ → R + has a remarkable property: it is strongly nonexpansive , by which we mean that F ( f + g ) ≤ F ( f ) + || g || ∗ K holds for all f, g ∈ C ∗ (setting f = 0, we see that strong nonexpansiveness implies non-expansiveness). Indeed, we have: F ( f + g ) = inf x ∈ X ( f + g )( x ) = inf x ∈ X ( f ( x ) + g ( x )) ≤ inf x ∈ X ( f ( x ) + sup x ∈ K g ( x )) = inf x ∈ X ( f ( x ) + || g || ∗ K ) = F ( f ) + || g || ∗ K .For any normed d-cone D we write L snesup ( D, R + ) for the collection of all Scott-continuoussuperlinear functionals F : D → R + that are strongly nonexpansive in the sense that: F ( x + y ) ≤ F ( x ) + || y || holds for all x, y ∈ D . The collection forms a Kegelspitze meet-semilattice, indeed it isa sub-Kegelspitze meet-semilattice of L sup ( D, R + ). One easily checks that L snesup ( C ∗ , R + )and L snesup ( K ∗ , R + ) are isomorphic as Kegelspitze meet-semilattices via the normed d-coneisomorphism between K ∗ and C ∗ . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 57 Putting all this together, we defineΛ K : S K −→ L snesup ( K ∗ , R + )to be the Kegelspitze meet-semilattice morphism given by the composition: S K Λ C ◦ u −−−−→ L snesup ( C ∗ , R + ) ∼ = L snesup ( K ∗ , R + )Theorem 4.5 then yields the desired functional representation theorem, adapting its hypothe-ses to Kegelspitzen: Theorem 4.6. Let K be a continuous full Kegelspitze. Then we have a Kegelspitze meet-semilattice morphism Λ K : S K → L snesup ( K ∗ , R + ) . It is given by: Λ K ( X )( f ) = def inf x ∈ X f ( x ) If, further, d - Cone ( K ) is convenient, then Λ K is an isomorphism.Proof. We note first that, for any X ∈ S K and f ∈ K ∗ , we have:Λ K ( X )( f ) = Λ C ( ↑ X )( (cid:101) f ) = inf x ∈↑ X (cid:101) f ( x ) = inf x ∈ X (cid:101) f ( x ) = inf x ∈ X f ( x )Next, as d - Cone ( K ) is continuous since K is, Theorem 4.5 tells us that Λ C is an order-embedding; so, as u is also one, so too is Λ K .For the isomorphism, assuming that d - Cone ( K ) is convenient, we need then only showthat Λ C ◦ u : S K → L snesup ( C ∗ , R + ) is onto. So take any strongly nonexpansive Scott-continuoussuperlinear F : C ∗ → R + . By Theorem 4.5 we know that there is a Y ∈ S C such thatΛ C ( Y ) = F , that is such that F ( f ) = inf y ∈ Y f ( y ) for all f ∈ C ∗ . Let X = Y ∩ K . Clearly, X is a Scott-compact convex set saturated in K . We want to show that X is non-emptyand Λ C ( u ( X )) = Λ C ( Y ), that is, inf y ∈ Y f ( y ) = inf x ∈ X f ( x ) for all f ∈ C ∗ .Since the d-cone K ∗ ∼ = C ∗ is assumed to be continuous, strong nonexpansiveness allowsus to apply the Main Lemma [28, Lemma 5.1(1)] to C ∗ . We learn that Λ C ( Y )( f ) = inf ϕ ( f ),where ϕ ranges over the Scott-continuous linear functionals on C ∗ such that Λ C ( Y ) ≤ ϕ ≤|| - || ∗ K . Using the hypotheses of continuity and reflexivity for C , and the discussion at thebeginning of this section, this can be rewritten in the form Λ C ( Y )( f ) = inf x ∈ Q f ( x ), where Q is the set of those elements x ∈ K that satisfy Λ C ( Y )( f ) ≤ f ( x ) for all f ∈ C ∗ . Notethat Q is non-empty, as, taking f to be constantly 0, we have inf x ∈ Q f ( x ) = Λ C ( Y )( f ) = 0(recalling that Y is non-empty).As Q is non-empty and Λ C ( Y )( f ) = inf x ∈ Q f ( x ), it only remains, therefore, to showthat X = Q . Clearly, X ⊆ Q . For the reverse containment, suppose that x ∈ Q ⊆ K .We cannot have x (cid:54)∈ Y as otherwise, by the Strict Separation Theorem [61, Theorem3.8], there is an r > f ( x ) ≤ f ( y ) > r for every y ∈ Y , whenceΛ C ( Y )( f ) = inf y ∈ Y f ( y ) ≥ r (cid:54)≤ f ( x ). So x ∈ K ∩ Y = X as required. We can now specialise to domains: Corollary 4.7. Let P be a domain. Then we have a Kegelspitze meet-semilattice isomor-phism Λ P : SV ≤ P ∼ = L snesup ( L P, R + ) It is given by: Λ P ( X )( f ) = def inf µ ∈ X (cid:90) f dµ Proof. Using Lemma 4.3, we can check that both Φ P and its inverse preserve strongnonexpansiveness, and so that Φ P cuts down to an isomorphism L snesup (( V ≤ P ) ∗ , R + ) ∼ = L snesup ( L P, R + )of Kegelspitze meet-semilattices. Next, from the discussion of the valuation powerdomainin Section 2.5, we know that d - Cone ( V ≤ P ) ∼ = V P is convenient, and so we can applyTheorem 4.6. One then displays Λ P as the following composition of Kegelspitze meet-semilattice isomorphisms: SV ≤ P Λ ( V≤ P ) −−−−−→ L snesup (( V ≤ P ) ∗ , R + ) Φ P −−→ L snesup ( L P, R + )Strong nonexpansiveness has a simple formulation for Scott-continuous homogeneousfunctionals F : L P → R + , viz. that F ( f + P ) ≤ F ( f ) + 1holds for all f ∈ L P . Clearly a strongly nonexpansive functional satisfies this condition,since || || ∞ = 1. Suppose conversely that the second condition is satisfied and take any f, g ∈ L P . For g = 0 there is nothing to prove. So let g (cid:54) = 0, and suppose that g is bounded,i.e., that || g || ∞ < ∞ . Then, using homogeneity and then monotonicity and then the simplifiedcondition, we have: F ( f + g ) = || g || F ( 1 || g || f + 1 || g || g ) ≤ || g || F ( 1 || g || f + P ) ≤ || g || ( F ( 1 || g || f ) + 1) = F ( f ) + || g || As every non-zero g ∈ L P is the directed sup of bounded non-zero such g ’s, using thecontinuity of F we then see that F is strongly nonexpansive.4.3. The convex power Kegelspitze. Here our representations employ functionals withvalues not in R + , but rather in P R + , the convex powercone of the extended nonnegativereals; this consists of the closed intervals a = [ a, a ], with a ≤ a in R + , ordered by theEgli-Milner order, where: [ a, a ] ≤ EM [ b, b ] if a ≤ b and a ≤ b , with addition given by[ a, a ] + [ b, b ] = [ a + b, a + b ], and scalar multiplication given by r [ a, a ] = [ ra, ra ]. Thesemilattice operation on P R + is [ a, a ] ∪ [ b, b ] = [min( a, b ) , max( a, b )]; the semilattice order iscontainment ⊆ . We define a norm on P R + by setting || a || = def a . Goubault-Larrecq calls such functionals subnormalised previsions in [9, 10, 11, 12]. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 59 We also use notation adapted from [28, Section 4] setting up, for any dcpo P , a bijectionbetween functions F : P → P R + and pairs of functions G, H : P → R + with G ≤ H . In onedirection, given such an F , we write F ( x ) and F ( x ) for the lower and upper ends of theimage F ( x ) of any x ∈ P , obtaining such a pair of functions F and F ; conversely, givensuch a pair of functions G and H , we set [ G, H ]( x ) equal to the interval [ G ( x ) , H ( x )]. Thefunction F is Scott-continuous if and only if both F and F are. In case we are consideringfunctionals F : D → P R + where D is a d-cone, then F is said to be ⊆ - sublinear , if F ishomogeneous and F ( x + y ) ⊆ F ( x ) + F ( y ) for all x, y ∈ D (which is equivalent to F beingsuperlinear and F sublinear), and it is said to be medial if we have F ( x + y ) ≤ F ( x ) + F ( y ) ≤ F ( x + y )for all x, y ∈ D .Let us recall the (diagonal) functional representation of the convex powercone P C of acoherent continuous d-cone C from [28, Sections 4 and 6]. It combines the representationsof the lower and upper powercones. Take a continuous coherent d-cone C with its its dual C ∗ and its convex powercone P C . Consider the map Λ C : P C → P R C ∗ + where:Λ C ( X )( f ) = def [ inf x ∈ X f ( x ) , sup x ∈ X f ( x )]Fixing f ∈ C ∗ , we obtain the map Λ C ( − )( f ) : P C → P R + , which is is the unique d-conesemilattice morphism P f : P C → P R + extending η R + ◦ f along the canonical embedding η C ,where, for any continuous coherent cone D , the canonical embedding η D : D → S D maps x to { x } (this follows from [28, Proposition 3.8]).Fixing X ∈ P C we obtain the Scott-continuous ⊆ -sublinear medial functionalΛ C ( X ) : C ∗ → P R + where Λ C ( X )( f ) = inf x ∈ X f ( x ) , Λ C ( X )( f ) = sup x ∈ X f ( x )The collection L ⊆ , med ( D, P R + ) of all Scott-continuous ⊆ -sublinear medial functionals F : D ∗ → P R + forms a d-cone semilattice, for any d-cone D ; indeed, it is a sub-d-conesemilattice of P R + D .In this way we obtain a d-cone semilattice morphismΛ C : P C −→ L ⊆ , med ( C ∗ , P R + )that represents the convex powercone by the Scott-continuous ⊆ -sublinear medial functionals