Duality Theory, Canonical Extensions and Abstract Algebraic Logic

DUALITY THEORY, CANONICAL EXTENSIONS AND ABSTRACT ALGEBRAIC LOGIC

MAR´IA ESTEBAN GARC´IA

Resumen. El principal objetivo de nuestra propuesta es mostrar que la LógicaAlgebraica Abstracta nos proporciona el marco teóricoadecuado para desarrollar una teor´ıauniforme de la dualidad y de las extensiones canónicaspara lógicasno clsicas. Una teor´ıatal sirve, en esencia, para definir de manera uniforme semánticas referenciales (e.g. relacionales, al estilo Kripke) de un amplio rango de lógicaspara las cuales se conoce una semántica algebraica.

Abstract. The main goal of our proposal is to show that Abstract Al- gebraic Logic provides the appropriate theoretical framework for devel- oping a uniform duality and canonical extensions theory for non-classical logics. Such theory serves, in essence, to deﬁne uniformly referential (e.g. relational, Kripke-style) semantics of a wide range of logics for which an algebraic semantics is already known.

1. Abstract Algebraic Logic Our work is located in the field of Abstract Algebraic Logic (AAL for short). Our main reference for AAL is the survey by Font, Jansana and Pigozzi [14]. We introduce now the definition of logic adopted in AAL. It consists, essentially, on regarding logic as something concerning validity of inferences, instead of validity of formuli. Let L be a logical language, defined as usual from a countably infinite set of propositional variables V ar. We denote by F mL the collection of all L -formulas. When we consider the connectives as the operation symbols of an algebraic similarity type, we obtain the algebra of terms, which is the absolutely free algebra of type L over a denumerable set of generators V ar. We call this algebra the algebra of formuli or the formula algebra and we denote it by FmL . Let us fix a logical language L from now on. We usually omit the subscript of F mL and FmL . A crucial notion in AAL is that of closure operator. C : P(X) −→ P(X), a function C on the power set of a set X, is a closure operator on X (cf. Definition 5.1 in [5]) when it satisfies the following conditions: (C1) for all Y ⊆ X, Y ⊆ C(Y ), (C2) for all Y,Y 0 ⊆ X, if Y ⊆ Y 0, then C(Y ) ⊆ C(Y 0), (C3) for all Y ⊆ X, C(C(Y )) = C(Y ). 1 2 MARÍA ESTEBAN GARCÍA

Conditions (C1)-(C3) are known as extensive, idempotent and isotone respectively. C is finitary or algebraic (cf. Definition 5.4 in [5]), when moreover satisfies: (C4) for all Y ⊆ X and all x ∈ X, if x ∈ C(Y ), then there is a finite Y 0 ⊆ω Y such that x ∈ C(Y 0). When X is the carrier of an algebra X, C is called structural when it satisfies: (C5) for all Y ∪ {x} ⊆ X and all h endomorphism on X, if x ∈ C(Y ) then h(x) ∈ C(h[Y ]).

Any closure operator C on X can be transformed in a relation `C on X as follows: for all Y ∪ {x} ⊆ X

Y `C x iﬀ x ∈ C(Y )

The properties that `C inherits from those of C as a closure operator, deﬁne what is called a consequence relation on X, i.e. a relation `C ⊆ P(X) × X such that:

(C1’) if x ∈ Y , then Y `C x, (C2’) if Y `C z for all z ∈ Z and Z `C x, then Y `C x. Clearly, any consequence relation `C on X deﬁnes a closure operator C` by

x ∈ C`(Y ) iff Y `C x. Now we are left to introduce the formal definition of logic in AAL. Given a logical language L , a logic (or deductive system) in the language L is a pair S = hFm, `S i, where Fm is the formula algebra of L and `S ⊆ P(F m)×F m is a substitution-invariant consequence relation on F m, i.e. `S satisfies the following conditions:

(C1’) if ϕ ∈ Γ, then Γ `S ϕ, (C2’) if ∆ `S γ for all γ ∈ Γ and Γ `S ϕ, then ∆ `S ϕ, (C3’) (structurality) if Γ `S ϕ, then e[Γ] `S e(ϕ) for any substitution e. Equivalently, we could say that a logic in the language L is a pair hFm, CS i, where Fm is the formula algebra of L and CS : P(F m) −→ P(F m) is an structural closure operator. This notion of logic is the standard framework of contemporary AAL. At a first sight, it might seem that only the so called ‘propositional’ or ‘sentential’ logics fall under the scope of this definition. Logics such as ordinary first order logic, quantifier logics or substructural logics seem to be left out. There have been, though, several approaches that accommodate all these logics in the framework of AAL (see Section 1.2 in [14] and its references). We focus on the definition given above, although it would be very interesting to investigate whether our results may be extended also to those wider approaches.

