Minimal lambda-theories by ultraproducts
DD. Kesner and P. Viana (Eds.): LSFA 2012EPTCS 113, 2013, pp. 61–76, doi:10.4204/EPTCS.113.8 c (cid:13)
A. Bucciarelli, A. Carraro & A. SalibraThis work is licensed under theCreative Commons Attribution License.
Minimal lambda-theories by ultraproducts
Antonio Bucciarelli Alberto Carraro
PPS, Universit´e Denis Diderot Paris, France { acarraro,buccia } @pps.univ-paris-diderot.fr Antonino Salibra
DAIS, Universit`a Ca’ Foscari Venezia, Italia ∗ [email protected] A longstanding open problem in lambda calculus is whether there exist continuous models of theuntyped lambda calculus whose theory is exactly the least lambda-theory lb or the least sensiblelambda-theory H (generated by equating all the unsolvable terms). A related question is whether,given a class of lambda models, there is a minimal lambda-theory represented by it. In this paper,we give a general tool to answer positively to this question and we apply it to a wide class of webbedmodels: the i-models. The method then applies also to graph models, Krivine models, coherentmodels and filter models. In particular, we build an i-model whose theory is the set of equationssatisfied in all i-models. Lambda-theories are congruences on the set of l -terms which contain b -conversion, providing (sound)notions of program equivalence. Models of the l -calculus are one of the main tools used to study thelattice of l -theories. After the first model, found by Scott in 1969 in the category of complete lattices andScott continuous functions, a large number of mathematical models for l -calculus, arising from syntax-free constructions, have been introduced in various Cartesian closed categories (ccc, for short) of domainsand were classified into semantics according to the nature of their representable functions, see e.g. [3,6, 23]. Scott continuous semantics [24] is the class of reflexive cpo-models, that are reflexive objectsin the category CPO , whose objects are complete partial orders and morphisms are Scott continuousfunctions. The stable semantics (Berry [7]) and the strongly stable semantics (Bucciarelli–Ehrhard [8])are refinements of the continuous semantics, introduced to approximate the notion of “sequential” Scottcontinuous function.Some models of l -calculus, called webbed models, are built from lower level structures called“webs” (see Berline [6] for an extensive survey). The simplest class of webbed models is the classof graph models, which was isolated in the seventies by Plotkin, Scott and Engeler [16, 23, 27] withinthe continuous semantics. The class of graph models contains the simplest models of l -calculus, isitself the easiest describable class, and represents nevertheless a continuum of (non-extensional) lambda-theories. Another example of a class of webbed models, and the most established one, is the class offilter models. It was isolated at the beginning of the eighties by Barendregt, Coppo and Dezani [4], afterthe introduction of the intersection type discipline by Coppo and Dezani [13]. Not all filter models live inScott continuous semantics: for example some of them lack the property of representing all continuousfunctions, and others were introduced for the stable semantics (see Paolini et al. [22], Bastonero et al.[5]).In general, given a class C of models, a natural completeness problem arises for it: whether the classis complete , i.e., for any lambda-theory T there exists a member of C whose equational theory is T .A related question, raised in [6] is the following: given a class C of models of the l -calculus, is there ∗ Work partially supported by the Fondation de Math´ematique de Paris. C ? If this is the case, we say that C enjoys the minimalityproperty . In [15] it was shown that the above question admits a positive answer for Scott’s continuoussemantics, at least if we restrict to extensional reflexive CPO-models. Another result, in the same spirit,is the construction of a model whose theory is lbh , a fortiori minimal, in the w -semantics (whichis different from Scott semantics). However, the proofs of [15] use logical relations, and since logicalrelations do not allow to distinguish terms with the same applicative behavior, the proofs do not carryover to non-extensional models. Similarly, in [10], it is shown that the class of graph models enjoys theminimality property.In this paper, we propose a method to prove that a given class of models enjoys the minimality prop-erty, based on two main ingredients: the finite intersection property (fip) and the ultraproduct property (upp). The fip is satisfied by a class C of models if for all models M , M in C there exists a model M in C whose equational theory is included in T h ( M ) ∩ T h ( M ) . The upp is satisfied in C if for everynon-empty family { M i } i ∈ I of members of C and for every proper ultrafilter U of sets on P ( I ) the ul-traproduct ( (cid:213) i ∈ I M i ) / U can be embedded into a member of C . We show in Theorem 3.1 that if theseconditions are satisfied, then C has the minimality property. An important technical device used in theproof of Theorem 3.1 is Lo´s Theorem: the ultraproduct of a family of models satisfies an (in)equationbetween l -terms if and only if the set of indexes of the component models satisfying it belongs to theultrafilter. Hence, proving the minimality property boils down to exhibiting an appropriate ultrafilter.As an application of this general method, we prove that the class of i-models introduced in [11]enjoys the minimality property. First of all, for every pair of i-models A , B we construct an i-model C such that T h ( C ) ⊆ T h ( A ) ∩ T h ( B ) . This result is obtained via a completion process applied to thecategorical product of A and B , adapted from [11]. In order to prove that the class of i-models enjoysthe upp, we exploit the fact that i-models are webbed models. Given an ultraproduct P of i-models, weconstruct the ultraproduct P ′ of the corresponding webs. It turns out that P ′ is a well defined web. Thenwe show that there exists an embedding from P to the i-model associated with P ′ . We also show how ourproof can be applied to smaller classes of webbed models, like graph models, Krivine models, coherentmodels, and filter models.Although we know that there exists a minimal i-model, its equational theory has not yet been char-acterized. Then the results of this paper do not give a solution to the longstanding open problem whichasks whether there exist continuous models of the untyped lambda calculus whose theory is exactly theleast l -theory lb .The paper is organized as follows. In Section 2 we provide