F : C ∗ → P R + . This property is called ‘canonicity’ in [28, Section 4.3], and ‘Walley’s condition’ in [11, Definition 7.1].Indeed, Walley includes this property among those of his ‘coherent previsions’ in his book on reasoning withimprecise probabilities [66, Section 2.6]. Theorem 4.8 ([28, Proposition 6.4, Theorem 6.8]) . Suppose that C is a continuous coherentd-cone. Then we have a d-cone semilattice morphism Λ C : P C −→ L ⊆ , med ( C ∗ , P R + ) , whichis an order embedding, where: Λ C ( X )( f ) = [ inf x ∈ X f ( x ) , sup x ∈ X f ( x )] Further, if C is convenient then Λ C is an isomorphism. We now consider a continuous coherent full Kegelspitze K . The continuous universald-cone C = d - Cone ( K ) is then also continuous and coherent (this last by Proposition 2.44).As in the proof of Theorem 3.11, the power Kegelspitze P K can be considered to be thecollection of all convex lenses in C that are contained in K , which latter is itself a convexlens of C . In this way P K can be seen as a Scott-closed convex ∪ -subsemilattice of P C .We then note that a convex lens X is in P K iff Λ C ( X ) is dominated by the normfunctional || - || ∗ K (for X is in P K iff ↓ X ⊆ K iff Λ C ( ↓ X ) ≤ || - || ∗ K , by the discussion inSection 4.1, and we have Λ C ( X ) = Λ C ( ↓ X )). Now, for any normed d-cone D , the collection L ≤ ⊆ , med ( D, P R + ) of all Scott-continuous and ⊆ -sublinear functionals F : D → P R + with F nonexpansive and medial forms a Kegelspitze semilattice, indeed it is a sub-Kegelspitzesemilattice of L ⊆ , med ( D, P R + ) (regarding the latter as a Kegelspitze semilattice).We therefore have a Kegelspitze semilattice morphismΛ K : P K −→ L ≤ ⊆ , med ( K ∗ , P R + )viz. the composition: P K Λ C (cid:22) K −−−−→ L ≤ ⊆ , med ( C ∗ , P R + ) ∼ = L ≤ ⊆ , med ( K ∗ , P R + )From Theorem 4.8 we then immediately obtain the following functional representationtheorem: Theorem 4.9. Let K be a continuous coherent full Kegelspitze. Then we have a Kegelspitzesemilattice morphism Λ K : P K −→ L ≤ ⊆ , med ( K ∗ , P R + ) which is an order embedding. It isgiven by: Λ K ( X )( f ) = def [ inf x ∈ X f ( x ) , sup x ∈ X f ( x ) ] Further, if d - Cone ( K ) is convenient then Λ K is an isomorphism. We specialise this result to domains. For any domain P (Φ c ) P : P R ( V ≤ P ) ∗ + → P R L P + where (Φ c ) P ( F ) = F (cid:55)→ F ◦ EXT P , is a d-cone semilattice isomorphism with inverse F (cid:55)→ F ◦ RES P . We then obtain: Corollary 4.10. Let P be a coherent domain. Then we have a Kegelspitze semilatticeisomorphism Λ P : PV ≤ P ∼ = L ≤ ⊆ , med ( L P, P R + ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 61 It is given by: Λ P ( X )( f ) = def [ inf µ ∈ X (cid:90) f dµ , sup µ ∈ X (cid:90) f dµ ] Proof. Checking that (Φ c ) P and its inverse preserve ⊆ -sublinearity and mediality, we seethat Φ cP cuts down to an isomorphism L ≤ ⊆ , med (( V ≤ P ) ∗ , P R + ) ∼ = L ≤ ⊆ , med ( L P, P R + )of Kegelspitze semilattices. As P is a coherent domain, then, following Section 2.5, wesee that V ≤ P is continuous and coherent, and that d - Cone ( V ≤ P ) is convenient. So wemay apply Theorem 4.9. One then displays Λ P as the following composition of Kegelspitzesemilattice isomorphisms: PV ≤ P Λ ( V≤ P ) −−−−−→ L ≤ ⊆ , med (( V ≤ P ) ∗ , P R + ) (Φ c ) P −−−−→ L ≤ ⊆ , med ( L P, P R + )5. Predicate transformers We are ready now to achieve a goal that we can summarise under the slogan ‘the equivalenceof state transformer and predicate transformer semantics’ . Let us begin by describing thegeneral framework for the lower and upper cases. In Section 3, in both these cases, wemodelled mixed probabilistic and nondeterministic phenomena by a monad S over a fullsubcategory C of the category of dcpos and Scott-continuous maps. The Kleisli category of S has as morphisms the Scott-continuous maps s : P → S ( Q )We name these state transformers .In Section 4, we considered functional representations of S , which led to a monad T withisomorphisms S ( P ) ∼ = T ( P ). This monad is a submonad of the continuation monad R R + P + ,and so its Kleisli category is faithfully embedded in the Kleisli category of the continuationmonad whose morphisms are the Scott-continuous maps t : P → R R + Q + These morphisms provide our general notion of state transformer. The collection ( R R Q ++ ) P of such state transformers can be regarded as either a d-cone join-semilattice or a d-conemeet-semilattice with respect to the pointwise structure obtained from R + , depending onwhether R + is viewed as a d-cone join-semilattice or a d-cone meet-semilattice.In this setting, it makes sense to think of R + as a space of truthvalues and thento call Scott-continuous maps f : P → R + on a dcpo P predicates , so that the functionspace R P + = L P becomes the dcpo of predicates on P . A predicate transformer is then aScott-continuous map p : L Q → L P and, as before, the collection ( L P ) L Q of such predicate transformers can be regarded aseither a d-cone join-semilattice or a d-cone meet-semilattice depending on how R + is viewed.There is an evident natural bijection PT : ( R R Q ++ ) P ∼ = ( L P ) L Q , where:PT( t )( g )( x ) = def t ( x )( g ) ( g ∈ L Q, x ∈ P )This bijection is both a d-cone join-semilattice isomorphism and a d-cone meet-semilatticeisomorphism, depending on which of the above semilattice structures are taken on thestate and predicate transformers. It is then our aim to characterise the ‘healthy’ predicatetransformers, that is, those p that correspond to the state transformers t : P → T ( Q ) ⊆ R R + Q + arising from the two monads for mixed nondeterminism.For the convex case there is a natural modification of this general framework wherethe role of R + is taken by over by P R + , the convex powercone over R + . As signalled inSection 4, the uniformity at hand is that, up to isomorphism, we are making use of thethree powercones H R + , S R + , and P R + . All three are based on R + , which is V , the freevaluation powerdomain on the one-point dcpo.In all cases considered here the ‘healthy’ predicate transformers do not preserve thenatural algebraic operations on the function spaces, that is, they are not homomorphisms.In particular, in the lower and upper cases they are respectively sublinear and superlinear.This phenomenon is explained from a general point of view in [24, 25].As indicated above, we restrict ourselves to predicate transformers for the powerKegelspitzen over domains. There are, nevertheless, related results for Kegelspitzen moregenerally. For example, in the lower and upper cases, one takes predicates on a Kegelspitze K to be elements of K ∗ , the sub-Kegelspitze of R K + of all Scott-continuous linear functionals.Predicate transformers are suitable Scott-continuous maps p : L ∗ → K ∗ and state transformers are linear Scott-continuous maps t : K → R L ∗ + In all three cases one obtains Kegelspitze isomorphisms between Kegelspitzen of statetransformers and Kegelspitzen of suitably healthy predicate transformers. This differsfrom the domain case, where one rather obtains Kegelspitze semilattice isomorphisms; thedifference arises as there seems to be no general reason why, for example, a dual Kegelspitze K ∗ should be a semilattice, whereas, if K = V P then K ∗ is L P which is a semilattice.5.1. The lower case. Consider two dcpos P and Q . For any state transformer t : P → R L Q + ,the corresponding predicate transformer PT( t ) : L Q → L P is given by PT( t )( g )( x ) = t ( x )( g )for g ∈ L Q and x ∈ P . One can check directly that || PT( t )( g ) || ∞ ≤ || g || ∞ for every g ∈ L Q if and only if t ( x ) ≤ || - || ∞ for every x ∈ P , and that PT( t ) is sublinear if and only if t ( x ) issublinear for every x ∈ P . Thus the state transformers t : P → L ≤ ( L Q, R + ) correspond IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 63 bijectively via PT to the nonexpansive sublinear predicate transformers p : L Q → L P . SoPT cuts down to a Kegelspitze join-semilattice isomorphism L ≤ ( L Q, R + ) P ∼ = L ≤ ( L Q, L P )taking the pointwise Kegelspitze join-semilattice structure on L ≤ ( L Q, R + ) P . Finally, tomake the link between state transformers and