2. Duality theory for non-classical logics The mathematical interest of studying Spectral-like and Priestley-style dualities goes back to Stone’s duality for Boolean algebras [28]. This duality DUALITY THEORY, CANONICAL EXTENSIONS AND AAL 3 has been generalized to distributive lattices in at least three ways (cf. [2] and its references). The approach initiated by Stone himself leads to a representation of distributive lattices in terms of topological spaces hX, τi that are sober, compactly-based (the collection of compact open sets forms a basis) and in which the collection of compact open sets is closed under finite intersections. Duals of algebraic homomorphisms are the so called Spectral functions, that are those maps whose inverse sends compact opens to compact opens. We use the name of Spectral-like dualities for those dualities having as objects of one of the categories, structures of the form hX, τ, . . . i, where hX, τi is a compactly-based sober topological space, and the suspension points indicate that we may have additional structure. A different approach initiated by Priestley [26] leads to a representation in terms of ordered Hausdorff topological spaces that are named Priestley spaces. These are ordered topological spaces hX, ≤, τi that are compact and totally order-disconnected (whenever a ≤ b, there exists a clopen upset U such that a ∈ U and b∈ / U). Duals of algebraic homomorphisms are order- preserving continuous maps. We use the name of Priestley-style dualities for those dualities having as objects of one of the categories, structures of the form hX, τ, ≤,... i, where hX, τ, ≤i is a compact totally order-disconnected ordered topological space, and the suspension points indicate again that we may additional structure. Recently, a third duality for distributive lattices, based on Pairwise Stone spaces has been studied in detail in [2]. Although we will restrict our study to the first two dualities, it would be very interesting to look into the last one as well. What makes Stone/Priestley duality be a powerful mathematical tool is that it allows us to use topology in the study of algebra (and vice versa). Many algebraic notions have its dual translation in terms of nice topological notions. It is precisely the fact that it is a dual equivalence of categories, i.e. the morphisms are reversed, what implies, for instance, that dual of injectivity is surjectivity (and vice versa), duals of subalgebras are order- quotients, and duals of homomorphic images are closed subsets. Regarding logic and theoretic computer science, Stone/Priestley duality has been used for different purposes: Rasiowa and Sikorski [27] applied Baire category theorem to the dual space of the Lindenbaum-Tarski algebra of the first-order logic to provide a topological proof of Gödel’scompleteness theorem for first-order logic; Jónssonand Tarski [23,24] proved relational completeness of modal logic with respect to general Kripke frames via duality (and canonical extensions), several years before Kripke invented his famous semantic for modal logics; Abramsky [1] used Stone duality for distributive lattices to connect specification languages and denotational semantics, thus linking lambda calculus and domain theory; more recently, Gehrke, Gri- gorieff and Pin [18] studied the connection between regular languages and monoids as another case of Stone duality. 4 MARÍA ESTEBAN GARCÍA

Figure 1. J´onsson-Tarski duality for Boolean Algebras

/ Modal o / BAO Stone/Priestley duality DGF / Kripke logic AAL o Frames

Algebraic Relational Semantics Semantics

Jónssonand Tarski’s paper on boolean algebras with operators, ushered a fruitful field of study: the study of the relation between algebraic semantics and relational semantics of logic via dual equivalences of categories. They idea, in brief, is that Stone/Priestley dualities provide us with the notion of Descriptive General Frames (DGF) as the dual topological spaces of Boolean Algebras with Operators (BAO). On the one hand, it is well known that BAO gives an algebraic semantics for Modal logic. On the other hand, when we forget about the topology, DGF turn out to be Kripke frames. This is summarized in the figure above: Although both Stone and Priestley approaches have been followed to generalize this pioneering work on representation of boolean algebras with operators, the Priestley-style approach has been held to be advantageous [21], mainly because it deals with Hausdorff (i.e. nicer) topological spaces. Re- cently, this field of study gained a renewed interest, specially in view of recent developments of the theory of canonical extensions (see [20] and its references). This theory, that we do not treat in detail here due to the lack of space, can be seen as an alternative (but equivalent) way of study such the relation between algebraic semantics and relational semantics of logic. In brief, instead of dualizing and then forget about the topology, what we do is completing the algebras, in such a way that then their dual spaces are discrete (they do not have any topology). Until mid-2000s, all categories of algebras (and homomorphisms) for which Stone/Priestley dualities had been studied where based on lattice-based algebras (i.e. algebras with a lattice reduct), in most cases distributive. This translates in what follows: all logics for which the relation between its algebraic semantics and its relational semantics had been studied via duality theory, were logics having well-behaved conjunction and disjunction connectives. In the recent literature we find further studies that extend the same ideas in mainly two different ways: On the one hand, categories whose objects are ordered algebras with well- behaved operations (e.g. residuated pairs) have been considered. These algebraic structures are the algebraic semantics of substructural logics, such as Lambek calculus and some of its extensions. In the development of this DUALITY THEORY, CANONICAL EXTENSIONS AND AAL 5 line of research, the theory of canonical extensions has played a key role [13, 29, 30]. A modular study of the relational semantics that follow from these studies was developed by Gehrke in [17], where such semantic models were called generalized Kripke frames. On the other hand, categories whose objects correspond to certain fragments of intuitionistic logic (that does not have conjunction and disjunction at the same time) have also been considered. The approach initiated by Stone has been followed in [6–8], whereas the approach initiated by Priest- ley has been followed in [3, 4, 9], among others.

3. The main goal of our proposal. Our aim is to generalize the work in [3, 4, 6–9] within the framework of AAL. We aim to identify the largest class of logics for which a Stone/Priestley duality can be given for the class of algebras that is their algebraic semantics. In order to identify such class of logics, we need to introduce some definitions. We say that a logic S is finitary when the closure operator CS is finitary, i.e. when for all Γ ∪ {ϕ} ⊆ F m, if Γ `S ϕ, then there is a finite subset Γ0 ⊂ Γ such that Γ0 `S ϕ. We say that a logic S has theorems when there is at least one formula ϕ ∈ F m such that ∅ `S ϕ. Let A be an algebra of the same type as S. We call a subset F ⊆ A an S-filter of A when for any homomorphism h from Fm to A and any Γ ∪ {ψ} ⊆ F m such that Γ `S ψ: if h(γ) ∈ F for all γ ∈ Γ, then h(ψ) ∈ F.

We denote by FiS (A) the collection of all S-filters of A. This collection A is a closure system. We denote by CS the consequence operator associated A with FiS (A). Thus for any subset B ⊆ A,CS (B) denotes the least S-filter of A that contains B. We denote by FiS (A) the lattice of all S-filters of A. A Let us consider now the specialization quasi-order ≤S on A associated with A CS , given by: A A A a ≤S b iff CS (b) ⊆ CS (a) A A We denote by ≡S the equivalence relation associated with ≤S . We may define the canonical class of algebras associated with the logic S in several ways. For instance, it may arise from an extension of the Lindenbaum-Tarski method. We choose, though, to present such class of algebras making use of A the relation ≡S : Definition 3.1. An algebra A, of the same type as S, is an S-algebra when A for any congruence θ of A, if θ ⊆ ≡S , then θ is the identity relation. We denote by AlgS the collection of all S-algebras. In the general framework of AAL, other classes of algebraic structures are canonically associated with S, such as reduced S-algebras, (reduced) S-models or (reduced) generalized S-models, each of which serves for a different purpose. In our case, 6 MARÍA ESTEBAN GARCÍA we restrict our attention to AlgS, although it would be interesting to investigate whether our results can be extended to a wider class of logics when we restrict to any of those different algebraic structures. Notice that when we refer to algebraic semantics of a logic we mean precisely to the canonical class of algebras associated with the logic. These and other tools studied in AAL motivate the study of mainly two hierarchies of logics, namely the Leibniz hierarchy and the Frege hierarchy. The Frege hierarchy is a classification scheme of logics under four classes defined in terms of congruence properties of the associated classes of algebras. Congruential logics are one of those classes, being the others fregean logics, fully fregean logics and selfextensional logics. The study of this classification, its structure and its relations with the Leibniz hierarchy started in the late 90’s, and has been intense in the last twenty years (see [15] and its references). We only need to introduce the definition of one of such classes: Definition 3.2. A logic S is called congruential, 1 when for every algebra A A of the same type, ≡S is a congruence of A. Next theorem gives an alternative definition of congruentiality: Theorem 3.3. A logic S is congruential if and only if for every algebra A of the same type: A A ∈ AlgS iff hA, ≤S i is a poset. From previous theorem we infer that for any congruential logic S, A is A an S-algebra if and only if ≡S is the identity relation. Let us introduce one more definition: Definition 3.4. A logic S is filter distributive when for all algebras A of the type of S, the lattice of S-filters FiS (A) is a distributive lattice. The class of filter distributive logics, first considered in [10] and studied also in [11, 25, 31], and includes a lot of well-known logics, for example any axiomatic strengthening of the intuitionistic logic, or any of its fragments having either the deduction-detachment theorem, or the property of strong disjunction. Summarizing, we are interested in the class of filter distributive congruential finitary logics with theorems, as the largest class of logics for which we can prove a Stone/Priestley duality between S-algebras and homomorphisms and certain dual spaces. In order to characterize such dual spaces, we draw on the Theory of Logical Calculi of Wójcicki [31].

4. Referential semantics Referential algebras were introduced in [31] by W´ojcicki as a tool for studying the link between relational semantics and algebraic semantics. The

1We follow here the terminology used in [19]. Congruential logics were previously called strongly selfextensional [16] and fully selfextensional [22]. DUALITY THEORY, CANONICAL EXTENSIONS AND AAL 7 underlying idea of referential algebras is simple and somehow straightfor- ward: it consists on assuming that truth values of the formuli depend on reference points. One of the possible interpretations of those reference points is that of possible worlds, in which case referential semantics reduces to relational semantics. Referential algebras define referential semantics for logics, that is, in essence, an abstract version of the general-frame-sylte semantics of many intensional logics. Without going into details, we highlight that in [31], selfextensional logics are identified as the widest class of logics that admits a referential semantics (see also [32]). In relation with this, we should mention the work in [22], that is the starting point of our work. In [22] Jansana and Palmigiano study as a fledged duality the formal connection between algebraic semantics and referential semantics that was outlined in [12] by Czelakowski. Moreover, the correspondence between selfextensional logics and reduced referential algebras addressed in [31] is formulated in [22] as a proper equivalence of categories in. An updated approach to this topic is carried out, using modern notation and terminology. Furthermore, they identify the category of reduced referential algebras that corresponds to congruential logics. It is also remarked that their duality serves as a general template where a wide range of Stone/Priestley dualities related with concrete logics can fit. But their construction is rather distant to those concrete examples. Our aim is to provide a construction closer to the various examples that we find in the literature. It only remains to note that the main difference between [22] and our work is that, their main tool for building a referential algebra from an S-algebra is the collection of all S-filters, whereas our main tools, as it will be said later on, are irreducible S-filters and optimal S-filters, respectively. Let us introduce now a few concepts from Wójcicki’s theory, following the formulation presented in [22]. As was previously mentioned, we make use of these concepts to characterize Stone/Priestley dual spaces of filter distributive congruential finitary logics with theorems.

Deﬁnition 4.1. A referential algebra is a structure X = hX, Bi where: (1) X is a non-empty set, and (2) B is an algebra of subsets of X.

For any referential algebra X = hX, Bi we deﬁne the relation X ⊆ X ×X as follows: x X y iﬀ ∀U ∈ B x ∈ U ⇒ y ∈ U

This relation is a quasi-order on X, and when it is moreover a partial order, the referential algebra is said to be reduced. In this case, we may denote X by ≤X , or even by ≤ when the context is clear. For any referential algebra X = hX, Bi we define a consequence relation `X ⊆ P(F m)×F m as follows: 8 MARÍA ESTEBAN GARCÍA

for all Γ ∪ {ψ} ⊆ F m \ Γ `X ψ iﬀ ∀h ∈ Hom(Fm, B), h(γ) ⊆ h(ψ). γ∈Γ

For any logic S of the same type of B, when `S ⊆ `X , X is said to be an S-referential algebra. It is easy to prove (check Remark 5.2. in [22]) that for each algebraic reduct B of a reduced L-referential algebra, B ∈ AlgL.

5. Our results: an AAL view of the duality theory for non-classical logics Let us conclude this abstract by briefly presenting our results: for each filter distributive congruential finitary logic with theorems S we define two categories: On the one hand, we define the category of S-Spectral spaces and S-Spectral morphisms. On the other hand, we define the category of S-Priestley spaces and S-Priestley morphisms. Then we prove that these categories are dually equivalent to the category of S-algebras and algebraic homomorphisms between them. Notice that we do not claim that filter distributive congruential finitary logics with theorems are the largest class of algebras for which such dualities can be defined. For the moment, we only prove sufficient conditions for having such theory. We present now the definitions of the mentioned categories and the state- ment of the main theorems, but we omit the proofs due to the lack of space. From now on let S be a filter distributive congruential finitary logic with theorems. We denote by AlgS the category that has S-algebras as objects and algebraic homomorphisms between them as morphisms. Definition 5.1. A structure hX, Bi is an S-Spectral space when: (Sp1) hX, Bi is a reduced S-referential algebra, ω T B (Sp2) for all U ∪ {V } ⊆ B, if U ⊆ V , then V ∈ CS (U), c (Sp3) κX := {U : U ∈ B} is a basis of open compact subsets for a topology

τκX on X,

(Sp4) the space hX, τκX i is sober. The main tool for getting an S-Spectral space from an S-algebra are the irreducible S-filters, that are the meet irreducible elements of the lattice of S-filters. It turns out that the collection of all irreducible S-filters is a closure base for A, i.e. we have an analogue of Birkhoff’s Prime Filter Lemma that involves irreducible S-filters. Notice that for proving such lemma we need the Axiom of Choice, as usual.

Deﬁnition 5.2. Let hX1, B1i and hX2, B2i be two S-Spectral spaces. A relation R ⊆ X1 × X2 is an S-Spectral morphism when: (SpR1) R ∈ Hom(B2, B1), (SpR2) R(x) is closed subset of hX2, τκ2 i for all x ∈ X1. Let us denote by SpS the category that has S-Spectral spaces as objects and S-Spectral morphisms as morphisms. DUALITY THEORY, CANONICAL EXTENSIONS AND AAL 9

Theorem 5.3. The categories AlgS and SpS are dually equivalent. For the Priestley-style duality we need to make a more involved analysis. First we study how any S-algebra, as a poset, can be embedded in a fancy way in a distributive semilattice called the S-semilattice of the algebra. Then from this fact and the Priestley-style duality for distributive semilattices given in [3] we obtain the following definitions: Definition 5.4. A structure hX, τ, Bi is an S-Priestley space when: (Pr1) hX, Bi is a reduced S-referential algebra, ω T B (Pr2) for all U ∪ {V } ⊆ B, U ⊆ V iff V ∈ CS (U), (Pr3) hX, τ, ≤i is a Priestley space, where ≤ is the order associated with the referential algebra hX, Bi, (Pr4) B is a family of clopen upsets for hX, τ, ≤i that contains X, (Pr5) X¯ := {x ∈ X : {U ∈ B : x∈ / U} is non-empty and up-directed} is dense in hX, τi, The main tool for getting an S-Priestley space from an S-algebra are the optimal S-filters. This is a notion that we introduce, inspired in what was called optimal meet filters in [3]. Optimal S-filters are, in brief, the S-filters that correspond with the optimal meet filters of the S-semilattice of the algebra. A more direct definition can be given, but it involves some other notions and spaces from the purposes of this abstract. It turns out that the collection of all optimal S-filters, is also closure base (this is proved using again the Axiom of Choice).

Definition 5.5. Let hX1, τ1, B1i and hX2, τ2, B2i be two S-Priestley spaces. A relation R ⊆ X1 × X2 is an S-Priestley morphism when: (PrR1) R ∈ Hom(B2, B1), (PrR2) If (x, y) ∈/ R, then there is U ∈ B2 such that y ∈ U and R(x) ⊆ U. Let us denote by PrS the category that has S-Priestley spaces as objects and S-Priestley morphisms as morphisms. Theorem 5.6. The categories AlgS and PrS are dually equivalent. Note that since we follow an abstract approach, and we do not fix any language, the new categories that we define encode, necessarily, the notion of S-algebra. This is a consequence of having that the dual spaces of S-algebras are S-referential algebras. This is the drawback of taking such an abstract perspective. However, we complement our study by a modular analysis of the dual formulation of some of the more common logical properties, such as property of conjunction, property of strong disjunction, property of deduction-detachment, property of inconsistent element and property of being closed under the introduction of a unary connective. From this analysis we conclude that for each particular logic S we do not necessarily need an algebraic structure in the dual side, rather we can have a purely topological structure. 10 MARÍA ESTEBAN GARCÍA

Our main conclusion is that the dualities presented in [2–4, 6–8], any of which is duality for a class of algebras that is the algebraic counterpart of a fragment of intuitionistic logic, are all particular instances of the ‘abstract’ dualities that we have just presented. Also well known dualities, such as Esakia duality for Heyting algebras or Stone duality for Boolean algebras with operators fall under the same scope. Moreover, we obtain a new Priestley-style duality for Hilbert algebras, the algebraic counterpart of the implicative fragment of intuitionistic logic, that simpliﬁes the one that had been already studied in the literature [9]. It would be very interesting to investigate further in this topic. More new dualities could be obtained, more logical properties could be considered and its dual could be studied, and a point-free but abstract approach based on canonical extensions could be studied in detail.

References [1] S. Abramsky. Domain theory in logical form. Annals of Pure and Applied Logic, 51:1– 77, 1991. [2] G. Bezhanishvili, N. Bezhanishvili, D. Gabelaia, and A. Kurz. Bitopological duality for distributive lattices and heyting algebras. Mathematical Structures in Computer Science, 20(03):359–393, 2010. [3] G. Bezhanishvili and R. Jansana. Priestley style duality for distributive meet- semilattices. Studia Logica, 98:83–123, 2011. [4] G. Bezhanishvili and R. Jansana. Esakia style duality for implicative semilattices. Applied Categorical Structures, 21(2):181–208, 2013. [5] S. Burris and H. P. Sankappanavar. A course in Universal Algebra. The millennium edition, 2000. [6] S. Celani. Representation of Hilbert algebras and implicative semilattices. Central European Journal of Mathematics, 4:561–572, 2003. [7] S. Celani and D. Montangie. Hilbert algebras with supremum. Algebra Universalis, 67:237–255, 2012. submitted to Algebra Universalis. [8] S. A. Celani, L. M. Cabrer, and D. Montangie. Representation and duality for Hilbert algebras. Cent. Eur. J. Math., 7(3):463–478, 2009. [9] S. A. Celani and R. Jansana. A topological duality for hilbert algebras. (preprint), 2012. [10] J. Czelakowski. Filter distributive logics. Studia Logica, 43:353–377, 1984. [11] J. Czelakowski. Algebraic aspects of deduction theorems. Studia Logica, 44(4):369– 387, 1985. [12] J. Czelakowski. Protoalgebraic Logics. Kluwer, 2001. [13] J. M. Dunn, M. Gehrke, and A. Palmigiano. Canonical extensions and relational completeness of some substructural logics. The Journal of Symbolic Logic, 70(3):713– 740, 2005. [14] J. M. Font, R. Jansana, and D. Pigozzi. A survey of abstract algebraic logic. Studia Logica, 74(1/2):13–97, Jun. - Jul. 2003. [15] J.M. Font. Beyond Rasiowa’s Algebraic Approach to Non-classical logics. Studia Log- ica, 82(2):179–209, 2006. [16] J.M. