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Duality for Lattice-Ordered Algebras and for Normal Algebraizable Logics Chrysa s Hartonas University of Leicester Dept of Mathematics and Computer Science University Road LE1 7RH Leicester, UK [email protected] February 7, 1997

Abstract

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (in uenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality. In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jonsson and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jonsson-Tarski additive operators. Representation of `-algebras is extended to full duality. In part III we discuss applications in logic of the framework developed. Speci cally, logics with restricted structural rules give rise to lattices with normal operators (in our sense), such as the Full Lambek algebras (FL-algebras) studied by Ono in [36]. Our Stone-type representation results can be then used to obtain canonical constructions of Kripke frames for such systems, and to prove a duality of algebraic and Kripke semantics for such logics.  The present paper is largely based on the author's dissertation at Indiana University, departments

of Mathematics and of Philosophy. The author has bene ted from useful discussions at I.U. with J.M. Dunn (Philosophy), K.J. Barwise (Mathematics) and Lawrence Moss (Mathematics). An early version appeared as a technical report at I.U. with title \Lattices with Additional Operators". Parts or preliminary versions have been presented at the universities of Toronto, Pittsburgh, Amsterdam, at the annual 1994 ASL meeting in Gainsville, Florida, and of course at Indiana and Leicester. Many thanks to the participants of the respective seminars for useful comments and suggestions.

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1 Introduction The aim of this paper is to develop a representation and duality theory for general lattices and for lattice-ordered algebras (`-algebras) over a lattice that may not be distributive. Both the motivation and intended application is: (1) to provide a systematic framework for constructing canonical frames for (possibly non-distributive) logical calculi and fragments theoreof 1 , and (2) to establish a duality of algebraic and Kripke semantics for such systems. Concrete examples of the construction and duality are given in the last part of the paper. The representation theory of Boolean algebras with operators has been rst developed by Jonsson and Tarski in [30], extending the Stone representation [41] of Boolean algebras (as the algebras of clopens of their ultra lter spaces). A number of authors2 have variously extended the Jonsson-Tarski results, in particular to the case of distributive lattices with additional operators3. `-algebras on a (non distributive) lattice arise naturally as the Lindenbaum algebras of logical systems that drop a combination of the structural rules, namely Weakening, Contraction, Exchange and Association. For a fairly broad algebraic approach to such systems we refer the reader to Ono [36], who introduces a notion of an FL-algebra (a Full Lambek algebra), as an algebra arising as the Lindenbaum algebra of some calculus obtained from the Gentzen system LJ for Intuitionistic logic (or from the system LK for Classical Logic). Various kinds of FLalgebras correspond to various combinations of structural rules dropped. We abstract a notion of a lattice-ordered algebra with normal operators (de nition 6.1, part II) and show how to construct canonical Kripke frames with (n + 1)-ary relations representing normal operators (section 7, part II). Representation and duality for `-algebras relies on having an appropriate representation theorem for general lattices, just like the tradition of similar theorems started in [30] relies on the Stone [41] representation of Boolean algebras and the Stone [42] or the Priestley [37] representation of distributive lattices. Part I is taken up with establishing a simple solution to this problem4 . Our lattice representation is inspired by and abstracts over both Goldblatt's [17] and Urquhart's [46]. It is based on the simple idea that since meets and joins are order-dual operators it should be possible, roughly speaking, to reduce the representation problem for lattices to that for semilattices. In part III, we return to what has been the motivation for the framework developed here. As an example, we show how our framework applies to the various FL-algebras 1 Such as (fragments of) Linear Logic, non-distributive

Relevance Logic and various extensions of the Lambek Calculus. 2 We defer more details for the introduction to part II. 3 We should mention at least Dunn's gaggle theory [13], [14] at this point, as our project can be seen as essentially completing the project that Dunn started, namely to provide a systematic construction of Kripke frames for various logical calculi. We go here beyond this project in that we also address the question of duality of algebraic and Kripke semantics for these systems. 4 The present author has addressed the lattice representation and duality problem before: In [3] with Gerard Allwein, extending Urquhart's [46] topological representation of general lattices to a full duality and in [27] with J. Michael Dunn. We present here a new solution, which is simpler in many respects.

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and point out some diculties of existing work relating to the possibility of an independent treatment of distinguished fragments of logical calculi. In particular, we discuss the question of the semantic clause for disjunction in the absence of distribution. The scope of the framework we develop is very comprehensive and we attempt to provide a characterization of the types of logical systems (normal algebraizable logics, de nition 9.2, part III) that the framework applies to. Our semantic treatment applies to such familiar systems as classical and intuitionistic logic and normal modal logics even though the canonical frames we construct are non-standard. The advantage, however, is that we need make no assumption of a choice principle which is indespensable in the case of standard canonical frames for these systems when using the Stone or Priestley representation theorems for Boolean algebras and distributive lattices.

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Part I Lattice Duality 2 Preliminaries Boolean algebras arise as algebras of subsets of a set, closed under the set-theoretic operations of union, intersection and complementation ( elds of subsets). The representation problem for a Boolean algebra is then to establish that every such algebra is isomorphic to a eld of subsets of some set (similarly for distributive lattices, with rings of subsets replacing elds). The solution to this problem given by Marshal Stone [41] is by now well known. Stone's original idea was to model an element b of a Boolean algebra by the set of homomorphisms h into the two-element Boolean lattice 2 such that hb = 1. Such homomorphisms are completely determined by the inverse image h?1 (1), which can be easily seen to be an ultra lter (a maximal lter). The same idea applies to the case of distributive lattices, modeling an element of a distributive lattice D by the morphisms h : D ! 2 that \make it true" (map it to 1). Such a map is also completely determined by h?1 (1), which is now a prime lter of the lattice. The idea has been generalized (cf Johnstone [29]) to the case of complete Heyting algebras (locales), where h?1 (1) prime lter (containing some element ai whenever it contains the join Wis a completely 5. a ) i2I i Unfortunately, for a general lattice L, and a lattice morphism h : L ?! 2, the inverse image of 1 is still a prime lter. Modeling then an element of the lattice by the set of lattice morphisms into 2 that map it to the top element 1, or equivalently by the set of prime lters containing it, forces an \interpretation" of joins as unions. We resort to the use of hemimorphisms, i.e. morphisms from a lattice L to the two-element lattice 2 that only preserve meets6 . We note that for a hemimorphism h : L ?! 2, where L is an arbitrary lattice, the inverse image h?1 (1) is a lter of the lattice and conversely every lter determines such a hemimorphism. The rationale behind the choice for hemimorphisms is the (obvious!) idea that a lattice is merely 5 Boolean algebras are particular kinds of rings. Stone represents a Boolean ring B as a function ring, considering the homomorphisms into Z2 (the two element Boolean ring). The question whether every ring can be represented as a function ring, in particular as a ring of functions in to the reals jR, has been addressed by various authors for particular classes of rings. Our motivation here is logical and we therefore seek to produce a representation of a lattice as a function lattice with functions into the two element lattice, re ecting a classical assumption of a two-valued semantics for the corresponding logical calculi. 6 The term \hemimorphism" was introduced by Halmos in his [23] and the concept was used for the representation of the necessity operator of modal logic. Sambin and Vaccaro [40] and later Dosen [9] made further use of this idea, still in the context of extending the Stone representation of Boolean algebras to the case of such algebras with, in addition, a necessity operator. Goldblatt [20] makes use of the same concept, considering distributive lattices with a class of join- or meet-hemimorphisms, extending the Jonsson-Tarski [30] framework.

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a \combination" of two meet semilattices, which are somehow \glued" together (after inverting the order of one of them). Our intuition has then been that it should be possible to reduce the lattice representation problem to that for (meet) semilattices, as long as we understand the essence of this \gluing" together. Hence, where the representation of Boolean algebras uses ultra lters and that of distributive lattices uses prime lters, we propose taking as the dual space of a general lattice the space whose points are the lters of the given lattice (equivalently, the hemimorphisms from the given lattice into the two-element lattice 2). Going from spaces to lattices now, lattices of clopen sets (or compact-opens of coherent spaces) are of course distributive and thus we need to invoke more structure on the space. Just as Boolean algebras (distributive lattices) arise as elds (rings, respectively) of subsets of some set, general lattices arise concretely as substructures of closure systems (intersection systems), where a closure system is a family F of subsets of a set X such that X 2 F and if Ai 2 F ; i 2 I , then Ti2I Ai 2 F . In the Boolean algebra case the associated closure operator can be taken to be the topological closure operator on sets of ultra lters (the closure system considered, in other words, is the family of closed subsets of the space). In Priestley's duality for distributive lattices, using partially ordered spaces, the closure operator involved is the map ( )", delivering the upper closure of a set. The particular closure operators are however additive, i.e. they distribute over unions and so the resulting lattices are distributive. The representation problem for general lattices is then to establish that every lattice can be viewed, up to isomorphism, as a collection of subsets in a closure system (on some set X ), closed under the operations of the system. We will then consider spaces with a closure system, where the associated closure operator may fail the additivity condition. In fact, one need look no further than partiallyordered spaces for the desirable setup. Closure operators are canonically obtained by composition of the two maps of a galois connection, or a residuated pair (just like, in a more general setup, monads are obtained by composition of adjoint functors). On the other hand, galois connections7 can be obtained from just any binary relation on a set (cf Birkho [6]) in a canonical way. If (X; ) is a set with a binary relation  X 2, the induced galois connection  a  ( the left and  the right galois maps) is de ned on subsets U; V  X by U = fx 2 X j U  xg and V = fx 2 X j x  V g, where U  x means that for every u 2 U; u  x, and similarly for x  V . In the sequel, we will refer to  and  as the Birkho galois connection. In the particular case where  is a partial order, the resulting galois connection is the familiar Dedekind-McNeile galois connection. Setting ? = , the operator ? is a closure operator which is not, in general, additive. The stable subsets A = ?A of X form a Wcomplete TlatticeSunder inclusion Swhere meets are intersections and joins are de ned by 2 A = fB j 2 A  B g = ?( 2 A ) .The intuition behind resorting to sets with a closure system, speci cally one induced by an appropriate galois connection, is again related to understanding how it is that a lati

I

i

i

7 A galois connection on partial orders P and Q is a pair of antitone such that for any elements p 2 P; q 2 Q we have q  fp i p  gq.

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I

i

i

I

i

maps f : P ! Q and g : Q ! P

tice is produced from two semilattices by gluing them together. Goldblatt's [17] and Urquhart's [46] have critically in uenced the author's intuitions on this point. Summing up, given an arbitrary lattice L, we construct its dual space by taking as points the hemimorphisms h : L ! 2 or, equivalently, the lters of the given lattice. Conversely, given a space (X; ) with a binary relation  on X , as the dual lattice of the space we take the family of stable compact-opens of the space, stability referring to the Birkho closure operator ? =  obtained by composition of the galois maps induced by the relation  as above. We call spaces as above closure spaces, since the structure on the space X we are interested in is that induced by the operator ? canonically associated with the relation  as explained. The reader familiar with Goldblatt's representation of ortholattices [17] will notice that our closure spaces generalize Goldblatt's orthogonality spaces (X; ?), where ? X 2 is a binary relation of orthogonality, or incompatibility.

3 The Stone Space of a Lattice For concreteness, we reiterate, slightly generalizing, our de nition of a closure space. De nition 3.1 By a closure space we mean a topological space X together with a closure operator ? on subsets of X , i.e. an operator satisfying U  ?U; U  V ) ?U  ?V; ??U  ?U . 2 Where L is a lattice8 , we let X = S (L) be the set of its lters (the hemimorphisms h : L ! 2). Let also  a  be the Dedekind-McNeile maps de ned on subsets of X as previously explained, and ? the closure operator induced by composition ? = . De ne a representation map h : L ! P (X ) in the natural way, by ha = fx 2 X ja 2 xg = Xa, and let X = h[L] = fXaja 2 Lg. It is immediate that Xa \ Xb = Xa^b, so that X is an intersection semilattice and intersection represents meet in L. For each a 2 L, denote by xa the principal lter xa = a" generated by a and observe that Xa = fx 2 X jxa  xg (where  is inclusion of lters). Dualize the inclusion xa  x and consider the order-dual X a of Xa, de ned by X a = fx 2 X jx  xa g. A small argument shows that the family X  = fX aja 2 Lg Tis also an intersection semilattice, where intersection now represents joins of L, i.e. X a X b = X a_b.9 What we have done so far is \decompose" the original lattice into two meet semilattices. Remains then to see how these can be \glued together" into a lattice, representing the original lattice. Note that the decomposition we have obtained re ects an elementary relation between meets and joins in a lattice, namely that they are order-dual operators (meaning simply that inverting the order of the lattice, meets appear as \joins" and vice-versa). The two semilattices we have obtained are pointwise order-dual, since their elements are the sets Xa = fxjxa  xg and X a = fxjx  xag. Order-duality of 8 We always assume that the lattice has an upper bound >. 9 Indeed, if x  xa ; xb, then x  xa ^ xb = xa_b and so x 2 X a_b. T xa ^ xb we obtain that x  xa and x  xb, hence x 2 X a X b.

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Conversely, too, from x  xa_b =

meets and joins is apparent when the lattice is endowed with a \dualizing" map :, that is to say an antitone map : : L ! Lop , inverting the order (Lop is L with ordering

ipped over) and with period 2, i.e. satisfying ::a = a for every element a 2 L. Then :(a ^ b) = :a _:b, i.e. meets are switched to joins (and conversely). This is what allows Goldblatt to sidestep the problem of directly representing joins by providing a representation of meets (as intersections) and a representation of the orthonegation operator (which has the above properties). In Hartonas and Dunn [27] we proposed taking the trivial duality id : L ! (Lop)op as a basis for the representation. The representation is considerably simpli ed, however, if we seek for such a dualizing map on the representation level, rather than on the lattice itself. In other words, having represented each of the two semilattices that our original lattice is made-up of, it is enough to nd a duality between the two. A duality, for partial orders P and Q, is a galois connection  a  with the stability requirement that q = q and p = p, for all p 2 P; q 2 Q (equivalently,  = ?1). Such a duality is what we vaguely referred to as the \glue" that produces a lattice out of two meet semilattices, in the sense of the following proposition.

Proposition 3.2 Each of X Wand X  is a fullT lattice, where meets are intersections and joins in X are de ned by Xa Xb = (Xa Xb), and dually for X .

Before turning to the proof, note that this proposition gives us a set-theoretic representation of joins which is of course not in terms of unions (since then the lattice would have to be distributive). For the proof, we verify that the galois connection  a  induced by the partial order on the lter space is a duality (that is,  = ?1 ) from X  to X, from which it will follow that both are full lattices. It is enough to verify that Xa = X a and X a = Xa. For the rst identity, if x 2 Xa, this means that x is a lower bound of Xa, which is the upper closure of the principal lter xa, and therefore x  xa , hence x 2 X a. Conversely, any lter contained in xa is below every lter in Xa , hence it is a lower bound of Xa. The other identity (with ) is shown similarly. Hence Xa = Xa and X a = X a and the galois connection  aT is indeed a duality. To show that (Xa Xb ) is the join of Xa and Xb we argue, more generally, that if S and K are meet semilattices (a dual argument applies to join semilattices) with a galois connection  a  from S to K , i.e.

 - op K  such that  = ?1 , then each of S and K is a full lattice, with joins in K de ned by k _ k0 = (k ^ k0). Indeed, from k ^ k0  k, it follows k = k  (k ^ k0), so that (k ^ k0 ) is an upper bound of k; k0. If k; k0  m, then m  k ^ k0, hence (k ^ k0)  m = m, so that (k ^ k0 ) is the least upper bound of k and k0 . 2 By de nition of ? =  and the argument above, ?Xa = Xa = Xa , so that the sets Xa = ha representing elements of the lattice L are stable sets. Furthermore, S



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Xa W Xb = (Xa T Xb ) = (Xa S Xb ) = ?(Xa S Xb ). This is simply by the fact that each of  andS codistributes over unions (because they form a galois connection), that T is to say (A B ) = A B , for any subsets A; B  X .

Corollary 3.3 Where L is a lattice, X = S (L) is the set of lters of L and h : L ! P (X ) is the representation map de ned above, L = h[L]. 2 h is clearly injective and the rest is by the previous proposition. By the imbedding h[L] ,! K , L is identi ed with a sublattice of the complete lattice K of stable subsets of its lter space. To characterize h[L] = X, we de ne a topology on the set X of lters taking as a subbasis the family S = fXa ga2L [ f?Xb gb2L

Proposition 3.4 The lter space X , with the topology induced by the subbasis S , is a Stone space.

For the proof we refer the reader to Goldblatt [17].

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Proposition 3.5 The stable compact-open subsets of X are exactly the sets Xa; a 2 L. Every Xa is (stable and) clopen, by the way the topology was de ned, hence compactopen, by the previous proposition, since the space is compact and Hausdor (the latter follows from total separation). For the converse, notice rst that the stable sets are exactly the upper closures T xT" of single points x in the space. Indeed, if U is a stable set of lters, let x = U = fz jz 2 U g. The intersection is never empty since for any z 2 U; > 2 z (recall that we assume a lattice always comes with a top element). Then y  U i y  x. Hence U = x#. Then U = U = x". T Consequently, every stable set U can be written in the form U = a2x Xa, for some x 2 X . Indeed, if U is stable, then U = x ", for some x 2 X . It is then clear that U = Ta2x Xa, where x is the base S point of U . It follows then that ?U = a2x(?Xa ). Hence, if U is clopen, by compactness, let a1; : : :; an 2 x such Tthat ?U can be written as the nite union of the sets ?Xai ; i = 1; : : :; n. Then U = ii==1n Xai . Letting a = a1 ^


^ an , it follows that U = Xa, qed. 2

Corollary 3.6 Every lattice L is isomorphic to the lattice L(S (L)) = X of stable

compact-opens of its dual closure space S (L).

3.1 Lower Bound

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Suppose L = (L; ; ^; _; >; ?) is a bounded lattice with top element > and bottom element ?. Since every lter contains > the set representing the top element is the set X of all lters which is always a stable set. On the other hand the set fxj? 2 xg is empty if only proper lters are allowed as values of x. But the empty set is not necessarily a stable set. Consider for example the lattice 8

>

?@@ ? a? @? b @ @@??

a^b= c

? The proper lters are f>g; fa; >g; fb; >g; fa ^ b; a; b; >g = (a ^ b)". Vacuously the set ; = X is the set of all (proper) lters. But X = f(a ^ b)"g = 6 ;. There is no reason to restrict to proper lters, however, and we allow for the lter space X to contain

the improper lter as well (the whole lattice). When considered as a lter, we use the notation ! for the whole lattice. Hence the set X? representing the bottom element is the singleton f! g. This will be of some signi cance when it comes to the semantic treatment of logical systems with a falsum object ?. Rather than declaring that for any world x; x 6j= ? we will allow for an inconsistent world, that is to say a world ! such that for any sentence ; ! j= . In particular, we will require of interpretations to satisfy j ?jj = f! g.

3.2 Representation and Choice

Both the Priestley or Stone representation of distributive lattices and the Stone representation of Boolean algebras make use of the axiom of choice, essentially in arguing that the map sending an element to the set of prime lters (ultra lters) containing it is injective (Prime Separation lemma). Urquhart's representation of general lattices, using maximal disjoint lter-ideal pairs also needs to make use of choice. Similarly, Hartung's extension [28] of Urquhart's theorem to a duality uses Zorn's lemma. The reader will have appreciated that our representation argument needs to make no use whatsoever of a choice principle. The downside of this is that we cannot obtain the Priestley and Stone dualities as special cases, when the lattice represented is distributive or a Boolean algebra. Hence our representation of such lattices is non-standard. The representation problem for general lattices, as we conceive it here, is to show that every lattice is a subsystem of an intersection (closure) system. This applies to distributive lattices and Boolean algebras as well. Admitedly, there are more obvious intersection systems for these cases (the system of all upper subsets of the prime spectrum of a distributive lattice, or the system of all closed subsets of the spectrum of ultra lters for Boolean algebras). A reader with foundational concerns will have nevertheless appreciated a solution of the lattice representation problem in ZF rather than in ZFC .

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4 Extension to Duality We show rst that the representation is functorial, that is that lattice homomorphisms f : L ! K can be also represented, by appropriately de ned morphisms of closure spaces. We de ne a stable continuous map f : (X; ?) ?! (Y;
) to be a continuous map f : X ! Y such that f ?1 takes stable compact-opens of Y to stable compact-opens of X . Apparently, the category Top of topological spaces and continuous maps is then a subcategory of ?Top, the category of closure spaces and stable continuous maps, since in the case of topological spaces the stable sets to be preserved by f ?1 are simply the closed sets. As a matter of notation, for a closure space (X; ?), X will always denote its dual lattice, that is the lattice of its stable clopens.

Proposition 4.1 Representation of lattices in their lter closure spaces is functorial. S (f ) f Furthermore, if L ?! K is a lattice monomorphism, then S (K ) ?! S (L) is a surjective

stable continuous map, where S (f ) = f ?1 . And if f is a lattice epimorphism, then S (f ) is injective.

If f : L ! K is a lattice homomorphism, let X = S (L) and Y = S (K ) be the respective lter spaces and observe that f ?1 takes lters of K to lters of L, hence f ?1 = S (f ) : Y ! X . We need to verify that its inverse, S (f )?1, maps X into Y where recall that X = fXaja 2 Lg and similarly Y = fYb jb 2 K g, and Xa is the set of L- lters containing a 2 L, Yb is the set of K - lters containing b 2 K . A direct calculation shows that S (f )?1Xa = Yfa , for each a 2 L. Also, by a straightforward calculation, S (f )?1(?Xa) = ?Yfa , and so it also follows that S (f ) : Y ! X is continuous, since it preserves subbasis elements. When f is injective, we may identify L with the sublattice f (L) of K and take f to be inclusion. Hence every lter of L is of the form L \ x = f ?1 (x), for some K - lter x. If f is surjective, then we may identify K with the quotient lattice L= of L, where  = ker(f ). Given then lters x; y of L= (i.e. of K ), since f ?1 x = fa 2 Lja 2 xg, it is clear that f ?1 x = f ?1 y implies x = y. That is f ?1 = S (f ) is injective, when f is a surjective lattice homomorphism. 2 We extend the representation theorem to a full Stone-type duality. The dual spaces of lattices are certain kinds of closure spaces with a Stone topology (compact, totally separated). A precise characterization of these spaces is given in the following de nitions.