the healthy predicate transformers, we setPT P,Q ( s ) = def PT(Λ Q ◦ s ), ( s : P → HV ≤ Q ), where Λ Q is as in Section 4.1, and calculatethat PT P,Q ( s )( g )( x ) = PT(Λ Q ◦ s )( g )( x ) = Λ Q ( s ( x ))( g ) = sup µ ∈ s ( x ) (cid:90) g dµ Combining the above discussion with Corollary 4.4 we then obtain: Corollary 5.1. Let P and Q be dcpos. To every state transformer s : P → HV ≤ Q we canassign a predicate transformer PT P,Q ( s ) : L Q → L P by: PT P,Q ( s )( g )( x ) = def sup µ ∈ s ( x ) (cid:90) g dµ ( g ∈ L Q, x ∈ P ) The predicate transformer PT P,Q ( s ) is sublinear and nonexpansive. The assignment PT P,Q is a Kegelspitze join-semilattice morphism ( HV ≤ Q ) P −→ L ≤ ( L Q, L P ) If Q is a domain then it is an isomorphism. Similar to the simplification of nonexpansiveness for functionals discussed in Section 4.1(and an immediate consequence of it), the condition of nonexpansiveness has a simpleformulation for homogeneous predicate transformers p : L Q → L P , viz. p ( Q ) ≤ P .5.2. The upper case. Consider two dcpos, P and Q . In agreement with the terminologyintroduced in Section 4.2 we will say that a predicate transformer p : L Q → L P is stronglynonexpansive if we have p ( f + g ) ≤ p ( f ) + || g || ∞ · P for all f, g ∈ L Q , where P is the constant function on P with value 1. Strongly nonexpansivepredicate transformers are nonexpansive. Indeed, for f = 0 the inequality yields p ( g )( x ) ≤|| g || ∞ for all x ∈ P , whence || p ( g ) || ∞ ≤ || g || ∞ . In the case of homogeneous predicatetransformers this can be simplified to the equivalent condition : p ( f + Q ) ≤ p ( f ) + P (for all f ∈ L Q )as follows immediately from the corresponding simplification for functionals in Section 4.2.For any state transformer t : P → R L Q + , we have PT( t )( g )( x ) = t ( x )( g ), ( g ∈ L Q, x ∈ P ).This firstly implies that t ( x ) is superlinear for every x if, and only if, PT( t ) is superlinear.It secondly implies that t ( x ) is strongly nonexpansive for every x if, and only if, PT( t ) isstrongly nonexpansive. For we have t ( x )( f + g ) ≤ t ( x )( f ) + || g || ∞ for every x ∈ P if, and only if, PT( t )( f + g )( x ) ≤ PT( t )( f )( x ) + || g || ∞ for every x ∈ P , that is, if, and only if,PT( t )( f + g ) ≤ PT( t )( f ) + || g || ∞ · P .We write L snesup ( L Q, L P ) for the set of strongly nonexpansive superlinear predicatetransformers. and note that it forms a sub-Kegelspitze meet-semilattice of ( L P ) L Q (takingthe pointwise Kegelspitze meet-semilattice structure on ( L P ) L Q ). So PT cuts down to aKegelspitze meet-semilattice isomorphism L snesup ( L Q, R + ) P ∼ = L snesup ( L Q, L P )Finally, to make the link between state transformers and the healthy predicate trans-formers, we set PT P,Q ( s ) = def PT(Λ Q ◦ s ) ( s : P → SV ≤ Q ), where Λ Q is as in Section 4.2(and assuming now that Q is a domain), and calculate that P T P,Q ( s )( g )( x ) = inf µ ∈ s ( x ) (cid:90) g dµ Combining the above discussion with Corollary 4.7 we then obtain:. Corollary 5.2. Let P be a dcpo and let Q be a domain. To every state transformer s : P → SV ≤ Q we can assign a predicate transformer PT P,Q ( s ) : L Q → L P by: PT P,Q ( s )( g )( x ) = def inf µ ∈ s ( x ) (cid:90) g dµ ( g ∈ L Q, x ∈ P ) The predicate transformer PT P,Q ( s ) is superlinear and strongly nonexpansive. The assign-ment PT P,Q is a Kegelspitze meet-semilattice isomorphism ( SV ≤ Q ) P ∼ = L snesup ( L Q, L P )5.3. The convex case. For this case we have to modify our framework. First we need afunction space construction. For d-cone semilattices C and D , the collection L mon ( C, D ) ofScott-continuous ⊆ -monotone maps from C to D is a sub-d-cone semilattice of the d-conesemilattice of all Scott-continuous maps from C to D equipped with the pointwise d-conesemilattice structure.In the modified framework, the rˆole of R + , considered as join- and meet-semilattices inthe lower and upper cases, is taken over by P R + , the convex powercone over R + . Predicateson a dcpo P are no longer functionals with values in R + but are now rather Scott-continuousfunctionals of the form f : P → P R + ; they form a d-cone semilattice with the pointwisestructure. We define a norm on predicates f : P → P R + by: || f || = def || f || ∞ (= (cid:87) x ∈ P f ( x )).Employing the notation of Section 4.3 we have a bijection f → ( f , f ) between predicatesand pairs of linear functionals g, h ∈ L P with g ≤ h . Note that ( f , f ) ≤ ( f (cid:48) , f (cid:48) ) if, and onlyif, f ≤ f (cid:48) and f ≤ f (cid:48) , that ( f , f ) ∪ ( f (cid:48) , f (cid:48) ) = ( f ∧ f (cid:48) , f ∨ f (cid:48) ), and that ( f , f ) ⊆ ( f (cid:48) , f (cid:48) ) if, andonly if, f ≥ f (cid:48) and f ≤ f (cid:48) .We take general state transformers to be maps: t : P → P R + R + Q IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 65 One might rather have expected, t : P → P R P R + Q + , uniformly replacing R + with P R + ; wechose our definition to be closer to the functional representation.Predicate transformers are taken to be Scott-continuous maps p : ( P R + ) Q → ( P R + ) P which are, in addition, required to preserve the partial order ⊆ , i.e., to be ⊆ -monotone.This requirement is a technical condition to achieve an isomorphism between generalstate transformers and predicate transformers (see below). The predicate transformersform a d-cone semilattice L mon (( P R + ) Q , ( P R + ) P ). Note that a predicate transformer p isnonexpansive if, and only if, || p ( f ) || ∞ ≤ || f || ∞ , for any predicate f .We link these predicate transformers to general state transformers via ‘predicate trans-formers of diagonal form’ which we take to be Scott-continuous functions: q : L Q −→ ( P R + ) P State transformers t : P → ( P R + ) L Q are connected to predicate transformers of diagonalform by the map T : ( P R L Q + ) P −→ (( P R + ) P ) L Q where T( t )( g )( x ) = def t ( x )( g ). This map is evidently an isomorphism of d-cone semilattices,with respect to the pointwise structures.To connect predicate transformers of diagonal form to predicate transformers we firstextend some definitions from predicates to functions F : D → ( P R + ) P , with D a d-cone. Let F be such a function. We define F , F : D → L P by setting F ( x ) = F ( x ) and F ( x ) = F ( x ),for x ∈ D . Then define a map P between the two kinds of predicate transformers:P : (( P R + ) P ) L Q −→ L mon (( P R + ) P , ( P R + ) Q )by P ( q )( f ) = def [ q ( f ) , q ( f )]). Lemma 5.3. P is an isomorphism of d-cone semilattices.Proof. It is routine to verify that P is a morphism of d-cone semilattices. To see that P isan order embedding suppose that P( q ) ≤ P( q (cid:48) ) and choose g ∈ L ∗ to show that q ( g ) ≤ q (cid:48) ( g ).Then we have [ q ( f ) , q ( f )]) ≤ [ q (cid:48) ( f ) , q (cid:48) ( f )]) where f = [ g, g ]. So q ( g ) ≤ q (cid:48) ( g ) and q ( g ) ≤ q (cid:48) ( g ),and so q ( g ) ≤ q (cid:48) ( g ), as required.To see that P is onto, choose a predicate transformer p to find a q with p = P( q ). Weclaim that p ([ f , f ]) = p ([ f , f ]) and p ([ f , f ]) = p ([ f , f ]). For the first of these claims, as[ f , f ] ≤ [ f , f ] we have p ([ f , f ]) ≤ p ([ f , f ]), since p preserves the order ≤ , and as [ f , f ] ⊆ [ f , f ]we have p ([ f , f ]) ≥ p ([ f , f ]), since p preserves the order ⊆ . The proof of the second of theseclaims is similar: as [ f , f ] ≤ [ f , f ] we have p ([ f , f ]) ≤ p ([ f , f ]), since p preserves the order ≤ , and as [ f , f ] ⊇ [ f , f ] we have p ([ f , f ]) ≥ p ([ f , f ]), since p preserves the order ⊆ .Defining q = g (cid:55)→ p ([ g, g ]), we then see that p = P( q ), as required. So we have a Kegelspitze semilattice isomorphism between general state transformersand predicate transformers:P ◦ T : ( P R L Q + ) P ∼ = L mon (( P R + ) P , ( P R + ) Q )and we seek the relevant healthiness conditions on the predicate transformers.Define a function F : D → ( P R + ) P , with D a d-cone and P a dcpo, to be medial if: F ( x + y ) ≤ F ( x ) + F ( y ) ≤ F ( x + y )for all x, y ∈ D , and define a function F : D → C , where D is a d-cone and C is a d-conesemilattice, to be ⊆ -sublinear if it is homogeneous and F ( x + y ) ⊆ F ( x ) + F ( y ), for all x, y ∈ D (this generalises the definition of ⊆ -sublinearity in Section 4.3, and in the casewhere C is ( P R + ) P , it is equivalent to F being sublinear and F being superlinear).Now fix a state transformer s and set q = T( s ) and p = P( q ). We have t ( x ) sublinearfor every x ∈ P iff q is sublinear iff p is sublinear and, similarly, t ( x ) is superlinear for every x ∈ P iff p is superlinear. So t ( x ) is ⊆ -sublinear for every x ∈ P iff q is ⊆ -sublinear iff p is ⊆ -sublinear. Next, t ( x ) is medial for all x ∈ P iff q is medial iff p is medial. Finally, t ( x ) isnonexpansive for all x ∈ P iff t ( x ) ≤ || - || ∞ for all x ∈ P iff || q ( g ) || ∞ ≤ || g || ∞ for all g ∈ L Q ,iff || p ( f ) || ∞ ≤ || f || ∞ for all predicates f , that is, iff p is nonexpansive.We write L ≤ , ⊆ , med (( P R + ) Q , ( P R + ) P ) for the set of ⊆ -monotone, ⊆ -sublinear, medial,nonexpansive predicate transformers. As is straightforwardly checked, it forms a sub-Kegelspitze semilattice of L mon (( P R + ) Q , ( P R + ) P ), and, from the above considerations, wesee that P ◦ T cuts down to a Kegelspitze semilattice isomorphism: L ≤ ⊆ , med ( L Q, P R + ) P ∼ = L ≤ , ⊆ , med (( P R + ) Q , ( P R + ) P )Finally, to make the link between state transformers and predicate transformers we setPT P,Q ( s ) = def P(T(Λ Q ◦ s )), ( s : P → PV ≤ Q ), where Λ Q is as in Section 4.3 (and assumingnow that Q is a coherent domain), and calculate thatPT P,Q ( s )( g )( x ) = [ inf µ ∈ s ( x ) (cid:90) g dµ, sup µ ∈ s ( x ) (cid:90) g dµ ]Combining the above discussion with Corollary 4.10, we then obtain: Corollary 5.4. Let P be a dcpo and let Q be a coherent domain. To every state transformer s : P → PV ≤ Q we can assign a predicate transformer PT P,Q ( s ) : P R Q + → P R P + by: PT P,Q ( s )( g )( x ) = def [ inf µ ∈ s ( x ) (cid:90) g dµ, sup µ ∈ s ( x ) (cid:90) g dµ ] ( g ∈ P R Q + , x ∈ P ) The predicate transformer PT P,Q ( s ) is nonexpansive, ⊆ -monotone, ⊆ -sublinear, and medial.The assignment PT P,Q is a Kegelspitze semilattice isomorphism ( PV ≤ Q ) P ∼ = L ≤ , ⊆ , med ( P R Q + , P R P + ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 67 The unit interval In this section we consider replacing the extended positive reals R + by the unit interval I .In the lower and upper cases, functional representations will involve maps to I ; I will playthe rˆole of truth values for predicates; and predicate transformers will be functions from L ≤ Q to L ≤ P . In the convex case, functional representations will involve maps to P I ; P I will play the rˆole of truth values; and predicate transformers will be functions from P I Q to P I P . As we shall see, the results obtained are the same as those with R + , except thatnonexpansiveness requirements are dropped.First, we slightly weaken the notion of a norm introduced in Section 4 deleting therequirement that nonzero elements have nonzero norm. A seminorm on a Kegelspitze K is defined to be a Scott-continuous sublinear map from K to R + and a seminorm on acone C is a Scott-continuous sublinear map from C to R + . A seminormed Kegelspitze is aKegelspitze equipped with a seminorm ||−|| , and similarly for seminormed cones. A function f : K → L between seminormed Kegelspitzen is nonexpansive if, for all a ∈ K we have: || f ( a ) || ≤ || a || and similarly for seminormed cones. A seminormed d-cone semilattice is a d-cone semilatticeequipped with a seminorm such that the operation ∪ is nonexpansive, by which we meanthat || a ∪ b || ≤ max( || a || , || b || ).We next need some function space constructions. For any Kegelspitze K and Kegelspitze L (Kegelspitze semilattice L ) we write L hom ( K, L ) for the Scott-continuous homogeneousfunctions from K to L . Equipped with the pointwise structure, L hom ( K, L ) forms a sub-Kegelspitze (respectively, sub-Kegelspitze semilattice) of L K ; further, for any d-cone C (d-cone semilattice C ), L hom ( K, C ) (regarding C as a Kegelspitze) forms a sub-d-cone(respectively sub-d-cone semilattice) of C K when equipped with the pointwise structure.For seminormed d-cones C and D , we write L ≤ ( C, D ) for the collection of all Scott-continuous, homogeneous, nonexpansive functions from C to D . Equipped with the pointwisestructure it forms a sub-Kegelspitze of L hom ( C, D ); further, if D is a seminormed d-conesemilattice, L ≤ ( C, D ) forms a sub-Kegelspitze semilattice of L hom ( C, D ).We have a basic function space isomorphism as an immediate consequence of theuniversal embedding in a d-cone of a full Kegelspitze given by Theorem 2.35. Let e : K → C be a universal Kegelspitze embedding (in the sense of Section 2) of a Kegelspitze K in ad-cone C . Then Theorem 2.35 tells us that, for any d-cone D , function extension f (cid:55)→ f yields a dcpo isomorphism L hom ( K, D ) ∼ = L hom ( C, D )with inverse given by restriction g (cid:55)→ g ◦ e along the universal arrow. Moreover, as restrictionpreserves the pointwise structure, the isomorphism is an isomorphism of d-cones; further,if D is additionally equipped with a semilattice structure, then the isomorphism is anisomorphism of d-cone semilattices. To connect nonexpansiveness and Kegelspitzen we make use of particular seminorms. Forany Scott-closed convex subset X of a cone C , we define the (lower) Minkoswki functional ν X : C → R + by: ν X ( a ) = def inf { r ∈ R + | a ∈ r · X } Minkowski functionals were previously considered in [47] and in [23]. Proposition 6.1. Let X be a Scott-closed convex subset of a d-cone C . Then: (1) ν X is Scott-continuous and sublinear. (2) If < ν X ( a ) < ∞ then, for some x ∈ X , we have a = ν X ( a ) · x . (3) X = { a ∈ C | ν X ( a ) ≤ } . (4) If C is a d-cone semilattice and X also a subsemilattice, then we have: ν X ( a ∪ b ) ≤ max( ν X ( a ) , ν X ( b )) Proof. (1) (a) For monotonicity, suppose a ≤ b ∈ C . Then if b ∈ r · X , we have a ∈ r · X , since r · X is a lower set and so ν X ( a ) ≤ ν X ( b ).(b) Having established monotonicity, for continuity it remains to show that ν X ( (cid:87) i a i ) ≤ (cid:87) i ν X ( a i ), for any directed set a i ( i ∈ I ) of elements of C . Suppose that (cid:87) i ν X ( a i ) 1[ to show that ν X ( r · a ) = r · ν X ( a ). Thisfollows from the observation that, for any positive s ∈ R + , a ∈ s · X iff r · a ∈ rs · X .(d) For subadditivity, choose a , b in C and r > ν X ( a ), s > ν X ( b ). Then a ∈ r · X and b ∈ s · X whence a + b ∈ r · X + s · X = ( r + s ) · X , since X is convex. So we have r + s > || a + b || and, since this holds for all r > ν X ( a ) and s > ν X ( b ), we concludethat ν X ( a + b ) ≤ ν X ( a ) + ν X ( b ).(2) As 0 < ν X ( a ) < ∞ there are sequences r n ∈ R + and x n ∈ X , with r n decreasing andpositive, such that a = r n · x n and ν X ( a ) = inf r n . So x n = r − n · a is an increasingsequence, and taking sups we see that sup x n = (sup r − n ) · a = ν X ( a ) − · a . We have x = def sup x n ∈ X , as X is Scott-closed, and so ν X ( a ) · x = a .(3) Evidently X ⊆ { a ∈ C | ν X ( a ) ≤ } . Conversely, suppose that we have a ∈ C with ν X ( a ) ≤ 1. If ν X ( a ) < a ∈ · X = X . Otherwise we have ν X ( a ) = 1. Inthis case, by the second part we have a = ν X ( a ) · x for some x ∈ X . Then, as ν X ( a ) = 1,we see that a ∈ X , as required.(4) As for subadditivity, choose any real number r > max( ν X ( a ) , ν X ( b )). Then a and b are both in r · X . Since X is supposed to be a subsemilattice and x (cid:55)→ r · x is asemilattice homomorphism, r · X is subsemilattice, too, so that a ∪ b ∈ r · X , thatis ν X ( a ∪ b ) ≤ r . Since this holds for all r > max( ν X ( a ) , ν X ( b )), we conclude that ν X ( a ∪ b ) ≤ max( ν X ( a ) , ν X ( b )). IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 69 So all Minkowski functionals are seminorms. For every full Kegelspitze K and everyuniversal Kegelspitze embedding K e −→ C , we write || - || K for the seminorm ν e ( K ) : C → R + ,and on K we use the same notation for the seminorm || a || K = def || e ( a ) || K . When we do notmention below which seminorm we use we mean the relevant one of these. We have thefollowing pleasant facts: Fact 6.2. For full Kegelspitzen K and L , let K e −→ C and L e (cid:48) −→ D be universal Kegelspitzeembeddings. Suppose that f : K → L and and g : C → D are homogeneous maps such thatthe following diagram commutes: C g (cid:45) DKe (cid:54) f (cid:45) Le (cid:48) (cid:54) Then both f and g are nonexpansive. Proof. It suffices to show that g is nonexpansive. So, for a ∈ C we have to show that || g ( a ) || L ≤ || a || K . This is certainly true if || a || K = + ∞ . Otherwise take any real number r > || a || K . Then a ∈ r · e ( K ). We deduce that g ( a ) ∈ g ( r · e ( K )) = r · g ( e ( K )) = r · e (cid:48) ( f ( K )) ⊆ r · e (cid:48) ( L ) which implies that || g ( a ) || L ≤ r . Since this holds for all r > || a || K , we have thedesired inequality.The next proposition is at the root of our results for the unit interval. It enablesnonexpansiveness requirements to be dropped when using the unit interval. First define K e −→ C to be a universal Kegelspitze semilattice embedding if K is a full Kegelspitzesemilattice, D is a d-cone semilattice, e preserves the semilattice operation, and K e −→ C is auniversal Kegelspitze embedding. Note that then the norm || - || K on C satisfies property (4)of Proposition 6.1. Proposition 6.3. For full Kegelspitzen K and L , let K e −→ C and L e (cid:48) −→ D be universalKegelspitze embeddings. Then there is a Kegelspitze isomorphism: L hom ( K, L ) ∼ = L ≤ ( C, D ) The isomorphism sends f ∈ L hom ( K, L ) to e (cid:48) ◦ f ; its inverse sends g ∈ L ≤ ( C, D ) to therestriction of g ◦ e along e (cid:48) ; and f and g are related by the isomorphism if, and only if, g ◦ e = e (cid:48) ◦ f .In case L e (cid:48) −→ D is additionally a universal Kegelspitze semilattice embedding, theisomorphism is a Kegelspitze semilattice isomorphism.Proof. Composing with the Kegelspitze embedding e (cid:48) and then function extension, viewedas a Kegelspitze isomorphism, we obtain a Kegelspitze embedding: L hom ( K, L ) e (cid:48) ◦− −−−→ L hom ( K, D ) · −→ L hom ( C, D ) We see from Fact 6.2 that every function in the range of the embedding is nonexpansive.Conversely let g : C → D be a nonexpansive Scott-continuous homogeneous function. Then,in particular, for every a ∈ K we have || g ( e ( a )) || ≤ || e ( a ) || ≤ b ∈ L such that g ( e ( a )) = e (cid:48) ( b ). So we have a function f : K → L such that e (cid:48) ( f ( a )) = g ( e ( a )) for all a ∈ K ; this function is Scott-continuous and homogeneous as g is and e and e (cid:48) are Kegelspitze embeddings. As g extends f ◦ e (cid:48) along e , the Kegelspitzeembedding f (cid:55)→ f ◦ e (cid:48) of L hom ( K, L ) in L hom ( C, D ) sends f to g , and so cuts down to abijection, and so a Kegelspitze isomorphism, between L hom ( K, L ) and L ≤ ( C, D ), withinverse as claimed.That f and g are related by the isomorphism if, and only if, g ◦ e = e (cid:48) ◦ f is clear.With the extra semilattice assumptions, D is a d-cone semilattice and so L ≤ ( C, D ) isa Kegelspitze semilattice; further, the isomorphism preserves the semilattice structure as e (cid:48) does.We will typically apply this result by first restricting to a subclass (e.g., to sublinearfunctions in the lower case) and then specialising to dcpos or domains.We could also obtain general results for Kegelspitzen K by adding to the assumptionsconsidered above the assumption that the evident embedding of K ∗ in d - Cone ( K ) ∗ is universal,where now by K ∗ we mean the Kegelspitze of Scott-continuous linear functions from K to I . By Proposition 2.38, an equivalent assumption, assuming d - Cone ( K ) ∗ continuous, isthat every Scott continuous linear function from K to R + is a directed sup of bounded suchfunctions.One obtains general functional representation and predicate transformer results, exceptfor predicate transformer results in the convex case. (The obstacle in that case is thatthe equational proof below that mediality transfers in the domain case is not available ata general level, since there seems to be no general reason why dual Kegelspitzen or duald-cones should be semilattices.)As remarked in the introduction, one might prefer a development not involving d-conesat all. Another improvement, perhaps easier to achieve, would be a development where theassumptions on Kegelspitzen involved only I .There is a pleasant induction principle for d-cones given by Kegelspitze universal embed-dings. Say that a property of a d-cone is upper homogeneous if it is closed under all actions r ·− with r ≥ 1. Then, for any full Kegelspitze K and any universal Kegelspitze embedding K e −→ C , if a property of C is closed under directed sups, and is upper homogeneous, thenit holds for all of C if it holds for all of e ( K ). This can be proved by reference to theconstruction of universal embeddings in Section 2.There is an n -ary version of this induction principle. Given n > K i e i −→ C i of full Kegelspitzen K i ( i = 1 , . . . , n ), if a relation on C , . . . , C n is closed underdirected sups and is upper homogeneous (in an evident sense), then the relation holds for allof C × . . . × C n if it holds for all of e ( K ) × . . . × e n ( K n ). This follows from the unary IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 71 principle since, as shown in Section 2.3, universal Kegelspitze embeddings are closed underfinite non-empty products.6.1. The lower case. For any full Kegelspitzen K and L we take L sub ( K, L ) to be thecollection of Scott-continuous sublinear functions from K to L equipped with the pointwisestructure and so forming a sub-Kegelspitze of L K , and a sub-Kegelspitze join-semilattice if L is a Kegelspitze join-semilattice.Let e : K → C and e (cid:48) : L → D be universal Kegelspitze embeddings, where, additionally, L is a Kegelspitze join-semilattice and D is a d-cone join-semilattice (when e is automaticallya Kegelspitze semilattice morphism and so e (cid:48) : L → D is a universal Kegelspitze semilatticeembedding). Then the Kegelspitze semilattice isomorphism of Proposition 6.3 restricts to aKegelspitze semilattice isomorphism L sub ( K, L ) ∼ = L ≤ ( C, D )as an f ∈ L hom ( K, L ) is sublinear iff e (cid:48) ◦ f is iff (using Theorem 2.35) e (cid:48) ◦ f is.We saw in Section 2.5 that the inclusion L ≤ P ⊆ L P is a universal embedding for anydcpo P , and it is easy to check that the norm || - || ∞ is the Minkowski seminorm, therebyensuring consistency with the previous two sections. Further, L ≤ P is a Kegelspitze join-semilattice and L P is a d-cone join-semilattice. The analogous remarks apply to the inclusion I ⊆ R + .So, for any dcpo P we obtain the Kegelspitze semilattice isomorphism: L sub ( L ≤ P, I ) ∼ = L ≤ ( L P, R + )and for any dcpos P and Q we obtain the Kegelspitze semilattice isomorphism: L sub ( L ≤ Q, L ≤ P ) ∼ = L ≤ ( L Q, L P )As immediate consequences of Corollaries 4.4 and 5.1 we then obtain: Corollary 6.4. Let P be a dcpo. Then we have a Kegelspitze join-semilattice morphism: Λ P : HV ≤ P −→ L sub ( L ≤ P, I ) It is given by: Λ P ( X )( f ) = def sup ν ∈ X (cid:90) f dν If P is a domain then Λ P is an isomorphism. Corollary 6.5. Let P and Q be dcpos. To every state transformer s : P → HV ≤ Q we canassign a predicate transformer PT P,Q ( s ) : L ≤ Q → L ≤ P by: PT P,Q ( s )( g )( x ) = def sup ν ∈ s ( x ) (cid:90) g dν ( g ∈ L ≤ Q, x ∈ P ) The predicate transformer PT P,Q ( s ) is sublinear. The assignment PT P,Q is a Kegelspitzejoin-semilattice morphism ( HV ≤ Q ) P −→ L sub ( L ≤ Q, L ≤ P ) If Q is a domain then it is an isomorphism. The upper case. Suppose we are given full Kegelspitzen K and L and universalKegelspitze embeddings e : K → C and e (cid:48) : L → D , where L has a top element (which wewrite as 1). Then we say that a function f : K → L is strongly nonexpansive if: f ( a + r b ) ≤ f ( a ) + r || b || K · a, b ∈ K and r ∈ [0 , g : C → D is strongly nonexpansive if: g ( x + y ) ≤ g ( x ) + || y || K · e (cid:48) (1)holds for all x, y ∈ C ; note that this last definition is consistent with the correspondingdefinitions in previous sections. Also, a homogeneous function g : C → D is stronglynonexpansive iff it is when considered as a function between Kegelspitzen.We write L snesup ( C, D ) for the collection of all Scott-continuous superlinear stronglynonexpansive functions g : C → D ; this collection forms a Kegelspitze with the pointwisestructure, and a Kegelspitze meet-semilattice if D is a d-cone meet-semilattice. We furtherwrite L snesup ( K, L ) for the collection of all Scott-continuous superlinear strongly nonexpansivefunctionals f : K → L ; this collection forms a Kegelspitze with the pointwise structure, anda Kegelspitze meet-semilattice if L is one.Now suppose, additionally, that L is a Kegelspitze meet-semilattice and D is a d-cone meet-semilattice (when e (cid:48) is automatically a Kegelspitze semilattice morphism and so e (cid:48) : L → D is a universal Kegelspitze semilattice embedding). Then the Kegelspitze semilatticeisomorphism of Proposition 6.3 restricts to a Kegelspitze semilattice isomorphism: L snesup ( K, L ) ∼ = L snesup ( C, D )which sends f to g = def f ◦ e (cid:48) . Regarding superadditivity, f is superadditive iff f ◦ e (cid:48) is iff (byTheorem 2.35) f ◦ e (cid:48) is. Also g is strongly nonexpansive iff g ( x + r y ) ≤ g ( x ) + r || y || K · e (cid:48) (1)for all x, y ∈ C and r ∈ [0 , g ( e ( a ) + r e ( b )) ≤ g ( e ( a )) + r || e ( b ) || K · e (cid:48) (1) for a, b ∈ K , r ∈ [0 , e (cid:48) ( f ( a + r b )) ≤ e (cid:48) ( f ( a ) + r || b || K · a, b ∈ K , r ∈ [0 , 1] (as g ◦ e = e (cid:48) ◦ f ), iff f is strongly nonexpansive. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 73 The Kegelspitzen I , R + , L ≤ P and L P ( P a dcpo) are all Kegelspitze meet-semilatticesand so the inclusions I ⊆ R + and L ≤ P ⊆ L P are universal Kegelspitze semilattice embed-dings. So, in particular, for any dcpo P we obtain a Kegelspitze semilattice isomorphism: L snesup ( L ≤ P, I ) ∼ = L snesup ( L P, R + )and for any dcpos P and Q we obtain a Kegelspitze semilattice isomorphism: L snesup ( L ≤ Q, L ≤ P ) ∼ = L snesup ( L Q, L P )As immediate consequences of Corolleries 4.7 and 5.2 we then obtain: Corollary 6.6. Let P be a domain. Then we have a Kegelspitze meet-semilattice isomor-phism Λ P : SV ≤ P ∼ = L snesup ( L ≤ P, I ) It is given by: Λ P ( X )( f ) = def inf ν ∈ X (cid:90) f dν Corollary 6.7. Let P be a dcpo and let Q be a domain. To every state transformer s : P → SV ≤ Q we can assign a predicate transformer PT P,Q ( s ) : L ≤ Q → L ≤ P by: PT P,Q ( s )( g )( x ) = def inf ν ∈ s ( x ) (cid:90) g dν ( g ∈ L ≤ Q, x ∈ P ) The predicate transformer PT P,Q ( s ) is superlinear and strongly nonexpansive. The assign-ment PT P,Q is a Kegelspitze meet-semilattice isomorphism ( SV ≤ Q ) P ∼ = L snesup ( L ≤ Q, L ≤ P )Finally we show that, as in previous cases, the strong nonexpansiveness condition canbe simplified for homogeneous functions. Fact 6.8. Let K e −→ C and L e (cid:48) −→ D be universal embeddings where both K and L have topelements and where || - || K is a norm on C . Then:(1) A homogeneous function f : K → L is strongly nonexpansive iff f ( x + r ≤ f ( x ) + r x ∈ K and r ∈ [0 , g : C → D is strongly nonexpansive iff g ( x + 1) ≤ g ( x ) + 1, forall x ∈ C Proof. We assume, without loss of generality, that the embeddings are inclusions.(1) We have to show that f ( a + r b ) ≤ f ( a ) + r || b || K · a, b ∈ K and r ∈ [0 , || b || K = 0 then, as || - || K is a norm, b = 0. Otherwise, setting s = def || b || K , we seeby Proposition 6.1 that b = s · c for some c ∈ K and so that b ≤ s · 1. Then, taking t = def − (1 − r ) s , we note that r ≤ t and calculate: f ( a + r b ) ≤ f ( a + r s · 1) = f ( r/t · a + t ≤ f ( r/t · a ) + t r/t · f ( a ) + t f ( a ) + r || b || K · (2) We have to show that g ( x + y ) ≤ g ( x ) + || y || K · x, y ∈ C . This is trivial if || y || K = ∞ . If it is 0 then, as || - || K is a norm, y = 0. Otherwise, setting s = def || y || K andnoting that s − · y ∈ K , we calculate: g ( x + y ) = r · g ( s − · x + s − · y ) ≤ r · g ( s − · x + 1) ≤ r · ( g ( s − · x ) + 1) = g ( x ) + || y || K · The convex case. We begin with some definitions. Given a full Kegelspitze K anda full Kegelspitze semilattice L , say that a function f : K → L is ⊆ -sublinear if it ishomogeneous and, for all a, b ∈ K and r ∈ [0 , 1] we have: f ( a + r b ) ⊆ f ( a ) + r f ( b )Note that a function g : C → D from a d-cone to a d-cone semilattice is ⊆ -sublinear,as defined in Section 5.3, iff it is when considered as a function from a Kegelspitze to aKegelspitze semilattice.Next, given Kegelspitzen K , L , and M and two functions d L , u L : L → M , we say that afunction f : K → L is medial (w.r.t. d L , u L ) if, for all a, b ∈ K and r ∈ [0 , 1] we have: f d ( a + r b ) ≤ f d ( a ) + r f u ( b ) ≤ f u ( a + r b )where f d = def d L ◦ f and f u = def u L ◦ f . Similarly, given d-cones C , D , and E and twofunctions d D , u D : D → E , we say that a function g : D → E is medial (w.r.t. d D , u D ) if, forall x, y ∈ C , we have: g d ( x + y ) ≤ g d ( x ) + g u ( y ) ≤ g u ( x + y )where g d = def d D ◦ g and g u = def u D ◦ g ; this is equivalent to it being medial when consideredas a function between Kegelspitzen, provided that it is homogeneous, as are d D , and u D .Now we suppose given: • full Kegelspitzen K and M and a full Kegelspitze semilattice L , • d-cones C and E and a d-cone semilattice D , • universal Kegelspitze embeddings K e −→ C and M e −→ E and a universal Kegelspitzesemilattice embedding L e −→ D , and • Scott continuous homogeneous functions d L , u L : L → M and d D , u D : D → E such thatboth the following two diagrams commute: D d D (cid:45) ELe (cid:54) d L (cid:45) Me (cid:54) D u D (cid:45) ELe (cid:54) u L (cid:45) Me (cid:54) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 75 Then, if f : K → L and g : C → D are Scott-continuous homogeneous functions suchthat e ◦ f = g ◦ e then f is ⊆ -sublinear iff g is, and f is medial iff g is. The proofs arestraightforward using the binary induction principle for universal Kegelspitze embeddings.We now suppose further that • M is both a Kegelspitze meet- and join-semilattice, and d L , u L are both Kegelspitze semi-lattice morphisms, with M taken, accordingly, as a Kegelspitze meet- or join-semilattice,and • E is both a d-cone meet-semilattice and join-semilattice, and d D , u D are both d-conesemilattice morphisms, with M taken, accordingly, as a d-cone meet- or join-semilattice.We write L ≤ ⊆ , med ( C, D ) for the set of Scott-continuous, ⊆ -sublinear, medial, nonexpansivefunctions from D to E ; equipped with the pointwise structure it forms a sub-Kegelspitzesemilattice of L ≤ ( C, D ). We further write L ⊆ , med ( K, L ) for the set of Scott-continuous, ⊆ -sublinear, medial functions from K to L ; equipped with the pointwise structure it formsa sub-Kegelspitze semilattice of L hom ( K, L ).As we have seen that ⊆ -sublinearity and mediality transfer along the Kegelspitze semi-lattice isomorphism of Proposition 6.3, we now see that that, under our several suppositions,this isomorphism restricts to a Kegelspitze semilattice isomorphism: L ⊆ , med ( K, L ) ∼ = L ≤ ⊆ , med ( C, D )Let us now consider the particular case where K e −→ C is the inclusion L ≤ P ⊆ L P ,for some dcpo P , L e −→ D is the inclusion P I ⊆ P R + , M e −→ E is the inclusion I ⊆ R + ,d P R + ( x ) = def x , u P R + ( x ) = def x , and d P I and u P I are defined similarly.Then we already know that e is a universal Kegelspitze embedding and it is evidentthat e is too; that e is follows from Proposition 2.38, and so it is evidently a universalKegelspitze semilattice embedding. It is then clear that all the above assumptions hold, andso we have a Kegelspitze semilattice isomorphism: L ⊆ , med ( L ≤ P, P I ) ∼ = L ≤ ⊆ , med ( L P, P R + )where, on the left nonexpansiveness is defined relative to the Minkowski seminorms on L P and R + . However these seminorms are the same as those considered before: we alreadyknow this for L P , and it is easy to show that for R + (i.e., to show that || x || P I = x ). As animmediate corollary of Corollary 4.10 we then obtain: Corollary 6.9. Let P be a coherent domain. Then we have a Kegelspitze semilatticeisomorphism Λ P : PV ≤ P ∼ = L ⊆ , med ( L ≤ P, P I ) It is given by: Λ P ( X )( f ) = def [ inf ν ∈ X (cid:90) f dν , sup ν ∈ X (cid:90) f dν ]To obtain a corresponding result for predicate transformers we change the aboveframework slightly: instead of supposing that K is a full Kegelspitze, C is a d-cone, and K e −→ C is a universal Kegelspitze embedding, we suppose that K is a full Kegelspitzesemilattice, C is a d-cone semilattice, and K e −→ C is a universal Kegelspitze semilatticeembedding.Then, for Scott-continuous homogeneous functions f : K → L , g : C → D where e ◦ f = g ◦ e , we additionally have that f is ⊆ -monotone iff g is. This is proved by theuniversal embedding induction principle, as usual, but noting that ⊆ -monotonicity canbe expressed equationally: for example g is ⊆ -monotone if, and only if, for all x, y ∈ C , g ( x ) ∪ g ( x ∪ y ) = g ( x ∪ y ).We now write L ≤ , ⊆ , med ( C, D ) for the set of Scott-continuous, ⊆ -monotone, ⊆ -sublinear,medial, nonexpansive functions from D to E ; equipped with the pointwise structure it formsa sub-Kegelspitze semilattice of L ≤ ( C, D ). We further write L mon , ⊆ , med ( K, L ) for the setof Scott-continuous, ⊆ -monotone, ⊆ -sublinear, medial functions from K to L ; equipped withthe pointwise structure it forms a sub-Kegelspitze semilattice of L hom ( K, L ).As we have seen that ⊆ -monotonicity, ⊆ -sublinearity, and mediality transfer along theKegelspitze semilattice isomorphism of Proposition 6.3, we now see that that, under ourseveral suppositions, this isomorphism restricts to a Kegelspitze semilattice isomorphism: L ≤ , ⊆ , med ( C, D ) ∼ = L mon , ⊆ , med ( K, L )To apply this result, we note that, for any dcpo P , the inclusion P I P ⊆ P R P + is auniversal Kegelspitze semilattice embedding (for a proof again use Proposition 2.38, nowfollowing the same lines as the proof that the inclusion L ≤ P ⊆ L P is universal) andthat the Minkowski seminorm on P R P + is the same as the norm defined before, i.e., that || f || P R P + = sup x ∈ P f ( x ).Now consider the particular case where K e −→ C is the inclusion P I Q ⊆ P R Q + , for somedcpo Q , L e −→ D is the inclusion P I P ⊆ P R P + , for some dcpo P , M e −→ E is the inclusion I P ⊆ R P + , d P R P + ( f ) = def f , u P R P + ( f ) = def f , and d P I P and u P I P are defined similarly. Thenall the above assumptions hold, and so we have a Kegelspitze semilattice isomorphism: L ≤ , ⊆ , med ( P R Q + , P R P + ) ∼ = L mon , ⊆ , med ( P I Q , P I P )and then as an immediate corollary of Corollary 5.4 we obtain: Corollary 6.10. Let P be a dcpo and let Q be a coherent domain. To every state transformer s : P → PV ≤ Q we can assign a predicate transformer PT P,Q ( s ) : P I Q → P I P by: PT P,Q ( s )( g )( x ) = def [ inf ν ∈ s ( x ) (cid:90) g dν, sup ν ∈ s ( x ) (cid:90) g dν ] ( g ∈ P I Q , x ∈ P ) The predicate transformer PT P,Q ( s ) is ⊆ -monotone, ⊆ -sublinear, and medial. The assign-ment PT P,Q is a Kegelspitze semilattice isomorphism ( PV ≤ Q ) P ∼ = L mon , ⊆ , med ( P I Q , P I P ) IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 77 Acknowledgements We are very grateful to Jean Goubault-Larrecq for his many helpful comments and suggestions. References [1] S. Abramsky and A. Jung. Domain Theory. Handbook of Logic in Computer Science, Volume 3. 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For any r ∈ [0 , 1] we write r (cid:48) for 1 − r .We write t = u , . . . , t = u (cid:96) T t = u for a given equational theory T to mean that t = u follows by equational reasoning from the theory T and the equations t = u , . . . , t n = u n .We need a proof-theoretic version of another lemma of Neumann: [45, Lemma 3] (withcorrected bounds). We first show a lemma that can also be derived from general results dueto Sokolova and Woracek [55, Theorem 4.4 and Example 4.6]: Lemma A.1. Let ∼ be a congruence on the barycentric algebra [0 , . Then if two elementsof the open interval ]0 , are congruent, so are any other two.Proof. Let ∼ be such a congruence and suppose that we have r ∼ s with 0 < r < s < 1. Set α = r/s < 1. Then αr ∼ αs = r ∼ s hence αr ∼ s . Repeating the argument yields α n r ∼ s ,for any n ≥ 0, and so we get arbitrarily close to 0 with elements congruent to s .In the other direction, we can define a ‘symmetric’ congruence relation ∼ (cid:48) , setting p ∼ (cid:48) q to hold if, and only if, p (cid:48) ∼ q (cid:48) , as the map p (cid:55)→ p (cid:48) is an (involutive) automorphism of [0 , s (cid:48) ∼ (cid:48) r (cid:48) and 0 < s (cid:48) < r (cid:48) < 1. So, by the above argument, we can get arbitrarilyclose to 0 with elements in the symmetric congruence relation with r (cid:48) , and so arbitrarilyclose to 1 with elements congruent to r .As congruence classes are convex, we then see that any two elements of the open interval]0 , 1[ are congruent. Lemma A.2. Let T be an equational theory extending B . Then for any terms t and u , andany < r < s < and < p < q < we have: t + r u = t + s u (cid:96) T t + p u = t + q u Proof. One fixes r and s with 0 < r < s < ∼ where p ∼ q iff t + r u = t + s u (cid:96) T t + p u = t + q u .The equational theory BSD (cid:48) has axioms those of B and S together with equations x ∪ ( y + r z ) = ( x ∪ y ) + r ( x ∪ z ) ( r ∈ ]0 , (cid:48) )stating that ∪ distributes over each of the + r . Recall that a join-distributive bi-semilattice [52]is an algebra with two semilattice operations ∩ and ∪ , with ∪ distributing over ∩ . Theorem A.3. The equational theory BSD (cid:48) is equivalent to that of join-distributive bi-semilattices. IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 81 Proof. In the following we just write t = u rather than (cid:96) BSD (cid:48) t = u . Substituting ( y + r z )for x in D (cid:48) (and using S and re-using D (cid:48) ) we get:( y + r z ) = (( y + r z ) ∪ y ) + r (( y + r z ) ∪ z ) = ( y + r ( y ∪ z )) + r (( y ∪ z ) + r z ) (A.1)for all r ∈ ]0 , y ∪ z for z (and using S and then B) we get: y + r ( y ∪ z ) = ( y + r ( y ∪ z )) + r (( y ∪ z ) + r ( y ∪ z )) = y + r ( y ∪ z )for all r ∈ ]0 , y + r ( y ∪ z ) = y + s ( y ∪ z ) (A.2)for all r, s ∈ ]0 , r ∈ ]0 , y + r z ) = ( y + r ( y ∪ z )) + r (( y ∪ z ) + r z ) (by Equation (A.1))= ( y + r (cid:48) ( y ∪ z )) + r (( y ∪ z ) + r (cid:48) z ) (by Equation (A.2), and using B)= ( y + r ( y ∪ z )) + r (cid:48) (( y ∪ z ) + r z ) (by the entropic identity)= ( y + r (cid:48) ( y ∪ z )) + r (cid:48) (( y ∪ z ) + r (cid:48) z ) (by Equation (A.2), and using B)= ( y + r (cid:48) z ) (by Equation (A.1), substituting r (cid:48) for r )and so we can apply Lemma A.2 again, and obtain: y + r z = y + s z for all r, s ∈ ]0 , ∪ and ∩ . To translate from the theory of join-distributive bi-semilattices into our theory one translates ∪ as x ∪ x and ∩ as (e.g.) x + / x . In the other direction, one translates ∪ as x ∪ x , + r as x ∩ x , for all r ∈ ]0 , and + as x and x , respectively.We next consider a weaker theory: we drop the idempotence of ∪ . Let C be thetheory of commutative semigroups, that is of algebras with an associative and commutativemultiplication operation (having dropped idempotence, the change to a multiplicativenotation is natural). The equational theory CCSA of convex commutative semigroup algebras has as axioms those of B and C together with the following distributive laws, stating thatmultiplication distributes over the + r : x ( y + r z ) = xy + r xz ( r ∈ ]0 , , 1] provides an example convex commutative semigroup algebra with theusual barycentric operations and multiplication.Let D ω be the finite probability distributions monad. We regard D ω X as consisting ofconvex combinations (cid:80) i =1 ,m p i x i of elements of X ; with the evident barycentric operationsit is the free barycentric algebra over X . Then the free convex commutative semigroup algebra over a given commutative semigroup M is provided by the barycentric algebra D ω M ,with the following multiplication:( (cid:88) i = 1 ,...,m p i x i )( (cid:88) i = 1 ,...,n q j y j ) = (cid:88) i = 1 ,...,mj = 1 ,...,n ( p i q j ) x i y j and with unit the inclusion (this construction is a variant of the standard group algebraconstruction). In particular, writing M + ω for the finite non-empty multisets monad, we seethat D ω M + ω X is the free convex commutative semigroup algebra over X , as M + ω is the freecommutative semigroup monad.We regard D ω M + ω X as consisting of all polynomials with no constants, with variablesin X , and with coefficients in [0 , 1] adding up to 1. In other words, it consists of allconvex combinations of non-trivial polynomials with variables in X . The barycentricoperations are the evident convex combinations of such polynomials, and multiplication isthe usual polynomial multiplication. The unit η : x → D ω M + ω X is the inclusion, and theextension f : D ω M + ω X → A to a CCSA-homomorphism of a map f from X to a convexcommutative semigroup algebra A assigns to any polynomial p ( x , . . . , x n ) in D ω M + ω X itsvalue p ( f ( x ) , . . . , f ( x n )) ∈ A as obtained using the CCSA operations of A .A convex commutative semigroup algebra A is complete if, for all CCSA-terms t and u we have: A | = t = u ⇒ (cid:96) CCSA t = u This holds if, and only if, distinct polynomials in D ω M + ω X can be separated by elements of A , that is, if for any such p ( x , . . . , x n ) (cid:54) = q ( x , . . . , x n ), with variables in x , . . . , x n , thereare a , . . . a n ∈ A such that p ( a , . . . , a n ) (cid:54) = q ( a , . . . , a n ). For example, [0 , 1] is complete inthis sense.We now focus on the convex semigroup algebra D ω P + ω X . We need two lemmas. Lemma A.4. Let X be a set with at least two elements. Then D ω P + ω X is complete.Proof. Let p ( x , . . . , x n ), q ( x , . . . , x n ) be distinct polynomials in D ω M + ω X , and choose r , . . . , r n in [0 , 1] separating them. Choose two distinct elements y , z in X . We can definea semigroup homomorphism h : P + ω X → [0 , 1] by: h ( u ) = (cid:40) u = { y } )0 (otherwise)Then h has an extension to a CCSA-homomorphism h : D ω P + ω X → [0 , a i ∈ D ω P + ω X to be { y } + r i { z } , for i = 1 , n , and noting that h ( a i ) = h ( { y } ) + r i h ( { z } ) = r i ,we see that: h ( p ( a , . . . , a n )) = p ( h ( a ) , . . . , h ( a n )) = p ( r , . . . , r n ) (cid:54) = q ( r , . . . , r n ) = h ( q ( a , . . . , a n ))and so that a , . . . , a n are elements of D ω P + ω X separating p and q . IXED POWERDOMAINS FOR PROBABILITY AND NONDETERMINISM 83 Lemma A.5. Let X be a nonempty set and let T be a subtheory of CCSA . Then D ω P + ω X is not the free T -algebra over X .Proof. First note that, for any p ∈ D ω M + ω X , we have p (cid:54) = pp . Then, if D ω P + ω X were thefree T -algebra over X , there would be a T -algebra homomorphism h : D ω P + ω X → D ω M + ω X ,as D ω M + ω X is a CCSA-algebra and so a T -algebra. Choosing any y ∈ P + ω X , we would thenfind h ( y ) = h ( yy ) = h ( y ) h ( y ), a contradiction.So, in particular, for non-empty X , D ω P + ω X is, as may be expected, not the freeCCSA-algebra. We can now prove: Theorem A.6. Let X be a set with at least two elements. Then D ω P + ω X is not the free T -algebra over X for any equational theory T with the same signature as that of CCSA .Proof. Suppose, for the sake of contradiction that D ω P + ω X is the free T -algebra over X for an equational theory T with the same signature as CCSA. Then, in particular, it isa T -algebra, and so all T -equations hold in it. But, by Lemma A.4 all equations holdingin it are in CCSA. So D ω P + ω X is the free algebra for a subtheory of CCSA. However, byLemma A.5, that cannot be the case.So we have shown that there is no algebraic (i.e., equational) account of the naturalrandom set algebras D ω P + ω X . It may even be that D ω ◦P + ω admits no monadic structure. Appendix B. A counterexample Referring to Lemma 2.41 and the preceding discussion, we give an example of a continuousKegelspitze E whose scalar multiplication does not preserve the way-below relation (cid:28) E .That is, there are elements a (cid:28) E b in E such that ra (cid:54)(cid:28) E rb for some r < , 1] with its upper (= Scott) topology; theopen subsets are the half-open intervals ] r, ≤ r ≤ 1. Let L be the continuous d-coneof continuous functions f : ]0 , → R + (in classical analysis one would have said that the f ∈ L are the monotone increasing lower semicontinuous functions). Such a function f has agreatest value, namely f (1). We write L ≤ and L ≤ for the Scott-closed, hence continuous,Kegelspitzen of functions f ∈ L such that f (1) ≤ f (1) ≤ 2, respectively. We write k for the constant function 1 on ]0 , E be the collection of those f ∈ L ≤ which can be represented as a convexcombination f = q · + (1 − q ) · h (0 ≤ q ≤ 1) of the constant function 2k with some h ∈ L ≤ . This provides our counterexample.We first claim that E forms a sub-Kegelspitze of L ≤ . It is clearly convex and containsthe least element of L ≤ . To show E is a sub-dcpo of L ≤ with the inherited ordering, we set E (cid:48) = def { f ∈ L ≤ | f (1)k ≤ f + 2k } and, noting that E (cid:48) forms a sub-dcpo, show that E and E (cid:48) coincide. It is clear that E ⊆ E (cid:48) . Conversely, suppose f ∈ E (cid:48) so that 2 f (1)k ≤ f +2k . If f (1) = 2,then f = 2k and so f ∈ E . If f (1) ≤ 1, then trivially f ∈ L ≤ ⊆ E . Thus we can supposethat 1 < f (1) < 2. We can rewrite the condition for membership in E (cid:48) as 2( f (1) − ≤ f and we let (cid:98) f = f − f (1) − . Since (cid:98) f (1) = f (1) − f (1) − 1) = 2 − f (1) ( (cid:54) = 0) , wehave (cid:98) f (1)2 − f (1) = 1, so that (cid:98) f − f (1) ∈ L ≤ . Letting q = f (1) − − q = 2 − f (1) and0 < q < 1, since 1 < f (1) < 2, and f becomes a convex combination of the constant function2k and the function (cid:98) f − f (1) ∈ L ≤ , namely f = 2( f (1) − + (cid:98) f = q · + (1 − q ) · (cid:98) f − q ,so that f ∈ E .We remark that E is not a full Kegelspitze. For example, writing χ s for the characteristicfunction of the interval ] s, 1] (0 < s < χ s ≤ / · (2k ). However, we cannot have χ s = 1 / · f for any f ∈ E , as we would then have f (1) = 2 and so, by the defining propertyof E (cid:48) , f = 2k . But, as χ s = 1 / · f , we have f ( r ) = 0 for any 0 < r < s .We next claim that E is continuous. For this, it is enough to show that each element f is the lub of a directed set of elements (cid:28) E below it. In case f (1) ≤ 1, any element in L ≤ below it is also in E , and we use the continuity of L ≤ to get a directed set of elements (cid:28) L ≤ below it, and so (cid:28) E below it, as E is a sub-dcpo of L ≤ .Otherwise f (1) > 1, and we can again set (cid:98) f = f − f (1) − . Since (cid:98) f (1) = 2 − f (1) < (cid:98) f ∈ L ≤ . Let ρ p,g = 2 p k + g For 0 < p < f (1) − g (cid:28) L ≤ (cid:98) f , we obtain a family of functions which clearly is directedand has 2( f (1) − + (cid:98) f = f as its least upper bound. Each of these functions belongs to E , since it can be written as a convex combination p · + (1 − p ) · g − p and since g − p ∈ L ≤ (the latter as g (1) ≤ (cid:98) f (1) = 2 − f (1) < − p ).It remains to show that ρ p,g (cid:28) E f . For this, let ( f i ) i be a directed family in E suchthat f ≤ (cid:87) ↑ i f i . Then f (1) ≤ (cid:87) ↑ i f i (1) so that there is an i such that p < f i (1) − 1, since p < f (1) − 1. We now restrict our attention to the indices i such that f i ≥ f i . Since these f i belong to E = E (cid:48) , they satisfy 2( f i (1) − ≤ f i . It follows that 2 p k ≤ f i (1) − ≤ f i ,that is 0 ≤ f i − p k ∈ L . As f − p k ≤ ( (cid:87) ↑ i f i ) − p k = (cid:87) ↑ i ( f i − p k ), we then have f − p k ≤ L (cid:87) ↑ i ( f i − p k ). Since g (cid:28) L ≤ (cid:98) f = f − f (1) − ≤ L f − p k , we also have g (cid:28) L f − p k , and so g ≤ f i − p k for some i ≥ i . Thus ρ p,g ≤ p k + g ≤ f i .Now that we have shown E to be a continuous Kegelspitze, it only remains to see that,as claimed, scalar multiplication does not preserve (cid:28) E . One the one hand, from the abovediscussion we have k (cid:28) E (for, setting f = 2k and p = 1 / 2, we see that (cid:98) f = ⊥ and p < f (1) − 1, and then that k = ρ / , ⊥ ). However, on the other hand, we have 1 / · k (cid:54)(cid:28) E k (for k = (cid:87) ↑ n> χ − n , but we have 1 / · k ≤ χ − n for no n > This work is licensed under the Creative Commons Attribution-NoDerivs License. To viewa copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/http://creativecommons.org/licenses/by-nd/2.0/