the preliminary notions and resultsneeded in the rest of the paper, in Section 3 we present the general method for showing that a given classof models of the l -calculus has the minimality property, and in Section 4 we apply this method to theclass of i-models. With regard to the lambda-calculus we follow the notation and terminology of [3]. By L and L o , re-spectively, we indicate the set of l -terms and of closed l -terms. We denote ab -conversion by lb .A l -theory is a congruence on L (with respect to the operators of abstraction and application) whichcontains lb . A l -theory is consistent if it does not equate all l -terms, inconsistent otherwise. The setof lambda-theories constitutes a complete lattice w.r.t. inclusion, whose top is the inconsistent lambda-theory and whose bottom is the theory lb . The lambda-theory generated by a set X of identities is the.Bucciarelli, A.Carraro &A.Salibra 63intersection of all lambda-theories containing X .It took some time, after Scott gave his model construction, for consensus to arise on the generalnotion of a model of the l -calculus. There are mainly two descriptions that one can give: the category-theoretical and the algebraic one. Besides the different languages in which they are formulated, thetwo approaches are intimately connected (see Koymans [18]). The categorical notion of model, that ofreflexive object in a Cartesian closed category (ccc), is well-suited for constructing concrete models,while the algebraic one is rather used to understand global properties of models (constructions of newmodels out of existing ones, closure properties, etc.) and to obtain results about the structure of the latticeof l -theories. The main algebraic description of models of lambda-calculus is the class of l -models ,which are axiomatized over combinatory algebras by a finite set of first-order sentences (see Meyer [21],Scott [25], Barendregt [3]). In the following we denote by k and s the so-called basic combinators . Ultraproducts result from a suitable combination of the direct product and quotient constructions. Theywere introduced in the 1950’s by Lo´s.Let I be a non-empty set and let { A i } i ∈ I be a family of combinatory algebras. Let U be a properultrafilter of the boolean algebra P ( I ) . The relation ∼ U , given by a ∼ U b ⇔ { i ∈ I : a ( i ) = b ( i ) } ∈ U ,is a congruence on the combinatory algebra (cid:213) i ∈ I A i . The ultraproduct of the family { A i } i ∈ I , noted ( (cid:213) i ∈ I A i ) / U , is defined as the quotient of the product (cid:213) i ∈ I A i by the congruence ∼ U . If a ∈ (cid:213) i ∈ I A i ,then we denote by a / U the equivalence class of a with respect to the congruence ∼ U . If all members of { A i } i ∈ I are l -models, by a celebrated theorem of Lo´s we have that ( (cid:213) i ∈ I A i ) / U is a l -model too, because l -models are axiomatized by first-order sentences. The basic combinators of the l -model ( (cid:213) i ∈ I A i ) / U are k / U and s / U , and application is given by x / U · y / U = ( x · y ) / U , where the application x · y is definedpointwise.We now recall the famous Lo´s theorem that we will use throughout this paper. Theorem 2.1 (Lo´s) . Let L be a first-order language and { A i } i ∈ I be a family of L -structures indexedby a non-empty set I an let U be a proper ultrafilter of P ( I ) . Then for every L -formula j ( x , . . . , x n ) and for every tuple ( a , . . . , a n ) ∈ (cid:213) i ∈ I A i we have that ( (cid:213) i ∈ I A i ) / U | = j ( a / U , . . . , a n / U ) ⇔ { i ∈ I : A i | = j ( a ( i ) , . . . , a n ( i )) } ∈ U . Information systems were introduced by Dana Scott in [26] to give a handy representation of Scottdomains. An information system is a tuple A = ( A , Con A , ⊢ A , n A ) , where A is a set and n A ∈ A , Con A ⊆ P f ( A ) is a downward closed family containing all singleton subsets of A , and ⊢ A ⊆ Con A × A satisfiesthe four axioms listed below:(I1) if a ∈ Con A and a ⊢ A b , then a ∪ b ∈ Con A (where a ⊢ A b def = ∀ b ∈ b . a ⊢ A b )(I2) if a ∈ a , then a ⊢ A a (I3) if a ⊢ A b and b ⊢ A g , then a ⊢ A g (I4) /0 ⊢ A n A We adopt the following notational conventions: letters a , b , g , . . . are used for elements of A (alsocalled tokens ); letters a , b , c , . . . are used for elements of Con A , usually called consistent sets ; letters4 Minimal lambda-theories by ultraproducts x , y , z , . . . are used for arbitrary elements of P ( A ) . We usually drop the subscripts from Con A and ⊢ A when there is no danger of confusion.A subset x ⊆ A is finitely consistent if each of its finite subsets belongs to Con A . We denote by P c ( A ) the set of all finitely consistent subsets of A . We define an operator ↓ A : P c ( A ) → P c ( A ) bysetting x ↓ A = { a ∈ A : ∃ a ⊆ f x . a ⊢ a } . We may drop the subscript when the underlying informationsystem is clear from the context. Note that ↓ is a monotone map satisfying the following conditions: x ⊆ x ↓ ; x ↓ ↓ = x ↓ and x ↓ = ∪ a ⊆ f x a ↓ . We call point any subset of A which is in the image of ↓ . It iswell-known that the set of points, partially ordered by inclusion, constitutes a Scott domain and any Scottdomain is isomorphic to the set of points of some information system.An approximable relation between two information systems A , B is a relation R ⊆ Con A × B satis-fying the following properties:(AR1) if a ∈ Con A and a R b , then b ∈ Con B (where a R b def = ∀ b ∈ b . a R b )(AR2) if a ′ ⊢ A a , a R b , and b ⊢ B b ′ , then a ′ R b ′ . Inf is the category which has information systems as objects and approximable relations as arrows.The composition of two morphisms R ∈ Inf ( A , B ) and S ∈ Inf ( B , C ) is (using the meta-notation) theirusual relational composition: S ◦ R = { ( a , g ) ∈ Con A × C : ∃ b ∈ Con B . ( a , b ) ∈ R and ( b , g ) ∈ S } . Theidentity morphism of an information system A is ⊢ A .The Cartesian closed structure of Inf is described in [26], and we recall it here for the sake of self-containment.In what follows we use the projection functions fst and snd of a set-theoretic Cartesian product overthe first and second component, respectively. The same notation is extended to finite subsets of theCartesian product. For example, fst ( a ) = { fst ( a ) : a ∈ a } . Definition 2.1.
The Cartesian product of A and B is given by A N B = ( A ⊎ B , Con , ⊢ , n ) whereA ⊎ B = ( { n A } × B ) ∪ ( A × { n B } ) n = ( n A , n B ) a ∈ Con iff fst ( a ) ∈ Con A and snd ( a ) ∈ Con B a ⊢ a iff fst ( a ) ⊢ A fst ( a ) and snd ( a ) ⊢ B snd ( a ) The terminal object is the information system ⊤ whose underlying set contains only one token. Definition 2.2.
