The two-sorted algebraic theory of states, and the universal states of MV-algebras
aa r X i v : . [ m a t h . L O ] A ug THE TWO-SORTED ALGEBRAIC THEORY OF STATES,AND THE UNIVERSAL STATES OF MV-ALGEBRAS
TOM ´AˇS KROUPA AND VINCENZO MARRA
Abstract.
States of unital Abelian lattice-groups—that is, normalised pos-itive group homomorphisms to R —provide an abstraction of expected-valueoperators. A well-known theorem due to Mundici asserts that the categoryof unital lattice-groups (with unit-preserving lattice-group homomorphisms asmorphisms) is equivalent to the algebraic category of MV-algebras, and theirhomomorphisms. Through this equivalence, states of lattice-groups naturallycorrespond to certain [0 , R (or just in [0 , universal state of any MV-algebra from the existence of freealgebras in multi-sorted algebraic categories. The significance of the universalstate of a given algebra is that it provides (an algebraic abstraction of) themost general expected-value operator on that algebra—a construct that is notavailable if one insists that states be real-valued. In the remaining part of thepaper, we seek concrete representations of such universal states. We begin byclarifying the relationship of universal states with the theory of affine repre-sentations of lattice-groups: the universal state A → B of the MV-algebra A is shown to coincide with a certain modification of Choquet’s affine represen-tation (of the unital lattice-group corresponding to A ) if, and only if, B issemisimple. An MV-algebra is locally finite if each one of its finitely generatedsubalgebras is finite; locally finite MV-algebras are semisimple, and Booleanalgebras are instances of locally finite MV-algebras. Our second main result isthen that the universal state of any locally finite MV-algebra has semisimplecodomain, and can thus be described through our adaptation of Choquet’saffine representation. Introduction
We are concerned in this paper with the algebraic theory of states of MV-algebrasand Abelian lattice-groups with a (strong order) unit, here called “unital Abelian ℓ -groups”. Usually, states are defined as normalised positive real-valued linearfunctionals on Riesz spaces with unit or, more generally, normalised positive grouphomomorphisms to R of unital Abelian ℓ -groups [10]. Part of their importancestems from the long-recognised fact that unital Abelian lattice-groups and their Mathematics Subject Classification.
Key words and phrases.
State, lattice-ordered Abelian group, MV-algebra, multi-sorted alge-bra, free object, universal state, affine representation. states provide an abstraction of bounded real random variables and of expected-value operators, respectively. To illustrate, the collection M b ( X, A ) of bounded A -measurable functions from X to R is an Abelian ℓ -group under pointwise additionand order; and the function X → R constantly equal to 1 is a unit of this ℓ -group,due to our boundedness assumption. If µ is a probability measure on A , then theexpected-value operator R X − d µ : M b ( X, A ) → R is a state of the unital Abelian ℓ -group M b ( X, A ).In this paper we allow states to take values in any unital Abelian ℓ -group, and notjust in the real numbers; thus, if G and H are such groups, a state of G with valuesin H is a positive group homomorphism from G to H that carries the unit of G to theunit of H . We shall see that this level of generality allows us to investigate universalconstructions that, while ubiquitous in algebra, are not available if one insists thatstates be real-valued. We shall be interested, specifically, in the existence of a“most general state”, or universal state , of a given unital Abelian ℓ -group. Wewill prove that such a universal state indeed always exists, as a consequence of thestandard fact that free algebras exist in algebraic categories. One obstruction tothis plan is that the category of unital Abelian ℓ -groups and their unit-preservinghomomorphisms is not algebraic with respect to its underlying-set functor. This isbecause the characteristic property of the unit 1, that its multiples n := 1 + · · · + 1( n times) should eventually exceed any given element, is not even definable in first-order logic, by a standard compactness argument. Nonetheless, a well-known resultof Mundici [20] tells us that the category of unital Abelian ℓ -groups is equivalent tothe algebraic category of MV-algebras [6] and their homomorphisms. We shall provea version of Mundici’s result for states, rather than just for unital homomorphisms,that will allow us to carry our programme out to within an equivalence of categories.To begin with, the theory of (real-valued) states has already been adapted toMV-algebras. The original reference for this is [22], where MV-algebraic states wereintroduced with the motivation of modelling the notion of “average truth degree”in many-valued logic. For a primer on states of MV-algebras see [24, 8]. In line withwhat is discussed above for lattice-groups, in this paper we consider states betweenany two MV-algebras; see Definition 2.1 below. Basic facts about the functor Γthat features in Mundici’s equivalence are recalled in Section 2. In Theorem 2.2we extend Mundici’s equivalence to one between MV st (the category whose objectsare MV-algebras and whose morphisms are states) and A (the category whoseobjects are unital Abelian ℓ -groups and whose morphisms are states). In Section 3states are treated as two-sorted algebras, cf. Definition 3.1. The elementary and yetkey Proposition 3.1 about their equational presentation is proved. The propositionenables us to identify the category of states ES with the category of models in Set ofa finitely axiomatised two-sorted equational theory. There is a corresponding, non-algebraic category S of states between unital Abelian ℓ -groups. Theorem 3.1 thenshows that the categories ES and S are equivalent. Section 4 deals with free objects(free states) in ES , and with universal states (see below for precise definitions).We describe the free object generated by a two-sorted set by universal states andbinary coproducts in the category of MV-algebras (Theorem 4.1). This constitutesour first main result.The second part of our paper is devoted to the issue of representing universalstates explicitly, insofar as this is possible. For this, in Section 5 we show how Cho-quet’s theory of affine representations [10, Chapters 5–7] relates to the construction WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 3 of a universal state. Specifically, Proposition 5.1 says that the codomain of a univer-sal state is Archimedean (or semisimple, in the case of MV-algebras) precisely whenit coincides with the extended form of the affine representation that we introduce.Thus, universal states with a semisimple codomain admit of a satisfactory concretedescription through affine representations. Our second main result, proved in Sec-tion 6, is that the codomain of the universal state of any locally finite MV-algebra(in particular, of any Boolean algebra) is semisimple. The proof uses the dual-ity between finitely presented MV-algebras and the category of compact rationalpolyhedra with piecewise linear maps with integer coefficients as morphisms.We assume familiarity with MV-algebras and unital Abelian ℓ -groups; see [6, 24]and [4, 10] for background information. We often adopt the standard practice inalgebra of omitting underlying-set functors, if clarity is not impaired. We shall alsoomit parentheses in application of functors and functions, writing e.g. F I in placeof F ( I ), when this improves readability. We assume N := { , , . . . } .2. Mundici’s equivalence, for states
Let A ℓ be the category that has unital Abelian ℓ -groups as objects and unital ℓ -homomorphisms as morphisms, and let MV be the category of MV-algebras andtheir homomorphisms. In [20, Theorem 3.9], Mundici established a categoricalequivalence between A ℓ and MV which we recall here, without proofs. To eachunital ℓ -group G we associate its unit interval (1) Γ( G,
1) := { a ∈ G | a } , equipped with the operations a ⊕ b := ( a + b ) ∧ , ¬ a := 1 − a. Then (Γ( G, , ⊕ , ¬ ,
0) is an MV-algebra [6, Proposition 2.1.2]. We will also makeuse of binary operations ⊙ and ⊖ on Γ( G,
1) defined by a ⊙ b := ¬ ( ¬ a ⊕ ¬ b ) ,a ⊖ b := a ⊙ ¬ b. The operations + , ⊕ , and ⊙ in Γ( G,
1) are related as follows (see [6, Lemma 2.1.3(i)]):(2) a + b = ( a ⊕ b ) + ( a ⊙ b ) , a, b ∈ Γ( G, . Since every unital ℓ -homomorphism f : G → H restricts to a homomorphism ofMV-algebras Γ( G, → Γ( H, A ℓ −→ MV . If the unit 1 is understood, we write Γ G in place of Γ( G, M an MV-algebra, wesay that a := ( a i ) i ∈ N ∈ M N is a good sequence in M if a i ⊕ a i +1 = a i for each i ∈ N ,and there is n ∈ N such that a n = 0 for all n > n . We shall write ( a , . . . , a k )in place of ( a , . . . , a k , , , . . . ); in particular, ( a ) is short for ( a, , , . . . ), given a ∈ M . Addition of good sequences ( a i ) i ∈ N and ( b i ) i ∈ N is defined by( a i ) i ∈ N + ( b i ) i ∈ N := ( a i ⊕ ( a i − ⊙ b ) ⊕ · · · ⊕ ( a ⊙ b i − ) ⊕ b i ) i ∈ N . TOM´AˇS KROUPA AND VINCENZO MARRA
The set of all good sequences in M equipped with + becomes a commutative monoid A M ⊆ M N with neutral element (0). By general algebra, the full inclusion of thecategory of Abelian groups into that of commutative monoids has a left adjoint;write η A M : A M → Ξ M for the component at A M of the unit of this adjunction.The monoid A M can be shown to be cancellative, so the monoid homomorphism η A M is injective. To describe the elements of Ξ M explicitly, let us say two orderedpairs of good sequences ( a , b ) and ( a ′ , b ′ ) are equivalent if a + b ′ = a ′ + b , and let us write [ a , b ] for the equivalence class of ( a , b ). Then Ξ M is defined asthe set of all equivalence classes of the form [ a , b ] equipped with the addition[ a , b ] + [ c , d ] := [ a + c , b + d ] , with the neutral element [(0) , (0)], and with the unary inverse operation − [ a , b ] := [ b , a ] . Moreover, the monoid A M is lattice-ordered by the restriction of the product orderof M N , and this lattice order on A M extends (in the obvious sense, through theinjection η A M ) to exactly one translation-invariant lattice order on Ξ M . Thus, Ξ M is an Abelian ℓ -group with unit [(1) , (0)]. Lemma 2.1.