De nition 4.2 Let K = hK; ; V; Wi be a complete lattice and C a sublattice. C is compact-dense in K i V 1. (compactness) For any a 2 C and familyVaj 2 K; j 2 J , if j 2J aj  a, then there is a nite index subset F  J such that j 2F aj  a 2. (density) C is meet-dense in K, that is to say every a V2 K can be written as the meet of elements c 2 C such that a is below c, i.e. a = c2C ;ac c. 10

De nition 4.3 A lattice space (`-space), is a closure Stone space (X; ; ?) (where ? is the Birkho closure operator induced by the binary relation ) with topology generated by the family C of stable compact-opens and their complements. In addition 1. Every stable set A is the closure of a unique point, that is A = ?(fxg), for some x 2 X . As a consequence the closure operator is injective on singletons in the sense that ?(fxg) = ?(fy g) implies x = y . 2. If A; B 2 C , then also ?(A [ B ) 2 C . 3. The family C of stable compact-opens is compact-dense in the complete lattice of stable sets, in the sense of the previous de nition. With lattice-space maps the maps of their underlying closure spaces we conclude with a duality theorem: Theorem 4.4 The functors S : Lat ?! LTopop; L : LTopop ?! Lat, sending a lattice to its dual space and a lattice space to its dual lattice of stable compact-opens, form a duality of categories. We have already shown half of the above, namely that for a lattice L we have  L = L(S (L)), where S (L) = X is the dual space of L and L(X ) = X is the lattice of stable compact opens of X . By construction of the space X (the lter space) it is clear that X is a lattice space in the sense de ned above. Now given a lattice space (X; ; ?) and where L(X ) = X is its family of stable compact-opens and Y = S (X) is the dual space of X we need to verify that there is a stable homeomorphism f : X  Y . First, we verify the following: Lemma 4.5 Every space in LTop arises as the dual space of a lattice and every morphism in LTop arises from a lattice homomorphism. Writing ?X for the collection of stable subsets of X , note that the map sending an element x 2 X to the closure of its singleton ?(fxg) is a bijection. Ordering X by x  y i ?(fy g)  ?(fxg), the closure operator ? induces a complete lattice structure on X , where _ \ ^ _ xi = ??1 ?(fxi g) and xi = ??1 ?(fxig) i2I

i2I

i2I

i2I

where we slightly abuse notation and consider ? as a map ? : X ! ?X . The fact that the lattice of stable clopens is compact-dense in the lattice of stable sets implies that the collection of points xi in the space generating the stable compactopens of the space are compact in the lattice X . Also that the collection of these points is join-dense in X (meaning that every point x 2 X can be written as the join of such points xi that lie below it, i.e. such that xi  x (where  is the induced ordering on X ). With this in mind, we turn to the proof of the lemma. Let X be a space in LTop (a lattice-space), S = fXigi2I [ f?Xj gj 2I the subbasis generating its topology, and X = fXi gi2I the set of stable compact-opens. Since 11

Xi T Xj ; ?(Xi S Xj ) 2 X, the set X is a sublattice of the complete lattice ?X of stable subsets of X . Partially order the index Tset I by i  j i Xi  Xj . Writing i ^ j for the element k of the index set such thatSXi Xj = Xk , and similarly for i _ j for the unique element s 2 I such that Xs = ?(Xi Xj ), it is clear that (I; ; ^; _) is a lattice. We rst show that X is homeomorphic to the dual closure space F = S (I ) of the lattice I. If xi is the (compact) point in X whose upper closure is the subset Xi , for i 2 I , then by xi  xj i Xj  Xi i j  i, the indexing map  : i 7! xi is a coimbedding of the lattice I into the complete lattice X . By the universal property of lter cocompletions, if F is the lter space of I , then  : I ! X extends to a complete lattice homomorphism ^ : F ! X , explicitly de ned on a lter f 2 F by ^(f ) = Wfiji 2 f g = Wfxi ji 2 f g. Since the points xi = i are compact elements of the lattice X (by de nition), it follows that ^ is injective. For suppose that ^(f ) =W ^(f 0), where f; f 0 are lters of I (elements of F ). If i 2 f , then xi  ^(f ) = ^(f 0 ) = fxj jj 2 f 0 g. Hence (by compactness of xi ) there are nitely many indices j1; : : :; jn , each in f 0, such that xi  xj1 _


_ xjn . Then j1 ^


^ jn  i and so i 2 f 0 . Thereby f  f 0 . The converse is similar. Next we verify that ^ is surjective. If x 2 X , let f  I be the set f = fijxi  xg. If i  j , for i 2 f , then Xi  Xj , by de nition of the ordering on the indices, hence xj  xi  x. So j 2 f , too. If i; j 2 f , then xi^j = xi _ xj  x, hence i ^ j 2 f and f is therefore a lter of I , hence in F . That ^(f ) = x follows from join-density of the set of points xi 2 X (the base-points of the W stable clopens of the space), since by de nition of ^ and choice of f we get ^(f ) = fxi jxi  xg. Therefore, ^ is a bijection. Next we verify that it is continuous and an open map, when F is endowed with the

representation topology, as this was previously de ned. Where Fi = ff 2 F ji 2 f g and Xi = fx 2 X jxi  xg observe that f 2 Fi ) i 2 f ) (i) = xi  ^(f ) ) ^(f ) 2 Xi . Conversely, if xi  ^(f ), apply ^?1 to get i" f , i.e. i 2 f and thereby f 2 Fi . Hence, ^?1 Xi = Fi and ^Fi = Xi so that ^ is bijective, continuous and an open map, so a homeomorphism ^ : F  X . The above argument also shows that ^ preserves stable clopen sets, so it is a morphism of closure spaces. For the second part of the lemma, if h : Y ! X is a morphism of lattice-spaces, by the previous argument we may assume X = S (A); Y = S (B ), where A and B are the index sets for the respective subbases of X and Y . De ne the map h^ : A ! B by h^ a = b i h?1 (Xa) = Yb . It is clear that h^ is a homomorphism. Now h = S (^h), for if not, let y 2 Y such that hy 6= ^h?1 y . By total separation of X , let ^h?1 y 2 Xa; hy 62 Xa. Thus a 2 ^h?1 y and so ^ha 2 y . If h^ a = b, then y 2 Yb and, by de nition of h^ , h?1 (Xa) = Yb . So y 2 h?1 (Xa ), hence hy 2 Xa follows. Thus h = S (^h) = h^ ?1 as claimed. 2

To complete the proof of theorem 4.4, we rst verify that the map L assigning the lattice of stable compact-opens to a space (X; ?) is indeed a functor, that is it preserves 12

closure space maps. Given h : (Y;
) ! (X; ?), we may assume X = S (L); Y = S (K ) and h = S (f ) = ? 1 f , for some lattices L; K and lattice homomorphism f : L ! K . We let L(h) = S (f )?1. TIf Xa; Xb 2 X are two stable clopen subsets of X , then L(h)(Xa T Xb) = L(h)Xa L(h)Xb , since inverse maps always preserve intersections. Furthermore, a direct calculation shows that L(hW)Xa = S (f )?1Xa = Yfa (in fact also L(h)(?Xa) = S (f )?1(?Xa) = ?Yfa ). Since Xa Xb = Xa_b, we have

_ _ _ S (f )?1 (Xa Xb ) = Yf (a_b) = Yfa_fb = Yfa Yfb = S (f )?1Xa S (f )?1Xb

and therefore L(h) = S (f )?1 is a lattice homomorphism from the dual lattice X = LS (L) (the lattice of stable compact-opens) of X = S (L) into the dual lattice Y = LS (K ) of Y = S (K ). In conclusion, given a lattice L, we have shown that L  = L(S (L)), where S is the functor mapping a lattice to its dual closure space, and L is the functor mapping a lattice space to its dual lattice of stable compact-opens. For the converse, given a lattice space X = (X; ?), we have constructed a lattice I isomorphic to the dual lattice X = L(X ) and have veri ed that there is a stable homeomorphism X  S (I ), hence we have X  S (L(X )), qed. 2

Congruences and Epimorphisms: The lattice duality we have presented extends

to a duality for the lattices of congruences and of sublattices. Congruences can be identi ed with (the kernels of) epimorphisms which are carried over to injective closure maps which, in turn, can be identi ed with their image subspaces. Leaving details to the interested reader, the lattice of congruences can thus be seen to be isomorphic to the lattice of subspaces. Similarly, the lattice of sublattices is represented as the lattice of quotient spaces.

13

Part II Duality for Lattice-Ordered Algebras 5 Preliminaries Extensions of Boolean algebras, such as closure algebras (S 4 algebras), normal modal algebras, projective algebras and relation algebras is what motivated the representation theory developed in Jonsson and Tarski [30]. Eversince, the Jonsson and Tarski paper has been revisited by various authors. Hansoul [24] presents a duality for Boolean algebras with the Jonsson-Tarski operators and extends to the case of a distributive underlying lattice, using the original Stone representation. Goldblatt [21, 20, 22] considers (Boolean algebras and) distributive lattices with ^- or _-hemimorphisms, that is operators preserving meets (respectively, joins) in each argument place and extends the Priestley duality to one for distributive lattices with such operators. In his \Metamathematics of Modal Logic" [19], Goldblatt presents the rst duality of logical interest. Based on Halmos [23] and inspired by [19], Sambin and Vaccaro [40], give a new representation and duality for the algebraic and Kripke semantics for normal modal logic. Dosen [9] makes this result sharper, dropping the assumption of normality. Priestley's representation for distributive lattices has also been extended in other directions. First, by Urquhart [47], for Ockham lattices, i.e. bounded distributive lattices with a dual homomorphic operator (a negation operator satisfying the De Morgan laws and switching the bounds), generalizing De Morgan lattices. And also by Martinez [33] for Wajsberg algebras, which coincide with the bounded commutative BCK-algebras . In a paper currently in progress Urquhart [48] extends the Priestley duality to a duality for Relevance Logic. Dunn [13], inspired also by the Jonsson-Tarski paper, presents a representation by relations of distributive lattices with a broad family of operators. In his [14], Dunn also points out the need to extend such representation theorems for cases where the algebras are merely partially ordered, and he provides relevant results. In all the above cases, extension to a representation of ordered structures is rmly based on the existing representation and duality of the underlying Boolean algebra or distributive lattice or partial order. In a few words, the objective of this second part then is to extend the lattice duality we presented to the case of lattices with additional operators. With regard to the Jonsson and Tarski paper, we dier in that we will consider general lattices rather than Boolean algebras (or distributive lattices), and in that we seek to extend the class of operators to be treated.10, which is the only class of operators considered11 in [30]. More speci cally, our interest in this part is to prove a dual equivalence between 10 For example, a structure such as an implicative

lattice hL; ; ^; _; ?!i does not t into the Jonsson and Tarski framework of additive operators. An operator f on a lattice is additive just in case it distributes over nite joins in each argument place. 11 Of course in the setup of Boolean algebras interde nability of operators using classical complementation allows for a treatment of the operator !.

14

`-algebras and certain relational structures. This stems from a logical, model-theoretic

concern, as we wish to be able to construct (in part III) Kripke structures for the semantics of a variety of logical systems.

6 Lattice-Ordered Algebras In this section we de ne a broad notion of a lattice-ordered algebra (`-algebra) with normal operators. We rst state an extension of our lattice representation for the case of `-algebras. We then proceed to characterize the kind of completion of an `-algebra we seek to provide. We conclude that every `-algebra has a completion in a two-sorted algebra. Where the operators involved are either additive or multiplicative the completion is in a single-sorted algebra. The following is our proposed general notion of an `-algebra. De nition 6.1 An `-algebra is a structure A = hL; ; ^; _; (fi)i2I i, where hL; ; ^; _i is a lattice and each fi ; i 2 I , is a normal operator on the lattice. Normality for an n-ary operator f means that it is a monotone operator

f : L(0) 


 L(n?1) ?! L(n) distributing over \joins" of L(i) in its i-th position, where for each i 2 n + 1; L(i) is either L or its opposite Lop . If L is complete and f distributes over arbitrary \joins", then we call f a completely normal operator.

In particular, every additive operator, in the sense of Jonsson and Tarski, is apparently normal. The operator  of the Full Lambek algebras studied by Ono [36] is additive. Negation is normal in our sense : : L ! Lop but not additive. The possibility operator is additive 3 : L ! L, while its order-dual 2 : Lop ! Lop is normal but not additive. Similarly, implication !: L  Lop ?! Lop is normal but not additive. Normality for the lattice operator ^ : L  L ! L is equivalent to distributivity of the lattice.12 In particular, a multiplicative operator is a map f : Ln ! L preserving nite meets in each argument place. Where f n is an n-ary operator and g m1 ; : : :; g mn are operators of arities m1; : : :; mn the composition operator f n [g m1 ; : : :; g mn ] is de ned on mi -tuples x1; : : :; xn in the usual way by f n [g m1 ; : : :; g mn ](x1; : : :; xn ) = f n (g m1 x1 ; : : :; g mn xn ). The following is immediate.

12 Strictly speaking our notion of normality is tied up to a particular normal presentation f : L(0) 


 L(n?1) ?! L(n) of the operator. For example we can regard ^ : Lop  Lop ! Lop as normal,

too. In point of fact, when referring to an operator as a normal operator, we always mean a speci c presentation as above. Alternatively, we assume that an operator comes with a xed distribution type (Dunn [13]). There is no harm created by the slight looseness of the de nition, however, and we prefer to phrase things as we did rather than as in [13] as the concept in our arguments becomes more clear in this way.

15

Lemma 6.2 If each of the operators f n and gm1 ; : : :; gmn is normal then the composite

map f n [g m1 ; : : :; g mn ] is normal, provided the gmi and f n have the same output type (a join or a meet). 2 The condition that the output type be the same cannot be dropped. For example the composite operator ( ) ! 2( ) is not normal since ! outputs a join while 2 outputs a meet. For any `-algebra the following is a consequence of our lattice representation. Theorem 6.3 Every `-algebra is isomorphic to an `-algebra of sets on the lattice of stable compact-opens of a Stone space. Proof: Using a convenient notation from our lattice representation and given a normal operator f on L, let F be the operator de ned on the stable compact-opens of the lterspace of the lattice (see proposition 3.5, part I) in the obvious way: F (Xa1 ; : : :; Xan ) = Xf (a1;:::;an ). The rest is immediate. 2

6.1 The Completion Problem

Next we investigate the completion problem for `-algebras, much in the spirit of the Jonsson-Tarski paper [30], where they provide a completion result for additive operators. Some of the results obtained will be used to prove a dual equivalence between `-algebras and a certain kind of relational frames. De nition 6.4 Let L = (L; ; ^; _; f ) be a lattice with a normal n-ary operator f and M a complete lattice. An n-ary operator F on M is an adjoint completion of f i there is a lattice imbedding  : L ,! M such that 1. for all a1 ; : : :; an 2 L, f (a1 ; : : :; an) = F (a1 ; : : :; an ) 2. in each argument place, F is monotone (antitone) and (co)distributes over arbitrary joins (meets), accordingly as f is monotone (antitone) and (co)distributes over nite joins (meets), respectively. Hence a completion map F is a completely normal operator with the same type of distribution properties as f . We will need to generalize the above de nition and specify a notion of adjoint completion in a possibly many-sorted algebra A. De nition 6.5 Let L = (L; ; ^; _; f ) be a lattice with a normal n-ary operator f : L(0) 


 L(n?1) ! L(n) . Then an (n + 1)-sorted algebra

^_ A = h(Ai; ; )i2n+1; F : A0 


 An?1 ! Ani is an adjoint completion of L i VW 1. (A ; ; ) is a complete lattice, for each i 2 I , i

16

2. there exist imbeddings i : L(i) ! Ai , for i 2 n + 1, such that for all ai 2 L(i) n f (a0 ; : : :; an?1 ) = F (1 a1 ; : : :; n?1 an?1 ) 3. where f (co)distributes over nite joins of L(i) , F distributes over arbitrary joins of Ai .

The question whether every `-algebra has a completion in the above sense is answered in the following theorem.

Theorem 6.6 Assume f : L0 


 Ln?1 ?! Ln is a normal operator (additive in each argument place). Then f has an adjoint completion f^ : I0 


 In?1 ?! In where Ii is the ideal lattice of Li .

Where yi is an ideal in Ii ; i 2 n, de ne ^ 1


yn = _fyfc


cn jyci  yi g fy 1

In

(1)

where yfc1


cn = [fc1


cn ] # is the principal ideal of Ln generated by the element fc1


cn and  is inclusion of ideals. We rst verify that f^ is increasing in each of its argument places. Since yi  yi0 implies that for any ci 2 Li ; yci  yi ) yci  yi0 , we have an inclusion of sets fyfc1


cn jyci  ^ 1


yi


yn  fy ^ 1


yi0


yn follows. yig  fyfc1


cn jyci  yi0 g hence by taking joins fy Next, we verify the following ^ c1


ycn = yfc1


cn . Lemma 6.7 Where yci = (ci)# is a principal ideal of Li; fy ^ c1


ycn = Wfyfd


dn jydi  yci g. Note that the principal ideal yfc


cn By de nition, fy 1 1 ^ c1


ycn . is in the set on the right hand side, hence yfc1


cn  fy Conversely, if yfd1


dn is in this set, by ydi  yci we get di  ci , hence fd1


dn  fc1


cn and therefore yfd1


dn  yfc1


cn . Hence ^ c1


ycn = _fyfd


dn jydi  yci g  yfc


cn fy 1 1

^ c1


ycn = yfc1


cn . so that fy With this in place, we essentially only need to verify that Lemma 6.8 If A  Li and (yai )a2A are the principal ideals over A in Li, then ^ 1


yi?1 yai yi+1


yn ^ 1


yi?1 (Wa2A yai )yi+1


yn = Wa2A fy fy We only need to show one direction, namely that 17

2

Wfy ^ 1


yi?1 (Wa2A yai )yi+1


yn fc1


cn jycj  yj g= fy ^ 1


yi?1 yai yi+1


yn  Wa2A fy

since the other direction follows by the monotonicity property of f^, previously veri ed. We x (c1; : : :; cn) 2 L1 


 Ln such that for each j = 1; : : :; n; ycj  yj and then W ^


y yi y


y ). From y  y we get show that yfc1


cn  a2A fy 1 i?1 a i+1 n cj j ^ c1


yci


ycn  fy ^ 1


yi?1 yci yi+1


yn yfc1


cn = fy

W

From yci  yi = a2A yai , by compactness, let a1; : : :; ak 2 A such that yci  yai 1 _


_ yai k = yai 1_


_ak . Then for this xed i, ci  a1 _


_ ak , hence fc1


ci


cn  fc 1


(a1 _


_ ak )


cn W j = j ==1k fc1


ci?1 aj ci+1


cn Consequently.

yfc1


ci


cn  y_jj==1k fc1


aj


cn W = jj ==1k yfc1


aj


cn W ^


y y y


y = jj ==1k fy cn c1 ci?1 aj ci+1 ^ 1


yi?1 yaj yi+1


yn  WWjj==1k fy ^ 1


yi?1 yai yi+1


yn  a2A fy Therefore, we have established that f^ distributes over arbitrary joins of principal ideals

in each of its argument places. The corresponding claim for distribution over arbitrary joins of any ideals now follows from the fact that the collection of principal ideals is join-dense in the ideal lattice, that is to say that every ideal can be written as the join of principal ideals contained in it. Hence, considering f^ with all but one argument places xed, it follows from the above that, for this restricted unary operation, there is a right adjoint g^. 2 Remark 6.9 Note that the adjoint completion f^ constructed is unique in the sense that if also f  : I (L1) 


 I (Ln) ?! I (K ) is another adjoint completion of f , then since f^ and f  agree on principal ideals, by the rst sublemma, and they preserve arbitrary joins of ideals, they must be identical, by join-density of principal ideals. 2

Example 6.10 Before proceeding with representation, we give an example of the previous theorem, applying it to an `-groupoid (L; ; ^; _; ), where  is assumed to distribute

over nite joins (hence it is normal in our sense, in fact even additive in the JonssonTarski sense). The de nitionWof the map f^, denoted now by ?, is the binary operation on ideals y; z de ned by y?z = fyab j ya  y and yb  z g (recall that for e 2 L, we denote by ye the principal ideal e# generated by e). The argument for the proof of theorem 6.6 W W shows that ? distributes over arbitrary joins of ideals ( i2I yi ) ? z = i2I (yi ? z ), and W similarly for y ? ( j 2J zj ). We will eventually need to produce an operator on lters. 18

Furthermore, for the case of an implicative lattice (L; ; ^; _; !), the induced operator ) takes as arguments an ideal and a lter and delivers a lter, since the \ideal space"

I (Lop) is really the lter space F (L).

We list here, for later use, the de nition of the map f^ for an additive n-ary operator, which generalizes the de nition of our example: ^ 1


yn = _ fyfc


cn j yci  yi g (2) fy 1

I (L)

We conclude this section with stating the completion result which follows immediately from theorem 6.6: Theorem 6.11 Every n-ary normal operator f : Ln ! L has a completion in an algebra with two sorts, I (L) and F (L). 2

6.1.1 The Duality of Closed Ideals and Filters

Our main interest in this part is to obtain relational structures equivalent to `-algebras. More speci cally, we aim at using our lattice representation to construct such relational structures and we thus need to be able to de ne operators f^ on lters for each normal operator on the lattice. In the general case of a normal operator f we have produced an extension to an operator F . However, depending on the distribution properties of f , the operator F may take as arguments both lters and ideals in dierent argument places. We will now extract an operator de ned only on lters. First, we investigate properties of a galois connection between lters and ideals which restricts to a duality of closed lters and closed ideals, as we show.

Proposition 6.12 If I (L) and F (L) are the ideal and lter lattices of a xed lattice L, then there is a galois connection ^ : F (L) ! I (L)op, ^ : I (L)op ! F (L) which restricts

to a bijection from closed lters to closed ideals

T

A closed lter is a lter x such that x = f[a)jx  [a)g and similarly for a closed ideal y = Tf(a]jy  (a]g. Throughout this proof we use x; z for lters and y; v for ideals, sometimes with subscripts. We write xa for the principal lter [a) generated by the element a and similarly ya for the principal ideal (a] generated by a. Let ^ be the extension of the map  : W xa 7! ya explicitly de ned by ^x = Tfyaja 2 xg and de ne the map ^ on ideals by ^y = fxjy  ^xg. Lemma 6.13 Each of ^ and ^ is antitone and ^xa = ya, ^ya = xa. 2 Lemma 6.14 The maps ^ and ^ form a galois connection from lters to ideals

^ F (L) ^ 

- I (L)op 19

Proof: We need to show that x  ^y i y  ^x. Assume rst that x  ^y and let b 2 y, so that yb  y. We need to get b 2 ^x. From yb  y and antitonicity of ^ we get ^y  ^yb = xb . Using the hypothesis x  ^y we then have that x  xb . By lemma 6.13 we get yb = ^xb  ^x and thereby b 2 ^x, as needed. For the converse assume that y  ^x and let b 2 x. We need b 2 ^y . From b 2 x we get xb  x and then by lemma 6.13 ^x  ^xb = yb . From hypothesis it follows that y  yb and then by lemma 6.13 we obtain that xb = ^yb  ^y and thus b 2 ^y as needed.

2

^ codistributes over arbitrary joins of lters, that is to say ^ Wi2I xi = Corollary 6.15  T ^x . Similarly, ^ codistributes over arbitrary \meets" of I (L)op, that is to say i2I i ^ Wi2I yi = Ti2I ^yi. 2 T Now de ne also the map g on ideals by gy = fx ja 2 y g. a

Lemma 6.16 The map g is right adjoint to ^, ^ a g, and thereby by uniqueness of

adjoints g = ^. Consequently, for any ideal y , ^y is a closed lter. We need to show that y  ^x i x  gy . Assume rst that y  ^x and let a 2 x, i.e. xa  x. Then ^x  ^xa = ya hence y  ^x  ya. Now notice rst that g is antitone and gya = xa. For antitonicity if y  v then fxa ja 2 y g  fxa ja 2 v g hence taking intersections we get

\ \ gv = fxaja 2 vg  fxaja 2 yg = gy

Also notice that xa 2 fxe je 2 ya g, since a 2 ya . We show it is the smallest principal lter in that set. If also xe is in this set, then e 2 ya which means that e  a. Hence xa  xe . Thus \ gya = fxe je 2 ya g = xa From y  ya we then get a 2 xa = gya  gy and so a 2 gy . T For the converse, assume that x  gy and let b 2 y . From x  gy = fxaja 2 y g and b 2 y we get x  xb . Hence we obtain b 2 yb = ^xb  ^x and so b 2 ^x. 2 Corollary 6.17 For any ideal y, ^y is a closed lter. For any lter x, ^x is a closed ideal. 2 Proposition 6.18 ^^x = x and ^^y = y, assuming x and y are closed. Proof: Notice that the inclusions x  ^^x and y  ^^y hold for any lter and ideal by the fact that ^ a ^ form a galois connection. Indeed these inclusions are equivalent to the usual de ning condition x  ^y i y  ^x. First we show that ^^y = y for any closed ideal y . By de nition of ^ and corollary 6.15 we have _ \ ^^y = ^ fx jx  ^yg = fy jy  y g = y a

a

a

20

a

using the fact that xa  ^y i y  ^xa = ya and the assumption that y is closed. By a similar computation we have

_ \ ^^x = ^ fy jy  ^xg = fx jx  x g = x a

a

a

a

where again we use the fact that ya  ^x i x  ^ya = xa and the closure assumption on x. 2

An operator f^ on lters: Using the galois connection ^ a ^ the maps f^i de ned in

the proof of theorem 6.6 can be transfered in the lter lattice F (L) = X , de ning f ] by composition with ^ and ^ in the appropriate places. To make this technically precise (even though the idea should be clear), let f : L(0) 


 L(n?1) ?! L(n) be an n-ary normal operator and (x1; : : :; xn ) an n-tuple of lters. We de ne the operator f ] on this n-tuple by f ] x1


xn = f^i(1x1 ; : : :; nxn) where i = id if L(i) = Lop, i = ^ if L(i) = L, and where is either identity or ^ , depending on whether L(n) = Lop or L(n) = L, respectively. For example, given the (normal) implication operator !: L  Lop ?! Lop , theorem 6.6 delivers an operator ! ^ : I (L)  F (L) ?! F (L). The operator !] is then de ned on a pair (x1; x2) of lters by x1 !] x2 = ^x1 ! ^ x2 . Now recall from the lattice representation that, given the Birkho closure operator ? = , the ?-stable sets are closures of single points. Simplifying notation to ?x for the more accurate ?(fxg), stable sets are of the form ?x; x 2 X . Furthermore, the closure operator does not confuse points, meaning that ?x1 = ?x2 implies x1 = x2 . Thereby, given a normal operator fi on the lattice, and having de ned the map fi] on lters as previously explained, we can extract an operator Fi on stable sets in a straightforward way, simply by de ning Fi (?x1 ; : : :; ?xn ) = ?ffi] (x1 ; : : :; xn)g.

7 Representation by Relations We show in this section how to canonically construct Kripke frames with (n + 1)-ary relations interpreting n-ary normal operators on a given `-algebra. The following de nition lists our basic assumptions on relational structures. De nition 7.1 A closure frame is a structure F = hX; ; (Ri)i2I i where 1. X is a set and  a binary relation on X 2. For each i 2 I; Ri is a relation on X of some arity n + 1 (depending on i) 3. If ? is the closure operator induced by the relation , then every stable set A is the closure of a unique point denoted by xA , i.e. A = ?xA , and therefore ?x = ?y implies x = y

21

Given an (n + 1)-ary relation R in a closure frame we extract an n-ary operator R de ned on stable13 sets A1 ; : : :; An by R A1


An = fx 2 X j 8z (Rx1


xn z implies z  x)g (3) where xi is the point generating the stable set Ai , i.e. ?xi = Ai ; i = 1; : : :; n. Lemma 7.2 If A1; : : :; An are stable sets then their R-image is a stable set. For an n-tuple x1 ; : : :; xn of points we let Rx1


xn be the set fy j Rx1


xn y g. The proof relies on the observation that RA1


An is the set of -\upper bounds" of the set Rx1


xn, where ?xi = Ai . Recall that the Birkho galois connection  a  induced by the binary relation  is de ned on subsets of the set X by U = fxjU  xg (U is the set of -\upper bounds" of U ) and V = fy j y  V g. Hence R A1


An = Rx1


xn and stability of R A1


An follows by de nition of the closure operator ? =  and the property of a galois connection that  =  and  = , since ?R A1


An = ?Rx1


xn = Rx1


xn = Rx1


xn = R A1


An . 2 De nition 7.3 Let FT be a closure frame and C a family od stable subsets such that S A; B 2 C implies A B; ?(A B) 2 C . Then the relation Ri is C -normal (or just normal if C is understood) if Ri is a normal operator on C in the sense of de nition 6.1. Theorem 7.4 Every normal n-ary operator f on a lattice is representable as the operation R , for some (n + 1)-ary relation R on the dual frame of the lattice. Assume f : L(0) 


 L(n?1) ?! L(n) is normal and let f ] be the induced operator on lters. A relation Rf = R is then de ned by Rx1


xnz i z  f ] x1


xn (4) Extracting the operator R de ned by equation 3 we have RA1


An = fxj8z (Rx1


xnz ! z  x)g = fxj8z (z  f ] x1


xn ! z  x)g = fxj8z (z  ff ]x1


xn g ! z  x)g = fxj8z (z 2 (ff ]x1


xn g) ! z  xg = fxj (ff ]x1


xn g)  xg = (ff ]x1


xn g) = ?(ff ] x1


xn g) In particular, the restriction of R on the stable clopens of the lter space satis es  R Xa1


Xan = Xfa1


an , where Xai is the set of lters containing the element ai and Xfa1


an is the set of lters containing the element fa1


an . 2 13 The relation R induces an operator R de ned on all subsets U1 ; : : : ; Un of X as the subset  R U1


Un consisting of all points x such that, for all points z; u1 ; : : : ; un , if ?Ui = ?ui and Ru1


un z , then z  x. We are really only interested in de ning an operator on stable sets and this is why we identify the action of R on sets U1 ; : : : ; Un with its action on their closures ?U1 ; : : : ; ?Un . From

our lattice representation, recall that the closure of a set U is obtained by rst collapsing it to a point u (which is the intersection of all members of U ) and then taking the upwards closure of that point. In eect then ?U = ?u, where u is the greatest lower bound of U . Thus we always have that R (U1 ; : : : ; Un ) = R (?U1 ; : : : ; ?Un ) = R (?u1 ; : : : ; ?un ).

22

8 The Duality of `-Algebras and `-Frames We have already shown how to canonically constract the dual frame of a lattice-ordered algebra, de ning a relation Rf for each normal operator f on the lattice by means of an auxiliary operator f ] on the points of the dual space of the lattice. We proceed, in the present section, to extend to a duality result, appropriately characterizing the kinds of frames we are interested in. Given a closure frame F = hX; ; (Ri)i2I i let R be the induced operator. Recall also that for stable sets A1; : : :; An the set R A1


An = A is stable, as it can be written in the form A = Rx1


xn where xi is the point generating the stable set Ai . Since A is stable let x be the point that generates it by A = ?x. This determines an n-ary function rR on points of the space de ned by

rR x1


xn = x i Rx1


xn = ?x (5) Given the map rR rephrasing 5 gives Rx1


xn = ?(frR x1


xn g) = (frR x1


xn g) but, in general, there is no way of recovering the relation R from this identity ( cannot be \cancelled"). We prescribe the functionality requirement that R is de nable from the map rR and the relation  by Rx1


xn y i y  rR x1


xn (6) We will have use of the following -converse R of the relation R  1


xn y i rR x1


xn  y Rx (7)

De nition 8.1 A relation R in a closure frame F is functional i condition 6 above holds. The frame F is a normal Tclosure frame S (ncf) i there is a family C of stable sets such that A; B 2 C implies A B; ?(A B ) 2 C and every relation on the frame is normal (de nition 7.3) and functional.

Remark 8.2 An ncf F can be alternatively speci ed in terms of the functions on points of the space F = hX; ; (ri)i2I i where the relations Ri can be recovered using 6 as a

de nition. We will not pursue this representation since the only use for the operators rR that we have is the -converse relations R i de ned above.

De nition 8.3 A re ned ncf (rncf) is a structure F = hX; ; (Ri)i2I ; Ci where F0 = hX; ; (Ri)i2I i is an ncf with respect to the speci ed family C . A topological rncf is an rncf F = hX; ; (Ri)i2I ; Ci where the family S = fAgA2C [ f?BgB2C is a subbasis for a Stone topology on X and C is the family of stable compactopens of the space. We refer to C as the dual algebra F of the frame F . Our representation results have established that an `-algebra A is isomorphic to the dual algebra of its dual frame, i.e. if F = A is the dual frame of A, then A  = (A ).   We need to establish now that if F is a topological rncf then F = (F) , i.e. that F is 23

isomorphic to the dual frame of its dual algebra. We rst introduce a natural notion of morphism of topological rncf's and, with some additional restrictions on the frame, we proceed to prove a duality result. De nition 8.4 Two re ned normal closure frames F 1; F 2 are similar i their dual algebras F1; F2 are similar. A morphism of similar topological rncf's is a continuous function f : X1 ! X2 such that f ?1 : F2 ?! F1 is a homomorphism of their dual `-algebras, that is to say f ?1 sends stable compact-opens of X2 to stable compact-opens of X1 and preserves intersections, joins and the operators Ri . To obtain a duality we further assume that the dual algebra of a topological rncf is compact-dense in the complete lattice of stable sets in the sense of de nition 4.2, part I. De nition 8.5 An `-frame is a topological rncf F such that the dual algebra F is compact-dense in the complete lattice F y of stable sets in F . An `-algebra A is similar to an `-frame i A is similar to the dual algebra F of the frame F . We let `ALG and `FRM be the categories of `-algebras and `-frames. Theorem 8.6 There is a duality F : `ALG ?! `FRMop; C : `FRMop ?! `ALG, where an `-algebra A is isomorphic to the dual algebra of its dual `-frame and an `-frame is isomorphic to the dual frame of its dual algebra. Fix an `-algebra L = hL; ; ^; _; (fi)i2I i. By previous arguments let fi] be the induced operator on lters and Ri the relation de ned by Rix1 ; : : :; xnz i z 2 ?fi] x1 ; : : :; xn and Ri the operator de ned by equation 3. We have thus obtained a normal closure frame F = hX; ; (Ri)i2I i, where  is inclusion of lters. The dual algebra of F is the algebra F = hfXaja 2 Lg; ; \; _; (Ri )i2I i, where Xa = fx 2 X ja 2 X g. We have previously veri ed that L  = (L ), where L = F is the dual frame of L. If h : L ! M is a homomorphism, where L = hL; ; ^; _; (fi)i2I i and M = hM;  ; ^; _; (gi)i2I i, we rst verify that F (h) = h?1 : G ! F is a morphism of `-frames, where F = hX; ; (Ri)i2I i and G = hY; ; (Si)i2I i are the dual frames of L and M, respectively. From the lattice duality (part I) h?1 = F (h) is continuous and its inverse F (h)?1 takes stable compact-opens to stable compact-opens, preserving interesctions and binary joins of stable compact-opens. It remains to verify that F (h)?1 : F ?! G preserves the operators on stable compact-opens induced by the relations on the frames. Recall that the elements of F are the sets Xa = fx 2 X ja 2 xg; a 2 L and similarly G consists of the sets Ye = fy 2 Y je 2 y g; e 2 M . For a xed index i we need to verify F (h)?1 Ri (Xa1 ; : : :; Xan ) = Si(F (h)?1 Xa1 ; : : :; F (h)?1 Xan ) (8) From the functoriality of the lattice representation F (h)?1 Xaj = Yhaj . Since Ri and Si interpret the operators fi and gi , respectively, we have Ri (Xa1 ; : : :; Xan ) = Xfi (a1 ;:::;an ) and Si(Yha1 ; : : :; Yhan ) = Ygi (ha1 ;:::;han ) . Then F (h)?1 Xfi (a1;:::;an) = Yhfi (a1 ;:::;an ) . Since h is a homomorphism we have hfi (a1; : : :; an) = gi(ha1; : : :; han, hence equation 8 holds. 24

We note also that by the same argument as in the proof of proposition 4.1 in part I, if h is a monomorphism then F (h) is a surjective map and if h is an epimorphism then F (h) is injective. For the duality part, if F = hX; ; (Ri)i2I ; Ci is an `-frame, then X = hX; ; ?i is a lattice space (where ? is the closure operator induced by the binary relation ). By lattice duality X is the dual space of a lattice L and the family C of stable compact-opens is the family of sets fXaja 2 Lg where Xa is the set of lters containing a. Furthermore, there is a lattice isomorphism L  = C . The isomorphism induces then operators fi ; i 2 I on L, de ned by fi (a1; : : :; an ) = b i Ri (Xa1 ; : : :; Xan ) = Xb . Thereby, the `-frame F is the dual frame14 of the `-algebra L = hL; ; ^; _; (fi)i2I i. A similar argument applies for morphisms h : F ! G of `-frames, showing that they arise from lattice-algebra homomorphisms, along the lines of the proof in lemma 4.5, part I. 2

Remark 8.7 We call an `-frame F = hX; ; (Ri)i2I ; Ci regular if the relation  is a partial order  on X . By duality every `-frame is isomorphic, in the sense of de nition 8.4, to a regular frame. In the sequel, therefore, we will assume that `-frames are regular since this is really no loss of generality.

duality and given an `-frame F we need to show that F  (F ) , that is that the frame is isomorphic to the dual frame of its dual algebra. What we have shown is that there is an `-algebra L isomorphic to the dual algebra of F and that F  L , that is to say F is isomorphic to the dual frame of this algebra. Since L  = L. = C this amounts to the desired isomorphism F  (F ), since F = C  14 For