The exponentiation of B to A is given by A ⇒ B = ( A ⇒ B , Con , ⊢ , n ) whereA ⇒ B = Con A × B n = ( /0 , n B ) { ( a , b ) , . . . , ( a k , b k ) } ∈ Con iff ∀ I ⊆ [ , k ] . ( ∪ i ∈ I a i ∈ Con A ⇒ { b i : i ∈ I } ∈ Con B ) { ( a , b ) , . . . , ( a k , b k ) } ⊢ ( c , g ) iff { b i : c ⊢ A a i , i ∈ [ , k ] } ⊢ B g The category SD of Scott domains and Scott continuous functions is equivalent to the category Inf of information systems, via a pair of mutually inverse Cartesian closed functors ( · ) + : Inf → SD and ( · ) − : SD → Inf .In particular for an information system A , we have that A + , the set of points of an informationsystem, ordered by inclusion, is a Scott domain. Moreover, the domains [ A + → B + ] and A + × B + areisomorphic (in the category SD ) to the domains ( A ⇒ B ) + and ( A N B ) + , respectively. Let A , B be information systems and let f : A → B be a function. We define two Scott continuousfunctions f • : A + → B + and f • : B + → A + as follows: f • ( x ) = { f ( a ) : a ∈ x } ↓ B ; f • ( y ) = { a : f ( a ) ∈ y } ↓ A .Bucciarelli, A.Carraro &A.Salibra 65for every point x of A and every point y of B . In [11] simple conditions are given under which f cangenerate a retraction pair ( f • , f • ) from A + to B + in the category SD , i.e., f • ◦ f • = id A + . Definition 2.3 ([11]) . Let A , B be information systems. A morphism from A to B is a function f : A → B satisfying the following property:(Mo) a ∈ Con A iff f ( a ) ∈ Con B Definition 2.4 ([11]) . A morphism f : A → B is a b-morphism (resp. f-morphism ) if it satisfies thefollowing property (bMo) (resp. (fMo))(bMo) if f ( a ) ⊢ B f ( a ) , then a ⊢ A a (fMo) if a ⊢ A a , then f ( a ) ⊢ B f ( a ) The “b” (resp. “f”) in the name of the axiom stands for backward (resp. forward). We leave to thereader the easy relativization of the various notions of morphism given in Definition 2.4 to the case inwhich f is a partial map. Proposition 2.2.
Let f : A → B be a b-morphism. Then ( f • , f • ) is a retraction pair from A + into B + .Proof. From (bMo) it follows f • ◦ f • = id A + . Definition 2.5. An i-web is a pair A = ( A , f ) where A is an information system and f : ( A ⇒ A ) → A is a b-morphism. The set of tokens of A is called the web of A . Proposition 2.3.
Let A = ( A , f ) be an i-web. Then A + is a reflexive object in the category SD .Proof. As anticipated, there is a continuous isomorphism q : ( A ⇒ A ) + → [ A + → A + ] and by Propo-sition 2.2 the domain ( A ⇒ A ) + can be embedded into A + via the retraction pair ( f • , f • ) . Therefore ( q ◦ f • , q − ◦ f • ) is the desired retraction pair in the category SD .We set A + = ( A + , q ◦ f • , q − ◦ f • ) and call A + an i-model . Of course, since A + is a reflexive objectin SD , then A + is also a l -model and closed l -terms are interpreted as elements of A + (i.e. as points of A ) as follows: J x K A + r = r ( x ) , where r is any map from Var into A + J l y . M K A + r = { f ( a , a ) : a ∈ J M K A + r [ y : = a ↓ ] } ↓ A J MN K A + r = { b ∈ A : ∃ a ⊆ f J N K A + r . ( a , b ) ∈ { ( a ′ , b ′ ) : f ( a ′ , b ′ ) ∈ J M K A + r } ↓ A ⇒ A } The l -model structure associated to the i-model A + is the following. The basic combinators are k A + = J l xy . x K A + and s A + = J l xyz . xz ( yz ) K A + , and the application operation is given by u · z = { b ∈ A : ∃ a ⊆ f z . ( a , b ) ∈ { ( a ′ , b ′ ) : f ( a ′ , b ′ ) ∈ u } ↓ A ⇒ A } for all points u , z .6 Minimal lambda-theories by ultraproducts An extended abstract type structure ( EATS , for short, [14, Def. 1.1]) is an algebra ( A , ∧ , → , w ) , where“ ∧ ” and “ → ” are binary operations and “ w ” is a constant, such that ( A , ∧ , w ) is a meet-semilatticewith top element w . In the following ≤ denotes the partial order associated with the meet-semilatticestructure. Recall from [14, Def. 2.12,Thm. 2.13] that the filter models living in Scott semantics areobtained by taking the set of filters of EATS s satisfying the following condition:( ∗ ) If V ni = ( a i → b i ) ≤ g → d , then ( V i ∈{ i : g ≤ a i } b i ) ≤ d .Given an EATS ( A , ∧ , → , w ) , the structure A = ( A , P f ( A ) , ⊢ , w ) , where a ⊢ a iff ( V a ) ≤ a , is aninformation system.If the EATS satisfies condition ( ∗ ), then the function f : P f ( A ) × A → A given by f ( a , a ) = ( V a ) → a is a b-morphism, and hence an i-web A = ( A , f ) . The corresponding filter model is exactly thei-model A + (see [12] for the details).In Larsen and Winskel [19] the definition of information system is slightly different: there is nospecial token n . We remark that the corresponding class of i-models generated by the two definitionsis the same. We adopt Scott’s original definition just for technical reasons. With Larsen & Winskel’sdefinition we can capture some other known classes of models, as illustrated below.A preordered set with coherence (pc-set, for short) is a triple ( A , ≤ , ≎ ) , where A is a non-empty set, ≤ is a preorder on A and ≎ is a coherence (i.e., a reflexive, symmetric relation on A ) compatible with thepreorder (see [6, Def. 120]). A pc-set “is” an information system A = ( A , P cohf ( A ) , ⊢ ) , where P cohf ( A ) is the set of finite coherent subsets of A and a ⊢ a iff ∃ b ∈ a . b ≥ a . A pc-web (see [6, Def. 153]) isdetermined by a pc-set together with a map f : P cohf ( A ) × A → A satisfying:(1) f ( a , a ) ≎ f ( b , b ) iff ( a ∪ b ∈ P cohf ( A ) ⇒ a ≎ b )(2) if f ( a , a ) ≤ f ( b , b ) , then a ≤ b and ( ∀ g ∈ b ∃ d ∈ a . g ≤ d ) .A pc-web is a particular instance of i-web and properties (1),(2) say exactly that f is a b-morphism.Krivine webs [6, Sec. 5.6.2] are pc-webs in which ≎ = A × A (so that P cohf ( A ) = P f ( A ) ). Total pairs [6, Sec. 5.5] are Krivine webs in which ≤ is the equality: in fact in this the requirement of f to be ab-morphism boils down to injectivity. Therefore a total pair is simply defined as a set A together withan injection i A : P f ( A ) × A → A ; the underlying information system is A = ( A , P f ( A ) , ∋ ) . The graphmodel associated to the total pair is then the i-model A + , obtained by taking the powerset of A (see [6,Def. 120]). There is usually some ambiguity in the terminology since by “graph model” sometimes ismeant the total pair (as in [9], for example) underlying the model itself. Given a class C of l -models, a natural question to be asked is whether there exists a member A of C suchits equational theory, hereafter noted Eq ( A ) , is contained in the theories of all other members of C : onesuch model A is called minimal in C . This point was raised in print by C. Berline [6] who was mainlyreferring to the classes of webbed models of l -calculus. If a positive answer is obtained, usually it isdone by purely semantical methods and Eq ( A ) does not need to be characterised in the syntactical sense:this is the case of Di Giannantonio et al. [15], in which the authors prove that the class all extensionalreflexive CPOs has a minimal model. Of course if one is able to gather enough information about Eq ( A ) ,then one may be in the position to answer the related completeness question for the class C : is lb (or lbh ) a theory induced by a member of C ? An example of result of this kind can be found again in [15],where the authors construct a model with theory lbh in the w -semantics..Bucciarelli, A.Carraro &A.Salibra 67In this section we give general conditions for a class C of l -models under which we have the guar-antee that C has a minimal model. In the forthcoming Section 4 we apply this general result to the classof i-models and some of its well-known classes of models. Definition 3.1.
A class C of l -models has the finite intersection property (fip, for short) if for every twomembers A , B of C , there exists a member C of C such that Eq ( C ) ⊆ Eq ( A ) ∩ Eq ( B ) . For example the class of all l -models has the fip, and in general every class closed under directproducts has the fip. Every subclass which is axiomatized over the l -models by first-order universalsentences has the fip, but of course these conditions do not hold in general for the classes of webbedmodels, e.g. for the i-models. We will see that they do hold for the filter models.The fip is a property which is weaker than the closure under direct products. Of course a class whichis closed under arbitrary (non-empty) direct products has a minimal model. The next definition isolates aproperty that, together with the fip, can overcome the lack of direct products and guarantee the existenceof minimal models. Definition 3.2.
A class C of l -models has the ultraproduct property (upp, for short) if for every non-empty family { A i } i ∈ I of members of C and for every proper ultrafilter U of sets on P ( I ) the ultraproduct ( (cid:213) i ∈ I A i ) / U can be embedded into a member of C . For example the class of all l -models has the upp, and in general every class closed under ultraprod-ucts has the upp. Every subclass which is axiomatized over the l -models by first-order sentences has theupp, but of course these conditions do not hold in general for the known classes of webbed models, e.g.for the i-models. Theorem 3.1.
Let C be a class of l -models having both the fip and the upp. Then C has a minimalmodel.Proof. Let I be the set of all equations e betweeen closed combinatory terms for which there exists amodel A in C such that A = e . For every e ∈ I , consider the set K e = { J ⊆ f I : e ∈ J } . Since K e ∩ K e ′ = { J ⊆ f I : e , e ′ ∈ J } 6 = /0 for all e , e ′ ∈ I , then there exists a non-principal ultrafilter U on P f ( P f ( I )) containing the family ( K e : e ∈ I ) . By the finite intersection property of the class C , for every J ⊆ f I there exists a model A J in C such that e Eq ( A J ) for every e ∈ J . Let { A J } J ⊆ f I be the family composedby these models and consider the ultraproduct P U = ( (cid:213) J ⊆ f I A J ) / U . Let e ∈ I be a closed equationand let X e = { J ⊆ f I : A J = e } . Then X e ⊇ K e ∈ U , so that X e belongs to the ultrafilter U . Since e is a closed first-order formula, by Lo´s Theorem 2.1 P U = e . Since e was an arbitrary equation in I , we have that P U = e for every e ∈ I , so that Eq ( P U ) ⊆ T A ∈ C Eq ( A ) . Finally, since the class C has the ultraproduct property, then there exists a model B in C such that P U embeds into B . ThenEq ( B ) = Eq ( P U ) ⊆ T A ∈ C Eq ( A ) ⊆ Eq ( B ) and we get the desired conclusion. Corollary 3.2.
Let C be a class of l -models which has the fip and is closed under ultraproducts. Then C has a minimal model. We conclude the section by giving some other general results that can be proved by just assumingthe fip and the upp for a class C of l -models. In particular we prove a compactness theorem for lambda-theories whose equations hold in members of C . We also prove that, if there exists an easy l -term in C ,then there exists a continuum of different equational C -theories. In other words, there are uncountablymany different lambda-theories induced by models of the class C . Theorem 3.3 (Compactness) . Let C be a class of l -models having the upp, and let E be a set of equa-tions between closed l -terms. If every finite subset of E is satisfied by a member of C , then E itself issatisfied by a member of C . Proof.
For every e ⊆ f E , let K e = { d ⊆ f E : e ⊆ d } and let A e ∈ C be a model satisfying e . Let U bea proper ultrafilter on P f ( P f ( E )) containing K e for every e ⊆ f E . Then the ultraproduct ( (cid:213) e ⊆ f E A e ) / U satisfies E . Finally by the upp there exists a model B in C such that ( (cid:213) e ⊆ f E A e ) / U embeds into B , andthus has the same lambda-theory. We conclude that B satisfies E .Let C be a class of l -models. A closed l -term M is C -easy if for every closed l -term N there existsa member B of C such that J M K B = J N K B . Theorem 3.4.
Let C be a class of l -models having the upp such that there exists a C -easy l -term. Thenthere exist uncountably many C -theories.Proof. Let M be a C -easy l -term. For n ≥
1, we let p n ≡ l x . . . x n . x n . We prove that for every n ≥ M p n is C -easy.Let X = ( N n ) n ≥ be an arbitrary infinite sequence of closed bh -normal l -terms and define E ( X ) = { M p n = N n : n ≥ } . Let K = { M p n = N n , . . . , M p n k = N n k } be a finite subset of E ( X ) . Without loss ofgenerality, we may assume that n < · · · < n k . Let y be a fresh variable and define inductively Z : = y I · · · I | {z } n − N n ; Z m + : = Z m I · · · I | {z } n m + − n m − N n m Now set Z = l y . Z k . Since M is C -easy, then there is a member A of C such that A | = M = Z . Therefore A | = M p n i = Z p n i = N n i for all i = , . . . , k so that K ⊆ Eq ( A ) . Since every finite subset of E ( X ) issatisfied by a member of C , then by Theorem 3.3 E ( X ) itself is satisfied by a member of C , i.e. thereexists a member A X of C such that E ( X ) ⊆ Eq ( A X ) . Moreover if X and Y are two different infinitesequences of closed bh -normal l -terms, then Eq ( A X ) = Eq ( A Y ) . The result then follows from the factthat there are uncountably many infinite sequences of closed bh -normal l -terms. In the present section we apply the general results developed in Section 3. In particular we prove thatthe class of i-models has both the finite intersection property and the ultraproduct property. Then wecomment on how these general results also apply to other well-known classes of webbed models.
The goal of the first part of this section is to prove that for every pair A , A of i-webs there exists ani-web B such that Eq ( B + ) ⊆ Eq ( A + ) ∩ Eq ( A + ) . Such result would be trivial if the categorical product A N A could always be endowed with a suitable structure of i-web, but this is not the case. The bestthat we can do in general is to make A N A into a partial i-web . A partial i-web in general is a pair A = ( A , f A ) , where f A : A ⇒ A ⇀ A is a partial b-morphism. In particular, A N A is a partial i-webif we set if we can set f ( a , a ) = ( n A , n A ) if a ⊆ { ( n A , n A ) } and a = ( n A , n A )( n A , f A ( snd ( a ) , snd ( a ))) if a ∪ { a } ⊆ f { n A } × A ( f A ( fst ( a ) , fst ( a )) , n A ) if a ∪ { a } ⊆ f A × { n A } A partial i-web does not give in general an i-model, but we can complete it to an i-web through a limitprocess that involves countably many extension steps..Bucciarelli, A.Carraro &A.Salibra 69We say that B is an extension of S , notation S (cid:22) B , if S ⊆ B , Con S = Con B ∩ P f ( S ) , ⊢ S = ⊢ B ∩ ( Con S × S ) . We say that B is an extension of S , notation S (cid:22) B , if S (cid:22) B and f S is the restriction of f B to Con S × S .Let us call B the result of the (yet undefined) completion process of A N A . Of course B must besomehow related to the original i-webs A and A . In particular, we want that for every closed l -term M if ( n A , b ) ∈ J M K B + (resp. ( a , n A ) ∈ J M K B + ), then b ∈ J M K A + (resp. a ∈ J M K A + ) because this willguarantee that Eq ( B + ) ⊆ Eq ( A + ) ∩ Eq ( A + ) . We will achieve this property by means of the notion of f-morphism of partial i-webs. Notation.
Let f : A ⇀ B be a partial function. We write do ( f ) to indicate the domain of f anddo ( f ) to indicate the complement of do ( f ) in A . We define f : P f ( B ) → P f ( C ) and e f : ( P f ( B ) × B ) → ( P f ( C ) × C ) as follows: f ( b ) = { f ( b ) | b ∈ b , b ∈ do ( f ) } and e f ( b , b ) = ( f ( b ) , f ( b )) . Hence e f : P f ( P f ( B ) × B ) → P f ( P f ( C ) × C ) . Definition 4.1 ([11]) . Let B , C be partial i-webs. An f-morphism from B to C is an f-morphism y : B → C satisfying the following additional property:(iMo) if ( a , b ) ∈ do ( f B ) , then ( y ( a ) , y ( b )) ∈ do ( f C ) and y ( f B ( a , b )) = f C ( y ( a ) , y ( b )) The following proposition explains that, in general, f-morphisms of i-webs “commute” well to theinterpretation of l -terms. Proposition 4.1 ([11]) . Let B , C be i-webs, let y : B → C be an f-morphism of i-webs, and let M be aclosed l -term. If a ∈ J M K B + , then y ( a ) ∈ J M K C + . We remark that the two projection functions fst and snd are f-morphisms of partial i-webs from A N A to A and A , respectively.Our goal now is to construct a series of triples { ( S n , y n , y n ) } n ≥ such that S n (cid:22) S n + and y in : S n → A i ( i = ,
2) is an f-morphism of partial i-webs such that y in + extends y in ( i = , ( S , y , y ) where S : = A N A , y = fst, and y = snd. All subsequent triples are constructed via an algorithm that, given ( S n , y n , y n ) as input,returns ( S n + , y n + , y n + ) . The union of all partial i-webs and all f-morphisms of partial i-webs finallygives an i-web S w (called completion ) and two f-morphisms y i w ( i = ,
2) of i-webs that allow to showthat Eq ( S + w ) ⊆ Eq ( A + ) ∩ Eq ( A + ) .The 0-th stage of the completion process, i.e., the triple ( S , y , y ) has already been described.Now assuming we reached stage n , we show how to carry on with stage n + Definition 4.2. • S n + = S n ∪ do ( f S n ) • Con S n + is the smallest family of sets x ⊆ f S n ∪ do ( f S n ) such that either(1) there exist a ∈ Con n and X ∈ Con S n ⇒ S n such that X ⊆ do ( f S n ) and x = a ∪ X and y in ( a ) ∪ f A i ( f y in ( X )) ∈ Con A i (i = , ) or(2) there exists X ∈ Con S n ⇒ S n such that x ⊆ f ( X ∩ do ( f S n )) ∪ ( f S n ( X ∩ do ( f S n ))) ↓ S n • a ⊢ S n + a iff either a ∩ S n ⊢ S n a or a ∈ a • n S n + = n S n • f S n + ( a , a ) = f S n ( a , a ) if ( a , a ) ∈ do ( f S n )( a , a ) if ( a , a ) ∈ do ( f S n ) undefined if ( a , a ) ∈ ( S n + ⇒ S n + ) − ( S n ⇒ S n ) • for i = , we set y in + ( a ) = ( y in ( a ) if a ∈ S n f A i ( y in ( b ) , y in ( b )) if a = ( b , b ) ∈ S n + − S n Theorem 4.2.
We have that(i) S n + = ( S n + , Con S n + , ⊢ S n + , n S n + ) is an information system such that S n (cid:22) S n + ,(ii) S n + = ( S n + , f S n + ) is a partial i-web such that S n (cid:22) S n + ,(iii) y in + : S n + → A i (i = , ) is an f-morphism of partial i-webs.Proof. (i) We show that S n + is an information system, checking the properties (I1)-(I4) (see beginningof Section 2.3).(I1) Suppose a ∈ Con S n + and a ⊢ S n + b . If a has been added to Con S n + by clause (1), then exists i ∈ { , } , a ′ ∈ Con S n and X ∈ Con S n ⇒ S n such that X ⊆ do ( f S n ) and a = a ′ ∪ X and y in ( a ′ ) ∪ f A i ( f y in ( X )) ∈ Con A i . Since a ⊢ S n + b , then b = b ′ ∪ X , for some b ′ ∈ Con S n such that a ′ ⊢ S n b ′ . Now y in is a morphism, so that y in ( b ′ ) ∪ f A i ( f y in ( X )) ∈ Con A i . Therefore b is added to Con S n + by clause(1).If a has been added to Con S n + by clause (2), then also b is added to Con S n + by the same clause.(I2) If a ∈ a , then a ⊢ S n + a by definition of ⊢ S n + .(I3) Suppose a ⊢ S n + { a , . . . , a k } and { a , . . . , a k } ⊢ S n + g . If g ∈ { a , . . . , a k } then clearly a ⊢ S n + g .Otherwise { a , . . . , a k } ∩ S n ⊢ S n g and since a ∩ S n ⊢ S n { a , . . . , a k } ∩ S n we can conclude using theproperty (I3) of S n .(I4) Immediate.Finally it is immediate to see that S n (cid:22) S n + .(ii) Note that the fact that S n (cid:22) S n + automatically implies S n ⇒ S n (cid:22) S n + ⇒ S n + . Now we provethat f S n + : S n ⇒ S n → S n + is a total b-morphism, so that it is automatically a partial b-morphism from S n + ⇒ S n + to S n + .(Mo) We must show that X ∈ Con S n ⇒ S n iff ( X ∩ do ( f S n )) ∪ ( f S n ( X ∩ do ( f S n ))) ∈ Con S n + . If X ∈ Con S n ⇒ S n ,then ( X ∩ do ( f S n )) ∪ ( f S n ( X ∩ do ( f S n ))) is in Con S n + by clasuse (2).Let x = ( X ∩ do ( f S n )) ∪ ( f S n ( X ∩ do ( f S n ))) ∈ Con S n + . If x is added to Con S n + by clause (1),then there exist i ∈ { , } , a ∈ Con n and Y ∈ Con S n ⇒ S n such that Y ⊆ do ( f S n ) and x = a ∪ Y and y in ( a ) ∪ f A i ( f y in ( Y )) ∈ Con A i . Therefore Y = ( X ∩ do ( f S n )) and a = ( f S n ( X ∩ do ( f S n ))) . Now wehave y in ( a ) ∪ f A i ( f y in ( Y )) = y in (( f S n ( X ∩ do ( f S n )))) ∪ f A i ( f y in (( X ∩ do ( f S n ))))= f A i ( f y in (( X ∩ do ( f S n )))) ∪ f A i ( f y in (( X ∩ do ( f S n ))))= f A i ( f y in ( X )) Since y in ( a ) ∪ f A i ( f y in ( Y )) is in Con A i by hypothesis, then so is f A i ( f y in ( X )) and since both f A i and y in are morphisms of information systems, then so is their composition f A i ◦ y in , meaning that X ∈ Con S n ⇒ S n .If x is added to Con S n + by clause (2), then evidently X ∈ Con S n ⇒ S n .(bMo) We must show that f S n + ( X ) ⊢ S n + f S n + ( a , a ) implies X ⊢ S n + ⇒ S n + ( a , a ) . There are two cases to bedealt with. If ( a , a ) ∈ do ( f S n ) , then f S n ( X ) ∩ S n ⊢ S n f S n ( a , a ) and we derive f S n ( X ∩ do ( f S n )) ⊢ S n .Bucciarelli, A.Carraro &A.Salibra 71 f S n ( a , a ) so that by (bMo) for f S n we have that X ∩ do ( f S n ) ⊢ S n ⇒ S n ( a , a ) and hence X ⊢ S n ⇒ S n ( a , a ) .If ( a , a ) do ( f S n ) , then ( a , a ) = f S n + ( a , a ) ∈ f S n + ( X ) , so that ( a , a ) ∈ X and thus X ⊢ S n ⇒ S n ( a , a ) .(iii) Now we prove that y in + ( i = , ) is an f-morphism of i-webs.(Mo) ( ⇒ ) Suppose x ∈ Con S n + . We consider the clauses (1) and (2) of the definition of Con S n + .If x is added by clause (1), i.e. x = a ∪ X for suitable a and X , then y in + ( x ) = y in ( a ) ∪ f A i ( f y in ( X )) ∈ Con A i , by clause (1) itself.If x is added by clause (2), then x ⊆ f ( X ∩ do ( f S n )) ∪ ( f S n ( X ∩ do ( f S n ))) ↓ S n , for some X ∈ Con S n ⇒ S n .Now let y = ( X ∩ do ( f S n )) ∪ f S n ( X ∩ do ( f S n )) . We first observe that y in + ( y ) = f A i ( f y in ( X ∩ do ( f S n ))) ∪ y in ( f S n ( X ∩ do ( f S n )))= f A i ( f y in ( X ∩ do ( f S n ))) ∪ f A i ( f y in ( X ∩ do ( f S n )))= f A i ( f y in ( X )) This proves that y in + ( y ) ∈ Con A i . Now using property (fMo) y in we obtain that y in + ( y ) ⊢ A i y in + ( x ) , and hence y in + ( x ) ∈ Con A i .( ⇐ ) By the very definition of Con S n + , in particular by the clause (1).(fMo) Suppose a ⊢ S n + a . If a ∈ a , then of course y in + ( a ) ⊢ A i y in + ( a ) . If a ∩ S n ⊢ S n a , then y in + ( a ) = y in + ( a − S n ) ∪ y in ( a ∩ S n ) ⊢ A i y in ( a ∩ S n ) ⊢ A i y in ( a ) (iMo) Let ( a , a ) ∈ S n ⇒ S n . Then y in + ( f S n + ( a , a )) = f A i ( y in ( a ) , y n ( a )) = f A i ( y in + ( a ) , y in + ( a )) , bydefinition of y in + and the fact that it extends y in .The completion of the triple ( A N A , p , p ) is the triple ( S w , y w , y w ) , where S w = ( S w , Con S w , ⊢ S w , n S w ) and S w = ( S w , f S w ) are given by the following data: S w : = S m < w S m Con S w : = S m < w Con S m ⊢ S w : = S m < w ⊢ S m n S w : = n A N A f S w : = S m < w f S m y i w : = S m < w y im ( i = , ) Lemma 4.3. S w is an i-web and y i w : S w → A i (i = , ) is an f-morphism of i-webs.Proof. Indeed S w is an information system as a consequence of Theorem 4.2(i). Moreover the map f S w is total and it is easy to prove that it is a b-morphism from S w ⇒ S w using the fact that for every n themap f S n + is a partial b-morphism (Theorem 4.2(ii)). Similarly one can prove that y i w is an f-morphismof i-webs from S w to A i ( i = ,
2) simply using the fact that for every n the map y in is an f-morphismfrom the partial i-web S n to the i-webs A i ( i = ,
2) (Theorem 4.2(iii)).
Theorem 4.4. Eq ( S + w ) ⊆ Eq ( A + ) ∩ Eq ( A + ) .Proof. Suppose M = N Eq ( A + ) ∩ Eq ( A + ) . Suppose, w.l.o.g., that M = N Eq ( A + ) . Then there exists a ∈ A such that a ∈ J M K A + − J N K A + . It is not difficult to check that a ∈ J M K A + implies ( a , n A ) ∈ J M K S + w , since S w extends A N A . Now suppose, by way of contradiction, that ( a , n A ) ∈ J N K S + w . Since y w ( a , n A ) = a , by Proposition 4.1 we have that a ∈ J N K A + , which is a contradiction. This proves that ( a , n A ) ∈ J M K S + w − J N K S + w , so that M = N Eq ( S + w ) .2 Minimal lambda-theories by ultraproductsIn Section 2.4.1 we indicate how some of the most known classes of webbed models are recovered asparticular instances of i-models (more details for Filter Models are in [12]). Along these lines the notionof partial i-web generalizes those of partial pair [6] (related to graph models) as well as the notions ofpartial webs of the other types.The idea of partial pair and of a completion for obtaining a graph model generalizes the constructionof the Engeler model and the of the Plotkin–Scott P w model. It was initiated by Longo in [20] andfurther developed and applied by Kerth [17]. Definition 4.2 is the core of a completion of i-webs thatfurther generalizes Longo and Kerth’s work. As such, it can be adapted case by case so that the entirecompletion adapts to the various instances of i-webs in the sense that if we start with partial pair, at theend we obtain a total pair, if we start with a partial pcs-web, we end up in a total pcs-web etc.Of course Theorem 4.4 proves the finite intersection property for the class of i-models, but in viewof the above discussion it can also give proofs of the finite intersection property for the subclasses ofmodels mentioned in section 2.4.1.For the particular case of graph models the fip was proved by Bucciarelli&Salibra [10, 9], via aconstruction that they call weak product which has the same spirit of our completion method. For theother classes of models the fip was not known to hold. For the particular case of filter models one mayprove the fip as a simple consequence of the closure of filter models under the contruction of directproducts, a result that does not appear in the literature and we do not sketch here. In this subsection we deal with the ultraproduct property for the class of i-models: for every non-emptyfamily { A i } i ∈ I of i-webs and every ultrafilter U on P ( I ) the ultraproduct ( (cid:213) i ∈ I A + i ) / U can be embeddedinto an i-model.Let J be a non-empty set and let { A j } j ∈ J be a family of information systems and let U be a properultrafilter on P ( J ) . Define a binary relation q U on (cid:213) j ∈ J A j by setting ( a , b ) ∈ q U ⇔ { j ∈ J : a ( j ) = b ( j ) } ∈ U . Note that q U is an equivalence relation on (cid:213) j ∈ J A j ; we write ( (cid:213) j ∈ J A j ) / U for the quotient of (cid:213) j ∈ J A j by q U . As a matter of notation, for every a ∈ (cid:213) j ∈ J A j we let a / U = { b ∈ (cid:213) j ∈ J A j : ( a , b ) ∈ q U } and for every finite subset a ⊆ f (cid:213) j ∈ J A j , we let a / U = { a / U : a ∈ a } , i.e., a / U is the finite subset of ( (cid:213) j ∈ J A j ) / U constituted by the q U -equivalence classes of the tokens of a . Since each element a ∈ a is a J -indexed sequence, we denote by a ( j ) the j -th projection of a and we let a ( j ) = { a ( j ) : a ∈ a } . Definition 4.3.
We define an information system P U = ( P U , Con U , ⊢ U , n U ) as follows:P U = ( (cid:213) j ∈ J A j ) / U n U = ( ll j . n A j ) / Ua / U ∈ Con U iff { j ∈ J : a ( j ) ∈ Con A j } ∈ Ua / U ⊢ U a / U iff { j ∈ J : a ( j ) ⊢ A j a ( j ) } ∈ UWe also define an i-web P U = ( P U , f P U ) by setting f P U ( a / U , a / U ) = ( ll j . f A j ( a ( j ) , a ( j ))) / U .
We leave to the reader the easy verification of the fact that P U and P U indeed are an informationsystem and an i-web, respectively.We conclude the second main theorem of the section, the one that deals with the ultraproduct prop-erty. Let { A j } j ∈ J be a family of i-webs, let U be an ultrafilter over P ( J ) and let P U be the i-web ofDefinition 4.3. Since P U is an i-web, then P + U is a reflexive Scott domain and hence a l -model. Onthe other hand each i-web A j gives rise to a reflexive Scott domain A + j , which is a l -model. Then ( (cid:213) j ∈ J A + j ) / U is an ultraproduct of l -models, and thus again a l -model..Bucciarelli, A.Carraro &A.Salibra 73 Theorem 4.5.