For any MV-algebra M , and any unital Abelian ℓ -group G , the fol-lowing hold.(1) The function ϕ M : M → ΓΞ M given by ϕ M ( a ) := [( a ) , (0)] , a ∈ M, is an isomorphism of MV-algebras.(2) The lattice-ordered monoids G + := { a ∈ G | a > } and A Γ G are isomor-phic through the function g : G + → A Γ G that sends a ∈ G + to the uniquegood sequence g ( a ) := ( a , . . . , a n ) of elements a i ∈ Γ G such that a = a + · · · + a n .(3) The function ε G : G → ΞΓ G defined by ε G ( a ) := [ g ( a + ) , g ( a − )] , a ∈ G, is an isomorphism of unital ℓ -groups, where g is as in item (2) above, and,as usual, a + := a ∨ , a − := − a ∨ .Proof. See Theorem 2.4.5, Lemma 7.1.5, and Corollary 7.1.6 in [6]. (cid:3)
A homomorphism h : M → N of MV-algebras lifts to a function h ∗ : A M → A N upon setting h ∗ (( a i ) i ∈ N ) := ( h ( a i )) i ∈ N . Then Lemma 2.1 and the universalconstruction of Ξ M from A M entail that h ∗ has exactly one extension to a unital ℓ -homomorphism Ξ h : Ξ M → Ξ N . We thereby obtain a functor(4) Ξ : MV −→ A ℓ . Theorem 2.1 (Mundici’s equivalence) . The functors
Γ : A ℓ → MV and Ξ : MV → A ℓ form an equivalence of categories. Remark 2.1.
In what follows we shall often tacitly identify M with ΓΞ M ⊆ Ξ M ,and thus speak of functions defined on M having an extension to Ξ M , etc. WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 5
In the rest of this section we lift the equivalence of Theorem 2.1 from homomor-phisms to states.
Definition 2.1 (States) . For MV-algebras M and N , a function s : M → N is a state ( of M with values in N ) if s (1) = 1, and for each a, b ∈ M with a ⊙ b = 0the equality s ( a ⊕ b ) = s ( a ) + s ( b ) holds, where + is interpreted in Ξ N . For unitalAbelian ℓ -groups G and H , a function s : G → H is a state ( of G with values in H )if s (1) = 1, and s is a group homomorphism that is positive , i.e., for each g ∈ G + we have s ( g ) ∈ H + . We write MV st for the category whose objects are MV-algebrasand whose morphisms are states, and A for the category whose objects are unitalAbelian ℓ -groups and whose morphisms are states.States of MV-algebras can be defined in several equivalent ways. For instance,a function s : M → N satisfying s (1) = 1 is a state if, and only if, for each a, b ∈ M with a ⊙ b = 0, the equalities s ( a ⊕ b ) = s ( a ) ⊕ s ( b ) and s ( a ) ⊙ s ( b ) = 0 hold—see[15, Proposition 3.3]. In Section 3 we will give an equational characterisation ofstates (Proposition 3.1). Remarks 2.1. (1) It is elementary that requiring the group homomorphism s : G → H to be positive is equivalent to asking that it be order-preserving.(2) States are a classical notion in the theory of partially ordered Abelian groups(see e.g. [10]), where they are most often assumed to have codomain R . For em-phasis, we refer to the latter states as real-valued . A significant example is theLebesgue integral R [0 , − d λ over the unital Abelian ℓ -group of all continuous func-tions [0 , → R . Let us stress that states do not necessarily preserve infima andsuprema and, therefore, in general they are not morphisms in A ℓ .(3) States of MV-algebras also are a well-studied notion (see e.g. [24, 8]), and theyare usually assumed to have codomain [0 , ⊆ R . We refer to the latter statesas real-valued . Analogously to the previous item, states of MV-algebras are notmorphisms in MV , in general. Lemma 2.2.
Let M and N be MV-algebras, and G and H be unital Abelian ℓ -groups.(1) Any state of unital Abelian ℓ -groups s : G → H restricts to a function Γ( s ) : Γ G → Γ H that is a state of MV-algebras.(2) Any state of MV-algebras s : M → N has exactly one extension to a stateof unital Abelian ℓ -groups Ξ( s ) : Ξ M → Ξ N .Proof. To prove the first item, observe that 0 a G entails 0 s ( a ) H , because s preserves the order; further, s (1) = 1 by definition. If 0 a , a G , and a ⊙ a = 0 in the MV-algebra Γ G , then (2) yields a ⊕ a = a + a .For the second item, let us regard s as a function ΓΞ M → ΓΞ N . Then s iseasily seen to be order-preserving, [24, Proposition 10.2]. For a, b ∈ ΓΞ M we have s ( a + b ) = s ( a ) + s ( b ) as soon as a + b belongs to ΓΞ M . Indeed, the latter happensprecisely when a ⊙ b = 0 by (2), and then s ( a + b ) = s ( a ⊕ b ) = s ( a ) + s ( b )by the definition of state. Now existence and uniqueness of Ξ( s ) is granted bythe general extension result [12, Proposition 1.5]. (The hypotheses of the citedproposition require G to be directed, and to have the Riesz interpolation property;it is classical and elementary that any ℓ -group satisfies these properties.) (cid:3) TOM´AˇS KROUPA AND VINCENZO MARRA
In light of Lemma 2.2 we consider the functorsΓ : A −→ MV st and Ξ : MV st −→ A that extend the homonymous ones in (3) and (4), respectively. Theorem 2.2 (Mundici’s equivalence, for states) . The functors
Γ : A → MV st and Ξ : MV st −→ A form an equivalence of categories.Proof. The isomorphism ϕ M : M → ΓΞ M from Lemma 2.1(1) is natural: if s : M → N is a state of MV-algebras, the naturality square M ΓΞ MN ΓΞ N ϕ M s ΓΞ( s ) ϕ N commutes by the definitions of Ξ and Γ.The isomorphism ε G : G → ΞΓ G is also natural: if s : G → H is a state of unitalAbelian ℓ -groups, the naturality square G ΞΓ GH ΞΓ H ε G s ΞΓ( s ) ε H commutes. Indeed, ΞΓ( s ) ε G and ε H s are states G → ΞΓ H , and it follows fromdirect inspection of the definitions involved that they agree on the unit intervalof G ; but then they agree on the whole of G , by (1) and (2) in Lemma 2.2. (cid:3) The two-sorted variety of states
For multi-sorted universal algebra see the pioneering [13, 5], and the textbookreference [1]. We are concerned with the two-sorted case only. Unlike [5], and like[1], we allow arbitrary multi-sorted sets as carriers of algebras—no non-emptynessrequirement is enforced. The difference is immaterial for the present paper, becauseeach of our two sorts has constants. We recall that the product category
Set := Set × Set of two-sorted sets and two-sorted functions has as objects the orderedpairs (
A, B ) of sets, and as morphisms f : ( A , B ) −→ ( A , B )the ordered pairs f := ( f , f ) of functions f : A → A ,f : B → B . Composition of morphisms and identity morphisms are defined componentwise.We consider two sorts R and E of random variables and degrees of expectation ,respectively, and operations as follows.(T1) Operations ⊕ : R → R , ¬ : R → R , and 0 : R ∅ → R . Thus, these areoperations of arities 2, 1, and 0, respectively, in the sort of random variables. WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 7 (T2) Operations ⊕ : E → E , ¬ : E → E , and 0 : E ∅ → E . Thus, these are opera-tions of arities 2, 1, and 0, respectively, in the sort of expectation degrees.They are purposefully denoted by the same symbols as their counterpartsin the sort R .(T3) One operation s : R → E from the sort of random variables to that ofdegrees of expectation.Items (T1)–(T3) define a two-sorted (similarity) type. For the sake of clarity, letus spell out that a two-sorted function ( m, n ) : ( M , N ) → ( M , N ) is a homo-morphism between algebras of this two-sorted type precisely when m : M → M and n : N → N are homomorphisms in the type of R and E , respectively, andmoreover the square(5) M N M N sm ns commutes. Definition 3.1 (States as two-sorted algebras) . A state is an algebra ( M, N ) ofthe two-sorted type (T1)–(T3) such that the following equational conditions hold.(S1) ( M, ⊕ , ¬ ,
0) is an MV-algebra.(S2) ( N, ⊕ , ¬ ,
0) is an MV-algebra.(S3) For every a, b ∈ M , s : M → N satisfies(A1) s ( a ⊕ b ) = s ( a ) ⊕ s ( b ∧ ¬ a ),(A2) s ( ¬ a ) = ¬ s ( a ), and(A3) s (1) = 1. Remark 3.1.
The presented axiomatisation (S3) is originally inspired by that ofinternal states [9]. It was used already in [16] in case of states whose domains areBoolean algebras; see also [15].Let us emphasise that in the above and throughout we denote a two-sorted algebrasimply by its underlying two-sorted set (
M, N ). All operations— ⊕ , s , and soforth—are tacitly understood. This is in keeping with standard usage in algebra.Item (S3) in Definition 3.1 amounts to requiring that s : M → N be a state: Proposition 3.1.
Let s : M → N be a function between MV-algebras M and N .The following are equivalent.(1) The function s is a state.(2) The function s satisfies (S3) in Definition 3.1:(A1) s ( a ⊕ b ) = s ( a ) ⊕ s ( b ∧ ¬ a ) ,(A2) s ( ¬ a ) = ¬ s ( a ) ,(A3) s (1) = 1 .Proof. We will need the following equations, valid in all MV-algebras:(MV1) b ∧ ¬ a = b ⊖ ( a ⊙ b ),(MV2) a ⊕ b = a ⊕ ( b ∧ ¬ a ),(MV3) a ⊙ ( b ∧ ¬ a ) = 0.In detail, (MV1) holds by the definition of ∧ , since b ∧ ¬ a = b ⊙ ( ¬ b ⊕ ¬ a ) = b ⊙ ¬ ( a ⊙ b ) = b ⊖ ( a ⊙ b ) . TOM´AˇS KROUPA AND VINCENZO MARRA
The operation ⊕ distributes over ∧ by [6, Proposition 1.1.6], so a ⊕ ( b ∧ ¬ a ) = ( a ⊕ b ) ∧ ( a ⊕ ¬ a ) = a ⊕ b, which proves (MV2). Finally, for (MV3), a ⊙ ( b ∧ ¬ a ) = a ⊙ ( b ⊖ ( a ⊙ b )) = ( a ⊙ b ) ⊙ ¬ ( a ⊙ b ) = 0 . Assume s is a state. Then (A3) holds by definition. Further, (MV2), (MV3),and the definition of state yield s ( a ⊕ b ) = s ( a ⊕ ( b ∧ ¬ a )) = s ( a ) + s ( b ∧ ¬ a ) . Then s ( a ) + s ( b ∧ ¬ a ) ∈ N , and so the + above agrees in fact with ⊕ by (2).Therefore, (A1) holds. Finally, since a ⊙ ¬ a = 0, we can write1 = s ( a ⊕ ¬ a ) = s ( a ) + s ( ¬ a ) , which proves (A2) because 1 − s ( a ) equals ¬ s ( a ) in Ξ N . Thus, s satisfies (A1)–(A3).Conversely, assume s : M → N has properties (A1)–(A3). We first prove that s is order-preserving. Assume a b , or equivalently, by definition, b = a ⊕ ( b ⊖ a ).Then (A1) yields s ( b ) = s ( a ⊕ ( b ⊖ a )) = s ( a ) ⊕ s (( b ⊖ a ) ∧ ¬ a ) > s ( a ) , where the last inequality follows from monotonicity of ⊕ in each coordinate [6,Lemma 1.1.4].Let a ⊙ b = 0. Then (MV1) gives b ∧ ¬ a = b and, by (A1), s ( a ⊕ b ) = s ( a ) ⊕ s ( b ) . It remains to check that s ( a ) ⊙ s ( b ) = 0, which implies s ( a ) ⊕ s ( b ) = s ( a ) + s ( b ) asis to be shown. Employing (A1), (A3), and the assumption a ⊙ b = 0, we get s ( ¬ a ) ⊕ s ( ¬ b ∧ a ) = s ( ¬ a ⊕ ¬ b ) = s ( ¬ ( a ⊙ b )) = s ( ¬
0) = 1 . Since s is order-preserving, and since ⊕ is monotone in each coordinate by [6,Lemma 1.1.4], we infer s ( ¬ a ) ⊕ s ( ¬ b ) > s ( ¬ a ) ⊕ s ( ¬ b ∧ a ) = 1. Then by (A2) thisgives s ( a ) ⊙ s ( b ) = ¬ ( s ( ¬ a ) ⊕ s ( ¬ b )) = 0 , which completes the proof. (cid:3) We consider the category ES (for “Equational States”) of states in the sense ofDefinition 3.1, and their homomorphisms. By its very definition, ES is the categoryof models in Set of a (finitely axiomatised) two-sorted equational theory. Thus, ES is a two-sorted variety—i.e., it is closed under homomorphic images, subalgebras,and products inside the category of all algebras of the type (T1)–(T3)—by theeasy implication in Birkhoff’s Variety Theorem. (See [2] for details on Birkhoff’sTheorem in the multi-sorted setting.) Let us recall that “homomorphic images”here are the codomains of those homomorphisms that are surjective in each sort,and that these are exactly the regular epimorphisms [1, Corollary 3.5].We further consider the category S of states whose objects are all states G → H of unital Abelian ℓ -groups G with values in any unital Abelian ℓ -groups H , andwhose morphisms are pairs of unital ℓ -homomorphisms G → G and H → H making the obvious square commute. Thus, the objects of S are exactly the arrows G → H in A . We define a functorΓ : S −→ ES WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 9 by setting Γ ( G → H ) := (Γ G, Γ H ), for G → H an object of S , where the operation s : Γ G → Γ H of the two-sorted algebra (Γ G, Γ H ) is defined as the restriction ofthe given state G → H (cf. item (1) in Lemma 2.2). Concerning arrows, given themorphism ( g, h ) : ( G → H ) → ( G → H ) in S —i.e., given the pair of unital ℓ -homomorphisms g : G → G and h : H → H that form the required commutativesquare with the states G i → H i , i = 1 , ( g, h ) := (Γ g, Γ h ), which isevidently a morphism in ES . We also define a functorΞ : ES −→ S by setting Ξ ( M, N ) to be the state Ξ s : Ξ M → Ξ N (cf. item (2) in Lemma 2.2),where s : M → N is the two-sorted operation of ( M, N ). For a morphism ( m, n )from ( M , N ) to ( M , N ), we let Ξ ( m, n ) := (Ξ m, Ξ n ); this is a morphism in S because Ξ is a functor. Theorem 3.1 (Equational characterisation of states) . The functors Γ : S → ES and Ξ : ES → S form an equivalence of categories.Proof. The morphism ( ϕ M , ϕ N ) : ( M, N ) → Γ Ξ ( M, N ) is an isomorphism in ES ,where the components of ( ϕ M , ϕ N ) are as in Lemma 2.1 and thus are isomorphisms;naturality is verified componentwise, and thus follows at once from Theorem 2.2.Similarly, for every state G → H in A , ( ε G , ε H ) : ( G → H ) → Ξ Γ ( G, H ) isan isomorphism, where ε G and ε H are as in Lemma 2.1, and naturality reduces tonaturality in each sort, which holds by Theorem 2.2. (cid:3) Free and universal states
For a set I , let us write F I for the free MV-algebra generated by I , and ι I : I → F I (6)for the “inclusion of free generators”, i.e., for the component at I of the unit of thefree/underlying-set adjunction F ⊣ | − | .By general algebraic considerations, the functor | − | : ES −→ Set that takes a state of MV-algebras to its carrier two-sorted set has a left adjoint F : Set −→ ES . The existence of this left adjoint has nothing to do with MV-algebras specifically,and is entailed by the existence of free algebras in varieties of multi-sorted algebras.The original reference for this latter result seems to be [13, Section 5], with a gen-eralisation established in [5, Section 7]. For a two-sorted set S , write η S : S → F S for the component at S of the unit of the adjunction F ⊣ | − | . Then η S is char-acterised as the essentially unique two-sorted function S → F S such that, for anytwo-sorted function f : S → ( M, N ) with a state in the codomain, there is exactlyone morphism h : F S → ( M, N ) in ES making the diagram S F S ( M, N ) η S f h commute. In algebraic parlance, F S is a state freely generated by the two-sortedset S ; it is evidently unique to within a unique isomorphism, and therefore any state satisfying the preceding universal property will be called the free state generatedby S . Lemma 4.1.
Consider the two-sorted set ( ∅ , S ) , where S is any set. Writing for the initial MV-algebra (=two-element Boolean algebra), ( , F S ) is the freestate in ES generated by ( ∅ , S ) , where the operation s is the unique homomorphism → F S .Proof. Given a two-sorted function ( f ! , f ) : ( ∅ , S ) → ( M, N ), where f ! : ∅ → M isthe unique function from ∅ to M , write h ! : → M for the unique homomorphism,and h : F S → N for the unique homomorphism such that f = h ι S , where ι S is as in (6). Further, consider the component-wise inclusion ι : ( ∅ , S ) ⊆ ( , F S ).Then ( h ! , h ) : ( , F S ) → ( M, N ) is a morphism in ES , because is initial in MV st .Also, ( f ! , f ) = ( h ! , h ) ι , and ( h ! , h ) is clearly unique with this property. (cid:3) If M is any MV-algebra, a state υ M : M −→ Υ M is said to be universal (for M ) if for each state s : M → N there is exactly onehomomorphism of MV-algebras h : Υ M → N satisfying hυ M = s . Universal states,when they exist, are evidently unique to within a unique isomorphism. Any statesatisfying the preceding universal property will therefore be called the universalstate ( of M ). Lemma 4.2.
Consider the two-sorted set ( S , ∅ ) , where S is any set. Then thefree state F ( S , ∅ ) generated by ( S , ∅ ) is (isomorphic in ES to) ( F S , Υ F S ) , where the operation s is the universal state υ F S of F S .Proof. Let us display the components of F ( S , ∅ ) as the pair of MV-algebras( A, B ). If M is any MV-algebra and is the terminal (=one-element) MV-algebra,we consider the object ( M, ) of ES whose operation s is the unique homomor-phism M → . Given any function f : S → M , the unique two-sorted function( f, !) : ( S , ∅ ) → ( M, ) whose first component is f (and whose second component,necessarily, is the only possible function ! : ∅ → ) has exactly one extension toa homomorphism ( h , !) : ( A, B ) → ( M, ). The second component of this homo-morphism is the only possible one ! : B → . Then h : A → M is a homomor-phism that extends f , and it is the unique such: if h ′ : A → M extends f then( h ′ , !) : ( A, B ) → ( M, ) extends ( f, !), and thus h ′ = h by the uniqueness of ( h , !).This shows that A is F S . In the rest of this proof we write F S and drop A .Let us next consider the operation s : F S → B of the object ( F S , B ), with theintent of proving that the state s is universal for F S . Consider any state t : F S → N , and the corresponding object ( F S , N ) of ES . Writing ι : S → F S for theinsertion of free generators, we consider the two-sorted function ( ι, !) : ( S , ∅ ) → ( F S , N ). The universal property of ( F S , B ) yields exactly one homomorphism(1 F S , h ) : ( F S , B ) → ( F S , N ) extending ( ι, !), the first component of which is,necessarily, the identity 1 F S : F S → F S . The second component h thereforesatisfies h s = t . Further, h is the only homomorphism with this property. Indeed,if h ′ : B → N satisfies h ′ s = t , then (1 F S , h ′ ) : ( F S , B ) → ( F S , N ) extends ( ι, !)and so h ′ = h by the universal property of ( F S , B ). In conclusion, s : F S → B WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 11 is the universal state υ F S of F S , and therefore B is Υ F S . This completes theproof. (cid:3) Lemma 4.1 establishes the existence of the universal state of a free MV-algebra.For the general case, we take a quotient. If f : A → B is a function, we writeKer f := { ( x, y ) ∈ A | f x = f y } . This notation is opposed to ker f := f − { } ,which we also use later on in the paper and which makes sense when A and B areeither MV-algebras or lattice-groups. Corollary 4.1.
Every MV-algebra has a universal state.Proof.
Let M be any MV-algebra, write F M for the free MV-algebra generated bythe set M , and let q : F M → M be the unique, automatically surjective homo-morphism extending the identity function M → M . By Lemma 4.2, F ( M, ∅ ) is( F M, Υ F M ), where the operation s is the universal state υ F M of F M . In the rest ofthis proof we write υ , tout court . Set θ = Ker q and θ = h{ ( υx, υy ) | ( x, y ) ∈ θ }i ,where h−i denotes the congruence generated by − on Υ F M . Then ( θ , θ ) is ev-idently a congruence on ( F M, Υ F M ), and it is the congruence generated by thetwo-sorted relation ( θ , ∅ )—indeed, any congruence on ( F M, Υ F M ) that contains( θ , ∅ ) must contain ( θ , { ( υx, υy ) | ( x, y ) ∈ θ } ), by the compatibility with υ , andthus must contain ( θ , θ ).We consider the quotient state ( M, Υ F Mθ ) of ( F M, Υ F M ) modulo the congruence( θ , θ ), with operation M → Υ F Mθ denoted s , F M Υ F MM Υ F Mθ N N υq q kt s h N and we verify that s is the universal state of M . (We write q : Υ F M → Υ F Mθ forthe natural quotient map.) For this, let t : M → N be any state. The composite tq : F M → N is a state, too. Since υ is universal for F M , there is exactly onehomomorphism k : Υ F M → N such that kυ = tq . Let us show Ker q ⊆ Ker k .Since Ker q is θ , it suffices to show that the inclusion holds for { ( υx, υy ) | ( x, y ) ∈ θ } , because the latter is a generating set of θ . Given ( υx, υy ) with ( x, y ) ∈ θ ,from q x = q y we obtain tq x = tq y , and thus kυx = kυy , that is ( υx, υy ) ∈ Ker k ,as we intended to show. By the universal property of quotients (applied to MV-algebras), there is exactly one homomorphism h : Υ F Mθ → N with hq = k . Thishomomorphism satisfies hs = t . Indeed, from hq = k and kυ = tq we obtain hq υ = tq , and therefore hsq = tq ; but q is epic, whence hs = t . Finally, if h ′ is any homomorphism that satisfies h ′ s = t , then h ′ sq = tq and so h ′ q υ = tq ;applying the uniqueness property of k , we infer h ′ q = k , and applying that of h we conclude h = h ′ . This completes the proof that s is universal for M . (cid:3) Corollary 4.1 entails at once by elementary category theory (see e.g. [17, Theorem2 in Chapter IV]) that the faithful, non-full inclusion functor | − | : MV −→ MV st has a left adjoint Υ : MV st −→ MV . (7)Thus, Υ M is the MV-algebra freely generated by | M | . Remark 4.1.
The faithful, non-full inclusion of A ℓ into the category of unital par-tially ordered Abelian groups (with morphisms the unital order-preserving grouphomomorphisms) is proved to have a left adjoint in [4, Appendice A.2]; the argu-ment there is for the non-unital case, but is easily adapted. Thus, the unital Abelian ℓ -group freely generated by any unital partially ordered Abelian group exists. Asa special case of this, one has that the unital Abelian ℓ -group freely generated bya unital partially ordered Abelian group that happens to be lattice-ordered exists .Upon applying the results in Section 2 to translate into the language of orderedgroups, Corollary 4.1 provides an alternative proof of this result that is streamlinedby the use of two-sorted algebraic theories. For clarity, we mention that Bigard,Keimel, and Wolfenstein in [4] distinguish between “universal” and free ℓ -groups:the former are in fact what we call “free”, as is now standard; the latter have thefurther property that the universal arrow is an order embedding. Our own usageof “universal” for the components of the unit of the adjunction Υ ⊣ | − | is meantas mere emphasis, in view of the probabilistic meaning of the construction.General free algebras in ES reduce to universal states and coproducts in MV ;see [21] for the latter. We also say “sum” for “coproduct”. We write + to denotebinary sums in MV . If A → A + B ← B is a coproduct, we call the two arrows thecoproduct injections (with no implication about their injectivity as functions), andwe often denote them in and in , respectively. Theorem 4.1.
For any two-sorted set ( S , S ) , the state ( F S , Υ F S + F S )(8) in ES with operation s equal to in υ F S , where υ F S is the universal state of F S ,and in : Υ F S → Υ F S + F S is the first coproduct injection, is the free stategenerated by S := ( S , S ) , the component of the unit at S being η S := ( ι S , in ι S ) with ι S i as in (6) , i = 1 , , and in : F S → Υ F S + F S the second coproductinjection.Proof. We consider a two-sorted function ( f , f ) : ( S , S ) → ( M, N ). With refer-ence to the diagram below, S F S Υ F S Υ F S + F S M N F S S f ι S h υ FS in h in ι S f we have: WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 13 • Exactly one homomorphism h : F S → M making the upper left trianglecommute, by the freeness of F S ; • the state F S → N given by the composition F S → M → N ; • exactly one homomorphism Υ F S → N making at its immediate left com-mute; • exactly one homomorphism F S → N making the lower right triangle com-mute, by the freeness of F S ; • and therefore, by the universal property of coproducts, there is exactly onehomomorphism h : Υ F S + F S → N such that precomposing it withthe coproduct injections in and in yields Υ F S → N and F S → N ,respectively.By construction, the pair ( h , h ) is a morphism in ES that satisfies ( f , f ) =( h , h )( ι S , in ι S ). That it is the unique such follows readily from the diagramabove. (cid:3) Universal states, and Choquet’s affine representation
We recall basic facts about the theory of affine representations. For back-ground information and references see [10, Chapters 5–7]. We formulate the resultsin the language of ordered groups, as is traditional; they may be translated forMV-algebras via the equivalence in Theorem 2.1.For a unital Abelian ℓ -group G , setSt G := { s : G → R | s is a real-valued state } ⊆ R G . Equip R with its Euclidean topology, R G with the product topology, and St G withthe subspace topology. Then St G , the state space of G , is a compact Hausdorffspace, which is moreover a convex set in the vector space R G . For every a ∈ G weconsider the function ˆ a : St G −→ R (9) s s ( a ) . The function ˆ a is continuous and affine; we write A (St G ) for the set of all continu-ous affine maps St G → R . Then [10, Theorem 11.21] says that A (St G ) is a unitalAbelian ℓ -group under pointwise addition and order, with unit the function con-stantly equal to 1. The map e G : G −−→ A (St G )(10) a ˆ a, induced by (9) is a state G → A (St G ). It is an isomorphism onto its rangeprecisely when G is Archimedean [10, Theorem 7.7]. Recall that G is Archimedean if for a, b ∈ G , na b for all positive integers n implies a
0. In any case, evenwhen G fails to be Archimedean, we call (10) the affine representation of G .For any compact Hausdorff space X , write C ( X ) for the unital Abelian ℓ -groupof continuous real-valued functions on X , operations being defined pointwise; thefunction constantly equal to 1 is the unit. For every unital Abelian ℓ -group G ,the inclusion A (St G ) ⊆ C (St G ) preserves the unit, the group structure, and thepartial order. In general, however, it fails to preserve the lattice structure. Let ustherefore consider the sublattice-subgroup b G of C (St G ) generated by the imageof G under the affine representation map e G in (10). Modifying the codomain of the affine representation accordingly, but retaining the same notation, we obtainthe state e G : G −−→ b G (11) a ˆ a, which we call the extended affine representation of G .Reformulating the notion of universal state of MV-algebras, a state υ G : G −→ Υ G of the unital Abelian ℓ -groups G will be called universal (for G ) if for each state s : G → H there is exactly one unital ℓ -homomorphism h : Υ G → H satisfying hυ G = s . We will relate the extended affine representation (11) of G with thecodomain Υ G of a universal state. In light of the universal property of v G there isexactly one comparison unital ℓ -homomorphism(12) q G : Υ G −→ b G that satisfies(13) q G v G = e G . Proposition 5.1.
Let G be a unital Abelian ℓ -group. The homomorphism q G in (12) is an isomorphism if, and only if, Υ G is Archimedean.Proof. First observe that q G is always surjective. Indeed, v G [ G ] and e G [ G ] gener-ate Υ G and b G as ℓ -groups, respectively. Therefore, by (13), the map q G throwsa generating set of Υ G onto a generating set of b G ; hence it is surjective. Further,recall that the radical ideal of a unital Abelian ℓ -group H is defined asRad H := \ { ker h | h : H → R is a homomorphism } , and that H is Archimedean if, and only if, Rad H = { } . It now suffices to provethat ker q G = Rad Υ G . Since b G is Archimedean by construction, and unital ℓ -homomorphisms such as q G preserve radical ideals, the inclusion Rad Υ G ⊆ ker q G isclear. For the converse inclusion, suppose by contraposition that x Rad Υ G , withthe intent of showing q G x = 0. By the hypothesis there is a unital ℓ -homomorphism h : Υ G → R such that hx = 0. Thus we have a real-valued state s := hv G : G −→ R . Evaluation of elements of b G at s produces a homomorphismev s : b G −→ R f f s ∈ R such that ev s e G = hv G . (14)To see that (14) holds, pick a ∈ G and compute: (ev s e G )( a ) = ev s ( e G ( a )) =( e G ( a ))( s ) = s ( a ), where the last equality is given by the definition (9). Since s = hv G , (14) holds.From (13–14) we deduce ev s q G v G = hv G , WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 15 which by the universal property of v G entailsev s q G = h. (15)Since hx = 0 by hypothesis, from (15) we infer q G x = 0, as was to be shown. (cid:3) Remark 5.1.
The question of when the Abelian ℓ -group freely generated by anArchimedean partially ordered Abelian group is itself Archimedean has long hadan important place in the theory of ordered groups. Bernau [3] gave an exam-ple of an Archimedean partially ordered Abelian group, which moreover happensto be lattice-ordered, such that the Abelian ℓ -group it freely generates fails to beArchimedean. Further, Bernau obtained in [3, Theorem 4.3] a necessary and suf-ficient condition for the Abelian ℓ -group freely generated by a partially orderedAbelian group to be Archimedean; the condition, which we do not reproduce heredue to its length, is known as the uniform Archimedean property of a partiallyordered group. Via the results in Section 2, Bernau’s example entails that theuniversal state of a semisimple MV-algebra may have a non-semisimple codomain.However, we shall see in the next section that this cannot happen when the semisim-ple MV-algebra in question is locally finite.6. The universal state of a locally finite MV-algebra
An MV-algebra is locally finite if each of its finitely generated subalgebras isfinite. By general algebraic considerations, any algebraic structure is the directunion of its finitely generated subalgebras [14, Theorem 2.7]. Thus, every locallyfinite MV-algebra is the direct union of its finite subalgebras. We are going toprove:
Theorem 6.1.
Let υ M : M → Υ M be the universal state of a locally finite MV-algebra M (cf. Corollary 4.1). Then Υ M is semisimple. The proof of this theorem will require a number of lemmas. For each integer i >
1, let M i := { , i , . . . , i − i , } be the finite totally ordered MV-algebra ofcardinality i + 1. By [6, Proposition 3.6.5], a finite MV-algebra A is isomorphic toa product M k × · · · × M k n , for uniquely determined integers k , . . . , k n > n > n = 0, A is the terminal MV-algebra { } .) For the rest of this sectionwe use A to denote such a finite MV-algebra. Lemma 6.1.
Let A = M k × · · · × M k n be a finite MV-algebra and N be anyMV-algebra. Write { a , . . . , a n } for the atoms of A .(1) Any state s : A → N is uniquely determined by its values at the atoms of A .(2) A function s : { a , . . . , a n } → N has an extension to a state A → N if,and only if, k s ( a ) + · · · + k n s ( a n ) = 1 , and in that case the extension isunique.Proof. The unital Abelian ℓ -group Ξ A is the simplicial group Z n with unit theelement ( k , . . . , k n ), and the atoms a i are the standard basis elements of Z n ; fromthis item 1 follows at once. As for item 2, the left-to-right implication followsdirectly from the definition of state. Conversely, assume k s ( a )+ · · · + k n s ( a n ) = 1.For any a ∈ ΓΞ A ⊆ Ξ A = Z n , we can write a = P ni =1 c i a i for uniquely determinedintegers c i . Setting s ′ ( a ) := P ni =1 c i s ( a i ), one verifies that s ′ is a state, and then,by item 1, s ′ is the unique state extending s . (cid:3) Set k := ( k , . . . , k n ), for short. If F n is the MV-algebra freely generated by theset { x , . . . , x n } , there is a finitely generated (hence principal, by [6, Lemma 1.2.1])ideal U ( k ) of F n determined by the partition-of-unity relation n X i =1 k i x i = 1 , (16)where addition is interpreted in the unital Abelian ℓ -group Ξ F n . In more detail,there is a term σ ( x , . . . , x n ) in the language of MV-algebras such that, for anyMV-algebra M , and for any elements b , . . . , b n ∈ M , σ ( b , . . . , b n ) = 0 holds in M if, and only if, P ni =1 k i b i = 1 holds in Ξ M . Thus, U ( k ) is the ideal of F n generatedby σ ( x , . . . , x n ), when the latter is regarded as an element of F n in the usual way.For an explicit computation of σ the interested reader can consult [23]. We write S k for the quotient algebra F n/U ( k ). By Lemma 6.1, the function a i x i , i = 1 , . . . , n, is extended by exactly one state υ A : A −→ S k , (17)where A = M k × · · · × M k n . Lemma 6.2.
The state (17) is the universal state of the MV-algebra A = M k ×· · · × M k n .Proof. If s : A → N is any state, and { a , . . . , a n } is the set of atoms of A , considerthe assignment x i s ( a i ), i = 1 , . . . , n . We have P ni =1 k i s ( a i ) = 1 in N , because s is a state. Then, by the definition of S k and the universal property of homomorphicimages in varieties, this assignment has exactly one extension to a homomorphism h : S k → N . For each a i ∈ A we have ( hυ A ) a i = s ( a i ), which entails hυ A = s because { a , . . . , a n } is a Z -module basis of Ξ A = Z n . Such h is unique, becauseif h ′ : S k → N satisfies h ′ υ A = s then h and h ′ must agree on the generating set { x , . . . , x n } of S k : indeed, h ′ υ A = hυ A entails h ′ x i = ( h ′ υ A ) a i = ( hυ A ) a i = hx i , i = 1 , . . . , n . (cid:3) For the proof of the last lemma we need, Lemma 6.5 below, we shall applygeometric techniques. We use the integer d > R d . A function f : R d → R is PL (for piecewise linear ) if it iscontinuous with respect to the Euclidean topology on R d and R , and there is afinite set of affine functions l , . . . , l u : R d → R such that for each x ∈ R d onehas f ( x ) = l i ( x ) for some choice of i = 1 , . . . , u . (We note in passing that theterminology “piecewise linear” is traditional, even though “piecewise affine” wouldbe, strictly speaking, more appropriate.) If moreover, each l i : R d → R can bechosen so as to restrict to a function Z d → Z , then f is a Z -map . (This terminologycomes from [24].) In coordinates, this is equivalent to asking that l i can be writtenas a linear polynomial with integer coefficients. For an integer d ′ >
0, a function λ = ( λ , . . . , λ d ′ ) : R d → R d ′ is a PL map (respectively, a Z -map ) if each one of itsscalar components λ j : R d → R is a PL function ( Z -map). One defines PL maps( Z -maps) A → B for arbitrary subsets A ⊆ R d , B ⊆ R d ′ as the restriction andco-restriction of PL maps ( Z -maps).A convex combination of a finite set of vectors v , . . . , v u ∈ R d is any vector of theform λ v + · · · + λ u v u , for non-negative real numbers λ i > P ui =1 λ i = 1. WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 17 If S ⊆ R d is any subset, we let conv S denote the convex hull of S , i.e. the collectionof all convex combinations of finite sets of vectors v , . . . , v u ∈ S . A polytope is anysubset of R d of the form conv S , for some finite S ⊆ R d , and a ( compact ) polyhedron is a union of finitely many polytopes in R d . A polytope is rational if it may bewritten in the form conv S for some finite set S ⊆ Q d ⊆ R d of vectors with rationalcoordinates. Similarly, a polyhedron is rational if it may be written as a union offinitely many rational polytopes. It is clear that the composition of Z -maps is againa Z -map. Rational polyhedra and Z -maps thus form a category, which we denote PL Z . This is a non-full subcategory of the classical compact polyhedral category PL whose objects are polyhedra and whose morphisms are PL maps. Remark 6.1.
The full subcategory of PL Z whose objects are rational polyhedralying in unit cubes [0 , d , as d ranges over all non-negative integers, is equivalentto PL Z , see [19, Claim 3.5]. An analogous remark applies to PL . We shall make useof these facts whenever convenient, without further warning.Given a subset S ⊆ R d , we write ∇ Z S for the set of all Z -maps S → [0 , ∇ Z S inherits from [0 ,
1] the structure of an MV-algebra, upon defining operationspointwise; in other words, ∇ Z S is a subalgebra of the MV-algebra C ( S, [0 , S → [0 , X ⊆ R d is a rationalpolyhedron then ∇ Z X is finitely presentable; see e.g. [24, Theorem 6.3], wherethe result is stated for X ⊆ [0 , d —the case X ⊆ R d is then a consequence ofRemark 6.1. Following tradition, from now on we say ‘finitely presented’ insteadof ‘finitely presentable’, even when the latter would be the proper expression. By MV fp we denote the full subcategory of MV whose objects are finitely presentedMV-algebras. Further, if P ⊆ R d and Q ⊆ R d ′ are rational polyhedra for someintegers d, d ′ >
0, a Z -map λ : P → Q induces a function ∇ Z λ : ∇ Z Q −−→ ∇ Z P given by f ∈ ∇ Z Q ∇ Z λ f λ ∈ ∇ Z P. It can be shown that ∇ Z λ is a homomorphism of MV-algebras, see e.g. [19, Lemma3.3]. We thereby obtain a functor ∇ Z : PL Z −−→ MV op fp . (18)To define a functor Max : MV op fp −−→ PL Z (19)in the other direction, let us assume that a specific finite presentation of an MV-algebra is given. That is, we are given a finite set R := { ρ i ∈ F n | i = 1 , . . . , m } ofelements of F n , the free MV-algebra generated by n elements x , . . . , x n , and weconsider the finitely presented quotient F n/ h R i , where h R i denotes the ideal of F n generated by R . Since every finitely generated ideal of any MV-algebra is principalby [6, Lemma 1.2.1], we may safely assume that h R i is generated by the single ele-ment ρ . Writing π i : [0 , n → [0 ,
1] for the i th projection function, the assignment x i π i , i = 1 , . . . , n , extends to exactly one homomorphism F n → ∇ Z [0 , n by theuniversal property of the free algebra, and this homomorphism is an isomorphismby [6, Theorem 9.1.5]. Thus, ∇ Z [0 , n is the MV-algebra freely generated by theprojection functions. Under this isomorphism, ρ corresponds to an element ρ ′ of ∇ Z [0 , n , and the assignment x i π i , i = 1 , . . . , n , extends to exactly one isomor-phism F n/ h ρ i → ∇ Z [0 , n / h ρ ′ i . Then we set Max F n/ h ρ i := ρ ′− { } . Here, thezero set ρ ′− { } of the Z -map ρ ′ is a rational polyhedron in [0 , n . When regardedas a topological space, by [6, Theorem 3.4.3] the polyhedron ρ ′− { } is homeomor-phic to the maximal spectral space of the MV-algebra F n/ h ρ i , whence the nameof the functor (19). In turn, the maximal ideals of any MV-algebra M are in bi-jection with the kernels of the homomorphisms M → [0 ,
1] ([24, Theorem 4.16]).Therefore the points of ρ ′− { } are in bijection with (the kernels of) the homomor-phisms F n/ h ρ i → [0 , h : F n/ h ρ i → F m/ h τ i . Precomposing with h , a morphism F m/ h τ i → [0 ,
1] is sentto a morphism
F n/ h ρ i , so that we obtain a mapping Max h : τ ′− { } → ρ ′− { } which can be proved to be a Z -map. This is the definition of the functor Max forfinitely presented MV-algebras; for the general case of a finitely presentable algebra,to complete the definition one chooses a finite presentation for each algebra. Theorem 6.2 (Duality theorem for finitely presented MV-algebras) . The functors ∇ Z : PL Z −→ MV op fp and Max : MV op fp −→ PL Z form an equivalence of categories.Proof. Two different proofs of this result are given in [18] and [19]. (cid:3)
Remark 6.2.
We will apply Theorem 6.2 in the sequel without explicit referenceto its numbered statement, freely using such expressions as “the dual rational poly-hedron X of the finitely presented MV-algebra A ”.If x ∈ Q d , there is a unique way to write out x in coordinates as x = (cid:18) p q , . . . , p d q d (cid:19) with p i , q i ∈ Z , q i > p i and q i relatively prime for each i = 1 , . . . , d . The positiveinteger den x := lcm { q , q , . . . , q d } is the denominator of x . For a non-negative integer t , a t -dimensional simplex (orjust t -simplex ) in R d is the convex hull of t + 1 affinely independent points in R d ,called its vertices ; the vertices of a simplex are uniquely determined. A face ofa simplex σ is the convex hull of a nonempty subset of the vertices of σ , and assuch it is itself a simplex. The relative interior of σ is the set of points expressibleas convex combinations of its vertices with strictly positive coefficients. A simplexis rational if its vertices are. The following notion is fundamental to the arithmeticgeometry of PL Z : A simplex σ ⊆ R d is regular if whenever x is a rational point lyingin the relative interior of some face τ of σ with vertices v , . . . , v l , then den x > P li =1 den v i . See [24, Lemma 2.7]. Regular simplices are also called unimodular inthe literature.Since the finite MV-algebra A = M k × · · · × M k n is finitely presented, it has adual polyhedron, which by duality must be the sum of the duals of M k i , i = 1 , . . . , n .The dual of each M k i is immediately seen to be a single rational point p (in some R d ) such that den p = k i . Thus, the dual of A is any finite set of points { p , . . . , p n } such that den p i = k i , i = 1 , . . . , n . A specific coordinatisation of such a rationalpolyhedron is obtained as follows. Let { e , . . . , e n } be the standard basis vectorsin R n . Then den e i k i = k i . Thus, { e k , . . . , e n k n } is the rational polyhedron dual to A . WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 19
The convex hull ∆ k := conv (cid:26) e k , . . . , e n k n (cid:27) is a regular simplex by direct inspection. For each i = 1 , . . . , n , we write π k i : ∆ k → [0 ,
1] for the restriction of the projection function π i : [0 , n → [0 , π k i ∈ ∇ Z ∆ k . Observe that n X i =1 k i π k i = 1 . (20) Lemma 6.3.
For any k = ( k , . . . , k n ) , the dual polyhedron of the finitely presentedMV-algebra S k is ∆ k . In more detail, writing x k i for the generators provided by thepresentation of S k , the assignment x k i π k i , i = 1 , . . . , n , extends to exactly onehomomorphism S k → ∇ Z ∆ k , and that homomorphism is an isomorphism.Proof. Applying the definition of the functor Max in (19), the polyhedron Max S k dual to S k is the solution set in [0 , n of the equation P ni =1 k i π i = 1 that presents S k (or equivalently, it is the zero set of σ ( π , . . . , π n ), where σ is as in the definitionof S k ). Computation confirms that this solution set is ∆ k ; and the remaining partof the statement is a direct consequence of the duality Theorem 6.2. (cid:3) Corollary 6.1.
There is exactly one state υ A : A −→ ∇ Z ∆ k (21) that extends the assignment a i π i , i = 1 , . . . , n, and this is the universal state of the MV-algebra A = M k × · · · × M k n .Proof. The existence and uniqueness of the extension follows from Lemma 6.1, uponrecalling (20). That the state in question is naturally isomorphic to υ A in Lemma6.2 follows at once from the isomorphism in Lemma 6.3. (cid:3) We record a well-known elementary property of Z -maps for which we could notlocate a reference (however, see the related [24, Lemma 3.7]). Lemma 6.4.
Let X ⊆ R d and Y ⊆ R d ′ be rational polyhedra, for integers d, d ′ > ,and let f : X → Y be a Z -map. For each x ∈ X ∩ Q d , the number f x is rationaland den f x divides den x ; in symbols, den f x | den x .Proof. The Z -map f : X → Y may be written in vectorial form as ( f , . . . , f d ′ ), witheach f i : X → R a Z -map. In the rest of this proof, assume all rational numbersare expressed in reduced form. Pick a point x = ( p i q i ) ∈ X ∩ Q d , and regard x as a column vector. By definition, each f i agrees locally at the point x with anaffine function R d → R with integer coefficients. Thus, there is a column vector c = ( c i ) ∈ Z d ′ together with a ( d ′ × d ) matrix M = ( z ij ) with integer entries suchthat f x = M x + c. (22)This shows that f x is rational if x is. Further, let us write f x = ( a i b i ) ∈ Q d ′ . By(22) we have a i b i = P dj =1 z ij p j q j + c i , so that b i | den x , and therefore lcm { b i } d ′ i =1 =den f x | den x . (cid:3) We can now prove the promised lemma.
Lemma 6.5. If h : A → B is any injective homomorphism between finite MV-algebras, Υ h : Υ A → Υ B is injective.Proof. Let { e /k , . . . , e n /k n } ⊆ R n be the dual finite rational polyhedron of A ,with den e i /k i = k i , i = 1 , . . . , n . Similarly, let { e /t , . . . , e m /t m } ⊆ R m be thedual of B , with den e i /t i = t i , i = 1 , . . . , m . Let us write h ∗ := Max h : { e /t , . . . , e m /t m } → { e /k , . . . , e n /k n } for the Z -map dual to h . By Lemma 6.4 we have k i | t j whenever h ∗ ( e j /t j ) = e i /k i .Moreover, the Z -map h ∗ has exactly one extension to an affine map α : ∆ t → ∆ k ,where t := ( t i ) mi =1 , by the elementary properties of affine functions and simplices.The computation in [24, Lemma 3.7] (or else, which is essentially the same, directcomputation of the matrix form of α with respect to standard bases) confirmsthat α is a Z -map. Since h is injective, it is a monomorphism, and thus h ∗ isan epimorphism by duality. But then a straightforward verification shows that h ∗ is a surjective function. This entails at once that its affine extension α is surjective,too. For any two continuous functions f, g : ∆ k → R with f = g we then have f α = gα , by the surjectivity of α . In particular, this means that the homomorphism ∇ Z α : ∇ Z ∆ k −→ ∇ Z ∆ t , which by the definition of the functor ∇ Z carries a Z -map f : ∆ k → [0 ,
1] to the Z -map f α : ∆ t → [0 , h is ∇ Z α to within a natural isomorphism. That is, in light of Corollary 6.1, weneed to check that the square A B ∇ Z ∆ k ∇ Z ∆ t υ A h υ B ∇ Z α commutes, which amounts to a straightforward application of the definitions whichwe leave to the reader. (cid:3) Finally:
Proof of Theorem 6.1.
Write Σ for the directed partially ordered set of all finitesubalgebras of M , so that M = S A ∈ Σ A (direct union). The functor Υ in (7) isleft adjoint and thus preserves colimits [17, Theorem 1 in Chapter V.5]. Moreover,by Lemma 6.5, each inclusion A → B with A, B ∈ Σ is sent by Υ to an injectivehomomorphism. Together with Corollary 6.1, this says that Υ M is a directedcolimit of semisimple MV-algebras with injective transition homomorphisms. Sincethe radical ideal of semisimple MV-algebras is trivial, the radical of the directedcolimit Υ M is trivial, too, by [6, Proposition 3.6.4]. Hence, Υ M is semisimple. (cid:3) Corollary 6.2.
For any locally finite MV-algebra M , the extended affine represen-tation of M (cf. Section 5) is the universal state of M . This happens, in particular,for all Boolean algebras.Proof. Theorem 6.1 and Proposition 5.1. The last assertion holds because of thestandard fact that finitely generated Boolean algebras are finite. (cid:3)
WO-SORTED ALGEBRAIC THEORY OF STATES, AND THE UNIVERSAL STATES 21 Further research
The present paper is part of a nascent programme aimed at exploring universalconstructions in probability theory. While we refrain from sketching that pro-gramme here, we do point out some connections between the line of research pur-sued in this paper, and many-valued logic.
Fuzzy Probability Logic
FP( L) overinfinite-valued Lukasiewicz logic was developed by H´ajek [11, Chapter 8.4], andlater extended by Cintula and Noguera to a two-tier modal logic aimed at modellinguncertainty [7]. The logic FP( L) formalises reasoning about properties of states,similarly to probabilistic logics designed for reasoning about probability. The mainfeature of FP( L) is a two-level syntax. Probability assessments are representedin the language by a unary modality Probably, which can be applied to Booleanformulæ only. The class of MV-algebras and states provides a possible completesemantics for FP( L); see [7] for more details. States as two-sorted algebras, asintroduced here, may provide an equivalent multi-sorted algebraic semantics to thelogic FP( L). Details will be pursued in further research.
Acknowledgments
Both authors are deeply grateful to an anonymous reviewer for his/her carefuland competent reading of a previous version of this paper, which contained twosignificant oversights in the proof of Corollary 4.1.The work of Tom´aˇs Kroupa has been supported from the GA ˇCR grant projectGA17-04630S and from the project RCI (CZ.02.1.01/0.0/0.0/16 019/0000765).
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Institute of Information Theory and Automation of the CAS, Pod Vod´arenskouvˇeˇz´ı 4, 182 08 Prague, Czech Republic, and Artificial Intelligence Center, Faculty ofElectrical Engineering, Czech Technical University in Prague, Czech Republic
E-mail address : [email protected] (V. Marra) Dipartimento di Matematica “Federigo Enriques”, Universit`a degli Studidi Milano, Via Cesare Saldini 50, 20133 Milano, Italy
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