25

Part III Semantic Analysis of Normal Algebraizable Logics As mentioned in the Introduction the motivation for developing the framework of parts I and II has been to establish a systematic procedure for constructing canonical frames for a variety of more or less familiar logical systems (and fragments thereof) as well as to obtain a duality for their algebraic and Kripke-style semantics. In section 10 we provide concrete applications of our framework by considering the Full Lambek Logic and various extensions of it by addition of structural rules. First we x a notion of normal algebraizable logic, instances of which are the various systems considered in section 10

9 Normal Algebraizable Logics Fix a sentential language L , de ned in the usual recursive way over a signature  of logical operators which contains at least ^ and _. We will only consider languages over a signature of nitary operators. The set of G-terms (Gentzen-terms) G over the language L (we drop the subscript  for simplicity) is inductively de ned as the smallest set containing all sentences of L and closed under pairing. The subterm relation on Gterms is de ned as the transitive closure of the relation: ?  ?. If ? = (?1 ; ?2) then ?1 ; ?2  ?. We denote the occurrence in ? of a subterm
of ? by ?[
]. An n-ary rule R is a sequence of length n + 1 of pairs of G-terms (sequents) where the rst n entries are the premises of the rule and the last entry is the conclusion of the rule. In particular a 0-ary rule is an axiom. A logic L over a set of rules R1; : : :; Rm is the smallest subset L  G  G closed under the rules, that is to say if (?1;
1); : : :; (?n;
n) 2 L and Ri; i 2 f1; : : :; mg, is a rule with premises (?1;
1); : : :; (?n;
n) and conclusion (?;
), then also (?;
) 2 L (in particular L contains all axioms in the above sense). We write a pair (?;
) in the more familiar form ? `L
(we will hereafter drop the subscript L). As usual, an L-derivation from sequents S1; : : :; Sn is a tree of sequents such that (1) the topmost sequents are the sequents S1 ; : : :; Sn, and (2) every sequent except the lowest one is a premise of a rule whose conclusion is also in the derivation. An Lproof is a derivation from axioms (initial sequents). Assuming a symmetric consequence presentation15 of L in a Gentzen-style system we single out the following structural rules that may be among the rules of the logic L, where we simplify notation to ?[?1 ; ?2] for the more accurate ?[(?1 ; ?2)]: 15 The logic L is in asymmetric consequence presentation just in case for every sequent ? `
we have
= , for some sentence  in L. It is in symmetric implicational presentation if sequents are restricted to the form  ` . And it is in classical theoremhood presentation if sequents are restricted to the form ; `  (which we write as ` ), in which case L  L. L is in symmetric consequence presentation, if sequents are of the most general form (?;
) where both ? and
may be non-trivial G-terms.

26

Left Rule

Right Rule

?[; ] `
(Le) ? `
[; ] (Re) ?[; ] `
? `
[; ] ?`
? `
(Weakening) ; ? `
(Lw) ? ` ;
(Rw) ; ] `
? `
[; ] (Rc) (Contraction) ?[ ?[] `
(Lc) ? `
[] ?[(; ); ] `
? `
[(; ); ] (Association) ?[; (; )] `
(La) ? `
[; (; )](Ra) We single these rules out because dropping a combination of them results in the introduction of additional connectives in the logic, enriching the underlying signature , as is well known. If L is closed under all structural rules, then our G-terms can be merely regarded as sets of sentences16 . Dropping contraction amounts to taking multisets as our G-terms, while dropping exchange as well results in our G-terms being taken to be sequences of sentences. G-terms as we have de ned them, as pairs of pairs of ... of sentences, correspond to dropping contraction exchange and association at the same time. As is well known,V in classical propositional logic the sequent ? `
is derivable just W in case the sequent ? `
is, that is i the conjunction of members of ? entails V the disjunction of members of
. Similarly in intuitionistic logic, where ? `  i ? ` . For a dierent example, a Classical Linear Logic sequent ? `
is derivable just in case (?) ` +(
), where if ? = h1; : : :; ni and
= h 1; : : :; mi, then (?) = 1 


 n and +(
) = 1 +


+ m . Similarly for Intuitionistic Linear Logic ? ` is derivable i (?) ` is. For a more involved example we mention systems studied in [4], where ? ` translates to 1 # (2 


 n ) ` where # and  are what we call \application" and \composition", respectively, connectives. We will assume then that the logic has \enough" connectives so as to group together the subterms of ? to a sentence  (?) and similarly for
to a sentence (
). More speci cally we assume that for a sentence ;  () = () =  and ? `
is a derivable sequent i  (?) ` (
) is. In all the examples mentioned above the underlying signature of logical connectives contains an implication operator ! and the logic has a deduction theorem, so that one of ? `
,  (?) ` (
), or `  (?) ! (
) is derivable i any other is. In the following de nition   abreviates the assertion that both  ` and `  are derivable. De nition 9.1 A logic L is standard if it is closed under the rules of Identity and Cut (Exchange)

] [] `  (Identity) (; ) 2 L (Cut) ? `
[ [?] `
[] 16 The presence of structural rules induces an equivalence relation on G-terms and factoring out by that equivalence is essentially the same as starting out with merely sets.

27

and under the rule of replacement: If  is an n-ary logical operator and i  i for i = 1; : : :; n, then (1; : : :; n)  ( 1; : : :; n)

For a sentence  we let [] = f j   g, the set of provable equivalents of . An immediate consequence of the de nition of a standard logic is that the logic is algebraizable in the standard Lindenbaum-Tarski sense. The Lindenbaum algebra of L is the algebra A = L= consisting of the equivalence classes [] and with induced operators A ([1]; : : :; [n ]) = [(1; : : :; n)]. Assuming Identity and Cut the relation of deductive entailment translates to a partial order. We always assume that the signature  contains the logical operators ^ and _ with the standard Gentzen rules for introduction and elimination so that A is a lattice-ordered algebra.

De nition 9.2 By a normal algebraizable logic we mean a logic L on a signature  = f^; _g[OP that is standard (de nition 9.1) and such that in the algebra L= = A (where  is provable equivalence) the algebraic operators induced from logical operators in OP

are normal in the sense of de nition 6.1. Furthermore, if  and  are the \translation" maps discussed above such that ? `
is derivable i  (?) ` (
) is, we assume that both  and  are normal operators (possibly composite, in the sense of lemma 6.2 ).

The framework we have developed in parts I and II can be applied to any normal algebraizable logic in the above sense17 , resulting in the construction of canonical frames for a Kripke-style semantic treatment and in a duality of algebraic and Kripke semantics for the logic. In the next section we provide a concrete example, applying our framework to the family of extensions by structural rules of the Full Lambek Logic.

10 Structural Extensions of the Full Lambek Logic

The systems we consider arise from the Gentzen system LJ for Intuitionistic Logic by dropping a combination of structural rules. Ono [36] gives a discussion of the algebraic models for such logical systems. We assume familiarity with the Gentzen-style presentation of Intuitionistic Logic and brie y review a few things from Ono [36]. 18 Intuitionistic Full Lambek Logic is the system obtained from the Gentzen system LJ for Intuitionistic19Logic by dropping the structural rules of exchange (e), weakening (w) and contraction (c) and augmenting the underlying signature of the logic with the 17 The

assumption of a logic presented in a symmetric consequence form is not essential and our de nition of normal algebraizability can be adapted to systems presented, for example, in an axiomatic form. All we need to assume is that the logic is algebraizable in the standard sense and that the logical operators give rise to normal operators in the Lindenbaum algebra. For examples of logical systems that are not algebraizable in the above sense and for a more general investigation of the question of algebraizability of a logic we refer the reader to Blok and Pigozzi [7]. 18 For lack of space we do not give a thorough review, nor do we present the proof systems associated to the various types of algebras. For details we refer the reader to Ono [36, 35]. 19 Note that the rule of association is maintained.

28

operators  (times) and (reverse implication) and the constant 1.20 Thus the signature of logical operators is  = f^; _; ; ; !; 1; >; ?g. The left and right introduction rules for  are given by ?[; ] `  ? ` 
` (R) ?[  ] `  (L) ?;
`   For purposes of comparison we list also the rules for conjunction ?[ ] `  (L ^) ? `  ? ` (R^) ?[] `  (L ^) 1 ?[ ^ ] `  ?[ ^ ] `  2 ? ` ^ The rules for _ and ! are the usual ones for intuitionistic logic. As is well known, the rules for ! introduction in combination with the rules for  entail that for any sentences ; ;  the sequent   `  is derivable i the sequent `  !  is. In the absence of the exchange rule the operator  is noncommutative and the implication operator splits in two, as well. Reverse implication has the following introduction rules ? `  ;
`  (L ) ?;  ` (R ) ; ?;
`  ?`  The constant 1 is introduced by the following rules/axioms ?` ?` ` 1 (R1) ?; 1 `  (L1) 1; ? `  (L1) In the absence of the exchange rule there are two versions of (L1), as above. The meaning of the empty side of the ` is given by the following immediate consequence of the above. Lemma 10.1 For any sentence  `  i 1 ` . 2 We may then equivalently adopt this as a de nition of an empty left hand side of the ` in which case the axiom (R1) becomes redundant as it is equivalent to the instance 1 ` 1 of the identity axiom. The constant ? has only a left introduction licensed by the axiom ?; ? ` . The constant > is regulated by the axiom ? ` >. By FL we denote ambiguously either the logical calculus described above or its Lindenbaum algebra. The following is a well known fact, where a; b; c are equivalence classes of sentences (under provable equivalence). Proposition 10.2 For all a; b; c; a  b  c i b  a ! c i a  c a, that is to say for each element e the operator ( )  e has the right residual ( ) e and similarly e  ( ) has the right residual e ! ( ). Furthermore, the operator ( )  ( ) distributes over (any existsing) joins in each argument place while the operators ( ) e and e ! ( ) distribute over existing meets. Finally, the operators a ( ) and ( ) ! a codistribute over binary joins. 2 20 In the

absence of multiple conclusions the constant 0, the dual of 1, becomes obsolete and we thus ommit it altogether.

29

The point that merits attention is that in LJ distribution of ^ over _ is a consequence of residuation of ^ with !, which no longer holds when either of c, e or w is dropped. Thereby the logical systems that arise are non-distributive, unless distribution is explicitly postulated (as in Relevance Logic, dropping w). Following Ono [36] we denote extensions of FL obtained by adding structural rules by FLr , where r is a subset of fc; w; eg. Thus FLc assumes contraction, FLcw assumes both contraction and weakening but drops exchange and FLcwe is LJ. There are therefore 23 = 8 distinct systems. Each system may assume or drop the distribution law, with the exception of FLecw = LJ for which distribution is a consequence of residuation of ^ with !, hence a family of 15 systems arises. Our semantic treatment can be extended to various fragments of these systems obtained by restricting the signature of available logical operators. That this is possible follows from the fact that our representation of operators does not rely on certain families of operators being simultaneously present but rather treats each operator separately. The following is an immediate consequence of the Gentzen rules for the respective systems. Lemma 10.3 1. In FLew , for all a; b; a  b  a ^ b. In FLc the converse holds, namely a ^ b  a  b. Hence in FLecw (that is to say, in LJ) the operators  and ^ coincide. 2. In FLw for any a and b the inequality b  a  a holds, while in FLc the inequality a  a  a holds. Hence in FLcw the operator  is idempotent. 2 Concluding with the general review on the operators in FL-algebras we recall that 1 is an identity element for the operator , that is to say a  1 = 1  a = a. The constant ? (falsum) is a bottom element in the Lindenbaum algebra, namely ?  a for all a. Based on the above observations the various FLr -algebras with r 2 fe; c; wg can be characterized. Such characterization can be found in Ono [36]. For reader's convenience we summarize it below.

10.1 Algebraic Semantics

We systematize now the characterization of the algebraic semantics for the various systems considered. For detailed discussion of the correspendance of logical and algebraic principles we refer the reader to Ono [36] and Dunn [14].

FL-Algebras: The operators ^; _; ; ; !; 1; >; ? are all distinct and the following is an axiomatization of an FL-algebra A = hA; ; ^; _; ; ; !; 1; >; ?i. FL1. (A; ; ^; _; ?) is a lattice with bottom element ? FL2. (A; ; ; 1) is a p-groupoid (that is  is a monotone, associative operation) with two-sided identity 1 30

FL3. (A; ; ; !) is a double implicational lattice, that is both and ! are monotone

in their consequent position and antitone in their antecedent position and they satisfy the axiom b  a ! c i a  c b FL4. (A; ; ; ; !) is a residuated p-groupoid, that is a  b  c i b  a ! c i

ac b FL5. a  (b _ c) = (a  b) _ (a  c) and (b _ c)  a = (b  a) _ (c  a) FL6. (a _ b) ! c = (a ! c) ^ (b ! c) and a ! (b ^ c) = (a ! b) ^ (a ! c) FL7. c (a _ b) = (c a) ^ (c b) and (b ^ c) a = (b a) ^ (c a)

Remark 10.4 The reader will have noticed some redundance in our axiomatization.

For example the residuation axiom FL4 implies monotonicity (FL2) and distribution over joins for  (FL5). Similarly, it implies the monotonicity (FL3) and distribution properties for and ! (FL6, FL7). We have in fact provided independent axiomatizations of operators, as we wish to be able to consider fragments of FL-algebras, removing some of the operators. This corresponds to considering fragments of the full Lambek logic, by restricting the underlying signature of logical operators.

FLe -Algebras: Assuming exchange forces commutativity on the operator . As a concequence the operators ; ! coincide and the signature we obtain is f^; _; ; ! ; 1; >; ?g, axiomatized as for FL-algebras (with the obvious changes) plus the exchange (commutativity) axiom FLe . a  b = b  a

FLw -Algebras: The signature is f^; _; ; ; !; 1; >; ; :g axiomatized as for FLalgebras, together with the following weakening axiom FLw . For all a; b; b  a  a FLc-Algebras: The signature of logical operators is the same as for FL-algebras with the additional contraction axiom FLc. For all a; b; a ^ b  a  b We point out the following instance of the above axiom: For all a; a  a  a

FLew -Algebras: The signature is reduced to f^; _; ; !; 1; >; ?g since  is commutative. The following axiom holds in addition to the weakening axiom FLw 1 FLew . For all a; b; a  b  a ^ b 31

FLec -Algebras: The signature is the same as for FLe and the contraction axiom FLc is assumed.

FLcw -Algebras: The axioms for FL-algebras together with the weakening axiom FLw and the contraction axiom FLc hold.

FLecw -Algebras are simply the Heyting algebras. Lemma 10.5 1. The operators ; ; ! are normal in the sense of de nition 6.1 2. In every extension by structural rules of the Intuitionistic Full Lambek Logic the sequent ? `  is derivable i the sequent  (?) `  is, where  is de ned by  ( ) = , for a sentence , and if ? = (
; ) then  (?) =  (
)   (). The rst part is immediate from the axioms listed above. For the second part, since the rule of association is assumed (which algebraically amounts to associativity of the operator ) a G-term ? can be regarded as a sequence ? = 1 ; : : :; n , hence  (?) = 1 


 n and the claim above follows simply from the left introduction rule for the times operator . 2

Proposition 10.6 (Algebraic Soundness and Completeness) Where r is a subset of fc; e; wg, the logic FLr is sound and complete with respect to FLr -algebras. It is immediate (by induction on the length of the proof) that if B is an FLr -algebra and j j an interpretation of sentences in B, then if ? `  is derivable the inequality j  (?)j B B j j B holds. For completeness, as usual, every non-derivable sequent ? 6`  is invalidated in the Lindenbaum algebra of the calculus since then [ (?)] 6 [] will have 2

to hold.

10.2 Kripke Semantics

10.2.1 FL-Frames

An FL-frame is a regular ncf (normal closure frame, de nition 8.1 and remark 8.7, part II) of the form F = hW; ; Ro; R ; R!; I; ?; Ci where W is a set of information states,  is a partial order on states such that for any set fxiji 2 I g of information states the largest state ui2I xi below each xi and the smallest state ti2I xi extending each xi both exist, 32

C is a collection of stable sets of states (the facts, or propositions of the frame) with W; I; ? 2 C , where ? = f!g, where ! is the inconsistent world and R ; Ro; R! are ternary relations on W . Conditions need to be imposed on the relations so that the frame becomes appropriate for some system obtained as structural extension of the Full Lambek logic. We rst recall from part II that the relation R (and similarly for R ; R!) induces an operator R (cf equation 3 and the proof of lemma 7.2), which we will denote here by , de ned on stable sets A; B by

RAB = A B = Rxy where A = ?x; B = ?y Recall also that associated to the relation R is a state transforming map r (equation 5) de ned by rxy = z i Rxy = ?z . For simplicity of notation and for mnemonic reasons we use the in x symbol ^, setting rxy = x^y . In our de nition of an `-frame (de nitions 8.1, 8.5) we assumed that the relation R can be recovered from r and  by the condition R xyz i z  rxy = x^y Similar considerations apply to the relations R (and R! ) where we write B ( A (respectively A ) B ) for R AB (respectively for R! AB ) and (respectively !) for the associated operator on states. We will also need to make use of the relation R  (equation 7) which is de ned in our particular case by R xyz i x^y  z , and similarly for the relations R and R ! . Recall nally that for every stable set A there is a unique point xA such that A is the closure of xA . In particular we let e be the state such that I = ?e. For clarity's sake let us also recall our convention that for a set U; U  x means that for every y 2 U; y  x and similarly for x  U . As has been our practice before the set fz jRxyz g will be denoted by Rxy . Finally, we let W 0 be the set of states x such that ?x is a proposition (?x 2 C ) of the frame. The following table contains a number of conditions we consider on frames where for convenience and clarity we use both the relations and their associated state transforming maps. In the table all states mentioned are in W 0 and then the conditions listed are conditions on the restrictions of the relations and state transformers to W 0 .

33

TABLE 1

1. Additivity

If x = x1 u x2; y = y1 u y2 , then T Rxy  z i =1 2 R x y  z R(x^y)z  u i Rx(y^z )  u Rxyz i Ryxz Ryxx R xy(x t y) Rxy(x t y) i;j

2. FL2 3. FL 4. FL 5. FL 6. FL 1 e

w

c

ew

); (

i

;

j

If x = x1 u x2; y = y1 t y2 , then 7. )-Normality R! xy  z i R!x y  z for all i; j = 1; 2 8. (-Normality R xy  z i R x y  z for all i; j = 1; 2 i

j

i

j

Res

9. ( and ) 10. (; ; )

R ! xzy i R yzx Rxyz i R !xzy i R yzx

I, ?

11. Identity 12. Bottom

Rex  y i 8u (u  x implies u  y) ?  A (that is ! 2 A), for any stable set A

Theorem 10.7 Where r  fe; c; wg a frame F is an FLr-frame i its dual algebra F is an FLr -algebra. We prove this theorem in a series of propositions and lemmas. Proposition 10.8 The following are equivalent to the conditions in the above table with the same number, where A; Ai; B; Bi; C are propositions of the frame (members of C ): W W W 1. (A1 A2) (B1 B2) = i;j =1;2 (Ai Bj ) 2. A (B C ) = (A B ) C 3. A B = B A 4. B A  A T 5. A B  A B T 6. A B  A B 34

For the additivity condition let x1; x2; y1; y2 and x = x1 u x2; y = y1 u y2 be the points in W0 generating W the sets Ai ; BWi and their joins. Speci cally, we let Ai = ?xi; Bi = ?yi , so that A1 A2 = ?x andS B1 B2 = ?y . Condition (1) in TABLE 1 is equivalent to the identity Roxy =  i;jW=1;2 Roxi yW j . By de nition of we have W Roxy = ?x ?y and since ?(x1W\ x2 ) = ?x1W ?x2 = A1 A2 (and similarly for B1 B2) it follows that RSoxy = (A1 A2) (SB1 B2). Similarly, since Ro xi yj = Ai Bj it follows that  i;j=1;2 Roxi yj = ? (Ai Bj ) so that (A1

_

A2 ) (B1

_

B2 ) = R xy = 

[

o

(A B ) = i

i;j

=1;2

_

j

(A B ) i

i;j

j

=1;2

For associativity, let A; B; C be stable sets in C generated by the points x; y; z , respectively. Since B C = ?(fy ^z g) = Ryz and A B = ?(fx^y g) = Rxy the associativity identity can be re-written as the identity Rx(y ^z ) = R(x^y )z which is precisely condition 2 in the table. For commutativity, A B = B A is equivalent to the identity x^y = y ^x (where A = ?x; B = ?y). Then Rxyz i z  x^y i z  y ^x i Ryxz. For the weakening inclusion, the condition Ryxx is equivalent to x  y ^x which, in turn, is equivalent to fy ^xg  fxg. Since also for a singleton fxg the identity fxg = fxg and  = ?, it follows that Ryxx is equivalent to B A  A, where B = ?y; A = ?x. The contraction axiom (5 in the table) can be equivalently expressed by the inequality x^y  x t y. Given that stable sets are closures of single points this is equivalent to ?(Tx t y )  ?(x^y ). If A = ?x; B = ?y , the above inclusion is equivalent to the inclusion A B  A B. The argument for the Exchange/Weakening condition is similar. 2

Lemma 10.9 Condition 11 in the above table is equivalent to the assertion that I is a two-sided identity for . The equations I A = A I = A are equivalent to the equations Rex = Rxe = ?x

(where A = ?x). This is simply by the proof of lemma 7.2. These identities express the fact that the upper bounds of the set Rex are the same as those of Rxe and they are equal to the closure of the singleton fxg, which is precisely the content of the condition on Identity in the table. 2

Lemma 10.10

1. The )-normality condition (7 in the table) is equivalent to the following two (co)distribution properties of ):

W

T

(a) ( i=1;2 Ai ) ) B = i=1;2 (Ai ) B ) T T (b) A ) ( i=1;2 Bi) = i=1;2 (A ) Bi ) and similarly for the (-normality condition (8 in the table). 2. Condition 9 in the above table is equivalent to the statement: A  C ( B i B  A ) C , for all propositions A; B; C .

35

The proof of the rst part is similar to the argument given for the distribution property of . For the second part assume A; B; C are stable sets generated by x; y; z . The condition R ! xzy is equivalent to the inclusion B  A ) C . This follows from R !xzy i x! ^ z  y. The latter is equivalent to the converse inclusion of the closures of the points y and x! ^ z , that is to say to the inclusion ?y  ?(x! ^ z ). By de nition of ! ^ , x! ^ z = u i ?u = R! xz = A ) C , hence R !xzy holds i B  A ) C . Similarly R yzx is equivalent to the inclusion A  C ( B . 2

Lemma 10.11 Condition 10 in the table is equivalent to the residuation condition A  C ( B i A B  C i B  A ) C The proof is similar to the argument for the second part of lemma 10.10.

2

In parallel with the FL-algebras of various types we list below FL-frames of the corresponding type with the conditions from table 1 axiomatizing the relations. FL-frames are of course `-frames and therefore the conditions axiomatizing `-frames (de nition 8.5, part II) are assumed. The proof of theorem 10.7 follows from the previous proposition and lemmas and by inspection of the de nition of an FLr -frame.

FL-Frames F = hW; ; R ; R; R!; I; ?i: The frames are axiomatized by conditions 2 and 10-12. In particular, ? = f! g, where ! 2 W is the inconsistent world. We point out that residuation of with ( and ) implies the distribution properties of

these operators, hence their equivalent conditions 1 and 9-10. Restricting to fragments of the frame that do not contain all three relations at once requires that we add 1, 7, 8 or 9, as appropriate.

FLe -Frames Fe = hW; ; R; R!; I; ?i: The frames are axiomatized by R = R!,

the commutativity and residuation conditions 3 and 10, and conditions 11-12.

FLw -Frames Fw = hW; ; R ; R; R!; I; ?i: Same axiomatization as for FL-frames, with the addition of condition 4. FLc-Frames Fc = hW; ; R ; R; R!; I; ?i: Same axiomatization as for FL-frames with the addition of the contraction condition 5.

FLew -Frames Few = hW; ; R; R!; I; ?i: The axioms for FLe and FLw -frames are

extended to include condition 6.

FLec -Frames, FLcw -Frames, FLecw -Frames are axiomatized by putting together the appropriate groups of axioms (for FLe , FLew , FLc and FLw -frames). 36

10.2.2 FL-Models

An FL-model is a pair M = (F ; j j ) where F is an FL-frame and j j is an interpretation map on sentences  such that j j 2 C and j 1j = I; j >jj = W; j ?jj = ? = f! g (we use ? both for the bottom element of the lattice and the falsum object of the logic as well as for the singleton set containing the inconsistent world but it should be clear from context what the notation means in each particular case). Furthermore j  ^ j = fxjx 2 j j and x 2 j j g = j j T j j j  _ j = fxj8y8z [(z 2 j j [ j j implies y  z ) implies y  x]g = j j W j j j   j = fxj8y8z 8u (R yzu; j j = ?y and j j = ?z implies u  x)g = j j j j j  ! j = fxj8y8z 8u (R! yzu; j j = ?y and j j = ?z implies u  x)g = j j ) j j j j = fxj8y8z 8u (R! yzu; j j = ?y and j j = ?z implies u  x)g = j j ( j j As usual models can be equivalently de ned by means of a satisfaction relation j= inductively de ned as follows, given an assignment of propositions j j 2 C for atomic sentences: x j=  i x 2 j j , for atomic  x j=  ^ i x j=  and x j= x j=  _ i 8y8z [(z j=  or z j= implies y  z ) implies y  x] x j=   i 8y; z; u [(Ryzu; 8y0 (y0 j=  implies y  y0 ); and 8z 0 (z 0 j= implies z  z 0 ) implies u  x ] x j=  ! i 8y; z; u [(R! yzu; 8y0 (y0 j=  implies y  y0 ); and 8z 0 (z 0 j= implies z  z 0 ) implies u  x ] x j=  i 8y; z; u [(R yzu; 8y0 (y0 j=  implies y  y0 ); and 8z 0 (z 0 j= implies z  z 0 ) implies u  x ]

The Semantics of Disjunction: Apparently, the clause x j= ' _ i x j= ' or x j=

is not appropriate for non-distributive logics. One direction is unproblematic, in the sense that we expect a witness w that is either a witness for ' (w j= ') or one for (x j= ) to be one for the disjunction (x j= ' _ ). But these cannot be all the witnesses for ' _ . This is for the simple reason that, considering the set j ' _ j of witnesses that model a disjunction, the above clause forces j ' _ j = j 'j [jj j . Since conjunction is also canonically interpreted by intersection, it is clear that the usual clause for disjunction has a built-in assumption of distribution. There must then be more witnesses for ' _ than the set of witnesses that support either ' or . To give an intuitive account of how to \compute" these extra witnesses, let us introduce some terminology and de ne a direct witness (for some sentence ) by induction on the structure of . If  is atomic we admit every witness for  as a direct one. Otherwise, we call x a direct witness if its supporting the truth of  is a consequence of its delivering a verdict (true, or false) for an immediate subsentence of . For example, every witness for a conjunction is direct, since x j= ' ^ just in case x j= ' 37

and x j= . For a non-example, consider the relational semantics for Relevance Logic. If  = '  , then z j=  just in case there are witnesses x; y such that x j= '; y j= and Rxyz (where R is the ternary relation interpreting relevant implication (cf. RoutleyMeyer [38], or Dunn [12]). The same applies of course to a modal language, with the clause for 2'. In the case of disjunction, the direct witnesses for ' _ are the witnesses supporting either of the two disjuncts, that is to say the witnesses in the set j 'j [jj j . By a cowitness for a sentence  we shall mean a witness that is not direct. The set of witnesses j ' _ j is then to be obtained from the set j 'j [jj j of direct witnesses by somehow \completing", or \closing" it by throwing in all the cowitnesses as well. This of course is exactly the way we modeled disjunction in our lattice representation, in part I. Using our results on modeling joins, and in accordance to the discussion above, the semantic clause for disjunction that we proposed is

x j= ' _

i 8y (if y  j 'j [ j j ; then y  x )

Apparently every x such that x j= ' or x j= satis es the above clause (recall that for a set U , the abbreviation y  U means that y  u for all u 2 U ). The role of the binary relation  in the frame is then precisely to provide us with a way of \computing" not only the direct witnesses for the disjunction, but the cowitnesses as well. In terms of the closure operator ? induced by the relation , it should be clear that j ' _ j = ?(j 'j [ j j ). For a dierent intuition on the semantic clause for disjunction let us read x  y as: the state (of information) x extends to the state y . Then x j=  _ i x extends every state y that can be extended to all states at which at least one of  or is veri ed.

10.3 Soundness and Completeness

Where r  fe; c; wg we write Fr for an FLr frame and Mr for an FLr model. Theorem 10.12 (Soundness) For every model Mr = hFr; j j i, where r  fe; c; wg, if ? `  then ? j=M . That is to say ? `  implies j  (?)j  j j . The proof, as usual, is by induction on the length of the derivation of the sequent ? ` . For length 1 the sequent must be initial, i.e. an axiom, hence of the form  `  in which case the claim is obvious. Otherwise the proof reduces to verifying that the rules are sound. Soundness of the exchange (if assumed) and association rules follows from proposition 10.8, parts 2 and 3. Soundness of the weakening rule, if assumed in the logic, follows by the same proposition, part 4. We give some more detail, as an example. Since ? `  and ; ? `  are equivalent to  (?) `  and ;  (?) ` , by lemma 10.5, we may assume that ? is a single sentence. Hence soundness of the rule reduces to showing that if j j  j j then j j j j  j j . This follows by monotonicity of and by part 4 of 10.8: j j j j  j j  j j . Note that the monotonicity property of is an immediate 38

consequence of additivity (proposition 10.8, part 1). Soundness of the contraction rule, if assumed, again follows by monotonicity and proposition 10.8, part 5. Indeed, since j j  j j j j it follows that j  (?[])j  j  (?[; ])j  j j . For the introduction rules for logical operators, soundness of the rules for conjunction and disjunction is immediate and that for  introduction follows from lemma 10.5. The rules of right introduction for and ! are sound by lemma 10.11. For the left introduction rules we do only the rule for . By lemma 10.5 we may assume that the ?'s and
's occuring in the rule are single sentences. Hence we assume that j 1j  j j ; j j j 2j  j j and need to verify that j j j 1j j 2j  j j . This follows by the following calculation j j j 1j j 2j  (j j ( j j ) j j j 2j  j j j 2j  j j where we used the fact that (j j ( j j ) jjj  j j . This fact holds by the residuation lemma 10.11 since it is equivalent to j j ( j j  j j ( j j . Soundness of the rules/axioms for the constants 1; >; ? is immediate. 2

Theorem 10.13 (Completeness) If r  fe; c; wg and ? `  is a nonderibale sequent of FLr , then there exists a frame Fr and an interpretation j j of sentences as stable sets such that j  (?)j 6 j j . The underlying set of the frame is the set of lters of the Lindenbaum algebra of the calculus where the inconsistent world ! is the improper lter. The closure operator is the operator induced by the relation of lter-inclusion, as explained in the section on lattice representation. The family C of propositions of the frame is the fa,ily of stable compact-open subsets of the frame. The interpretation j j is de ned by

j j = W[] = fw 2 W j [] 2 wg where [] is the equivalence class of . The relations R; R ; R! are de ned as in the section on representation of normal operators. Speci cally, we de ne operators ^ ; ^; ! ^ on worlds ( lters) by

\ x^y = fxabjx  xa and y  xb g _ x! ^ y = fxa!b jx  xa and xb  y g _ y ^ x = fxb a jx  xa and xb  yg

(9) (10) (11)

where for an element e of the Lindenbaum algebra xe = e" is the principal lter generated by e. Then where ] is any of ^ ; ^; ! ^ we de ne R]xyz i z  x]y . We can then establish completeness after proving proposition 10.14.

Proposition 10.14 The canonical frame F = hW; ; R ; R; R!; I; ?; Ci is an FLr -

frame.

39

By proposition 10.7 it is enough to verify that the dual algebra

hC ; (; ; ); I; ?; W i of the canonical frame is an FLr -algebra. Now elements of C are the sets W[] = j j where [] is the equivalence class of the sentence  and w 2 W[] i [] 2 w. By theorem 7.4, part II, R(j j ; j j ) = j j jj j is identical to j   j = W[ ] and similarly for the other operators. Hence associativity of and residuation with (; ) follows immediately from the respective properties in the Lindenbaum algebra. Similarly, I = j 1j is clearly a 2-sided identity for , since 1 is one for . If e 2 r (that is the logic assumes exchange), then  is commutative and then so is

by the same argument as above. The other cases are shown in the same straightforward manner. 2 Having established that the canonical frame is an FLr -frame, completeness of the logic FLr now follows easily. As our canonical interpretation we have chosen the map j j sending a sentence  to the set X[] of lters that contain its equivalence class. Since ? `  i  (?) `  i [ (?)]  [] i X[ (?)]  X[] it follows that if ? `  is a nonderivable sequent of FLr , then the inclusion X[ (?)]  X[] fails, which is to say that j  (?)j 6 j j and this establishes completeness of the logic. Hence the proof of theorem 10.13 is complete. 2

10.4 The Duality of Algebraic and Kripke Semantics

The frames we have considered in our semantic analysis are regular ncf's (de nition 8.1 and remark 8.7) with relations R; R and R! to interpret the respective operators. In the construction of the canonical frame we have applied the representation technique of part II by which it follows that an FLr -algebra A is isomorphic to a subalgebra of its dual frame. In our completeness argument we have shown that the dual frame of an FLr -algebra is an FLr -frame. Again in part II we veri ed that the representation extends to morphisms of such algebras. To obtain a duality we apply the argument of part II and restrict the class of frames considered to FLr -frames that are also topological and re ned ncf's, that is to say to frames Fr = hW; ; R ; R; R!; I; ?; Ci, where C is the dual algebra of the frame (de nition 8.3). By the duality argument of part II every such frame is isomorphic to the dual frame of its dual algebra C and every morphism of frames (de nition 8.4) arises from a homomorphism of the dual algebras (theorem 8.6). Hence a duality of algebraic and Kripke-style semantics for the logical systems FLr is established.

11 Further Extensions and Open Problems We considered substructural systems obtained by dropping a combination of structural rules from the Gentzen system LJ for Intuitionistic Logic. The same techniques as we 40

have used apply to systems similarly obtained from the system LK for Classical Logic, extending the signature of available operators to include a grouping operator + for the right-hand side of sequents and an identity element 0 for +. The rules for + and 0 are these familiar from Linear Logic and it is not hard to see that + distributes over meets in each argument place, hence it is a normal operator in our sense and thereby the arguments we have presented apply to yield a relation R+ representing it in the sense of an induced operator . Similarly, the reader probably can apply himself the framework we have presented to (normal) modal extensions of these systems, recently investigated by Dosen [11]. For lack of space we have not considered a comparison of our semantics with alternatives that have been proposed in the literature, such as Ono and Komori [34]. An advantage of our approach is that our framework applies to fragments of the systems considered such as the implication fragment of FLr -logics. Proof-theoretic aspects of this fragment are studied in Ono [35]. In Ono and Komori [34], just as in Troelstra [45] and in Abrusci [1], one needs to assume that both  and ! (or  and both ; ! in the case of [1]) are present at the same time, as the argument for the construction of the canonical frame makes use of residuation of  and !. We have insisted in part III that the signature of logical operators includes both ^ and _ and the reader may well wonder if this is an essential assumption. The only reason for our doing so is that the problem of the Kripke semantics for non-distributive logical systems and of fragments theoreof appeared to be harder to deal with. As we already mentioned, Dunn [14] addresses the question of a representation of partial orders or meet semilattices (perhaps with an involution) with additional operators. What we wish to point out is that our techniques apply in a rather straighforward way to these cases. We have been consistently exploiting the algebraic structure of the lter space in order to de ne operators on points of the space. This can be done in the same way in the case of the lter space of a semilattice or a partial order (where \ lter" means simply an upper closed set). In Hartonas and Dunn [27] we have provided representation and duality theorems for partial orders and semilattices which can be then exploited to obtain a duality of algebraic and Kripke semantics for appropriate systems. Normality of operators, which has been a central theme in the case of systems with ^ and _ is no longer an issue. From that respect the enterprise for these cases appears to be quite simpler.

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