There exists an embedding of combinatory algebras from the l -model ( (cid:213) j ∈ J A + j ) / U intothe l -model P + U .Proof. The proof is rather technical and cumbersome. For this reason we state and prove a particularcase that only deals with graph models.We let x , y , . . . range over elements of (cid:213) j ∈ J A + j , so that x ( j ) ∈ A + j is a point of the graph model A j .We write x / U for the equivalence class of x w.r.t. the congruence on (cid:213) j ∈ J A + j given by x ∼ U y ⇔ { j ∈ J : x ( j ) = y ( j ) } ∈ U , i.e., x / U = { y ∈ (cid:213) j ∈ J A + j : x ∼ U y } .Recall that ∼ U is the relation on (cid:213) j ∈ J A j given by a ∼ U b ⇔ { j ∈ J : a ( j ) = b ( j ) } ∈ U . We definea map f : ( (cid:213) j ∈ J A + j ) / U → P + U as follows: f ( x / U ) = { a / U : a ∈ (cid:213) j ∈ J A j , ∀ j ∈ J . a ( j ) ∈ x ( j ) } It is easy to show that the definition of f is independent of the choice of the representatives of ∼ U -equivalence classes as, for all y ∈ x / U , we have { j ∈ J : y ( j ) = x ( j ) } ∈ U .We prove that f is injective. Suppose x / U = y / U and let Z = { j ∈ J : x ( j ) = y ( j ) } . Define X = { k ∈ J : x ( k ) ⊆ y ( k ) } and Y = { k ∈ J : y ( k ) ⊆ x ( k ) } . Then X ∩ Y = Z U . This means that it is notpossible that both X and Y belong to the ultrafilter U . Assume that X U . Then for every k ∈ J − X wehave x ( k ) y ( k ) , so that for each k ∈ J − X there exists an element g k ∈ A k such that g k ∈ x ( k ) − y ( k ) .Let d ∈ (cid:213) j ∈ J A j be an arbitrary sequence and let b ∈ (cid:213) j ∈ J A j be defined by b ( i ) = g i for i ∈ J − X and b ( i ) = d ( i ) for i J − X . By definition of f we have b / U ∈ f ( x / U ) , while b / U f ( y / U ) , so that f ( x / U ) = f ( y / U ) .Now we prove that f is homomorphism of combinatory algebras. We start proving that f preservesapplication. We have f ( x / U ) · f ( y / U ) = { a / U : ∃ a / U ⊆ f f ( y / U ) . f P U ( a / U , a / U ) ∈ f ( x / U ) } = { a / U : ∃ a ⊆ f (cid:213) j ∈ J A j . ∀ g ∈ a . ∀ j ∈ J . g ( j ) ∈ y ( j ) and ∀ i ∈ J . f A i ( a ( i ) , a ( i )) ∈ x ( i ) } = { a / U : ∀ j ∈ J . ∃ a ⊆ f y ( j ) . f A j ( a , a ( j )) ∈ x ( j ) } = { a / U : ∀ j ∈ J . a ( j ) ∈ { b ∈ A j : ∃ a ⊆ f y ( j ) . f A j ( a , b ) ∈ x ( j ) }} = { a / U : ∀ j ∈ J . a ( j ) ∈ x ( j ) · y ( j ) } = f (( x · y ) / U )= f ( x / U · y / U ) We now regard the basic combinators. Recall that by definition for each j ∈ J we have k A + j = J l xy . x K A + j = { f A j ( a , f A j ( b , b )) : b ∈ a } . Then f ( k ( (cid:213) j ∈ J A + j ) / U ) = f (( k (cid:213) j ∈ J A + j ) / U )= { a / U : a ∈ (cid:213) j ∈ J A j , ∀ j ∈ J . a ( j ) ∈ k A + j } = { f P U ( a / q U , f P U ( b / U , b / U )) : b / U ∈ a / U } = J l xy . x K P + U = k P + U Similarly f ( s ( (cid:213) j ∈ J A + j ) / U ) = s P + U .4 Minimal lambda-theories by ultraproductsWe remark that in the general case in which all the A j ( j ∈ J ) and P U are i-webs the map f : ( (cid:213) j ∈ J A + j ) / U → P + U is defined as f ( x / U ) = { a / U : a ∈ (cid:213) j ∈ J A j , ∀ j ∈ J . a ( j ) ∈ x ( j ) } ↓ P U .We remarked at the end of Section 4.2 that the fip can be derived for subclasses by suitably modifyingthe general construction detailed for i-models. Also the upp holds for the various classes of models. Herewe proved it for graph models, because it looks it looks very clear for this case, but the proof can beadapted (adding details and complication) to the other cases.Summing up, graph models, pcs-models, Krivine models, filter models and in general i-models haveboth the fip and the upp. For this reason Theorem 3.1 applies to all these classes, producing a minimalmodel in each case. It is known that there exist filter-easy terms [1] as well as graph-easy terms [2](for example ( l x . xx )( l x . xx ) ), and every graph-easy term is also pcs-easy and Krivine-easy, since thelatter classes contain the graph models. Therefore Theorem 3.3 and Theorem 3.4 both hold for all theseclasses, saying that each one of them induces a continuum of lambda-theories. We have presented a method for proving that a given class of models of the l -calculus has a minimalelement, i.e., an element whose l -theory is the intersection of all the l -theories represented in theclass. We have applied this method to the class of i-models, a subclass of Scott models defined in [11],containing several well-known instances of “webbed” models like the graph-models and the filter modelsliving in the category of Scott domains.Various extensions of this work can be explored, both toward the proof that the whole class of Scottmodels has the minimality property, and more generally toward the application of the method to otherclasses of models of the l -calculus.Concerning the former extension, a preliminary result would be the finite intersection property for thewhole class of Scott models, the completion method described in Section 4 being adapted to i-models.More generally, it is interesting to notice that webs, even beyond i-webs, are first-order axioma-tisable, hence closed by ultraproducts (by the way, this observation is an alternative way of showingthat Definition 4.3 is sound). By providing a first-order axiomatisation of sentences like A + (cid:15) M = N ,for given terms M , M and web A , we could invoke Lo´s theorem for showing that ( (cid:213) j ∈ J A j ) / U ) + and ( (cid:213) j ∈ J A + j ) / U have the same theory, and hence for deriving a strong form of the ultraproduct propertyfor the class of models corresponding to the considered webs.We conclude this section by providing an outline of a first-order axiomatisation of reflexive informa-tion systems. Let A = ( A , Con A , ⊢ A , n A ) be an information system. A can be defined as a first-orderstructure as follows: for every n ≥
1, let C n be an n -ary predicate and R n + be an ( n + ) -ary predicatewhose intended meanings are: C n ( a , . . . , a n ) ↔ { a , . . . , a n } ∈ Con A . and R n + ( a , . . . , a n , b ) ↔ { a , . . . , a n } ⊢ A b . Then, it is very easy to axiomatise information systems as universal Horn formulas:1. ∀ a . C ( a ) ;2. ∀ a . . . a n . C n ( a , . . . , a n ) → C k ( a i , . . . , a i k ) if k ≤ n and 1 ≤ i j ≤ n ;3. ∀ a . . . a n b . R n + ( a , . . . , a n , b ) → C n + ( a , . . . , a n , b ) ;.Bucciarelli, A.Carraro &A.Salibra 754. ∀ a . . . a n b . R n + ( a , . . . , a n , b ) → R n + ( a s ( ) , . . . , a s ( n ) , b ) , for every permutation s ;5. ∀ a . . . a n b . . . b k g . ( V ≤ i ≤ k R n + ( a , . . . , a n , b i )) ∧ R k + ( b , . . . , b k , g ) → R n + ( a , . . . , a n , g ) ;6. ∀ a . . . a n . C n ( a , . . . , a n ) → R n + ( a , . . . , a n , a i ) ;7. R ( n ) , for a constant n .In a similar but more complicated way it is possible to find a first-order axiomatisation of what is an ex-ponent and a reflexive object in the category Inf . Thus, an untraproduct of reflexive information systemsis again a reflexive information system. It deserves to be studied how first-order closure properties ofinformation systems can be transferred to the category SD of Scott domains. References [1] F. Alessi, M. Dezani-Ciancaglini & F. Honsell (2001):
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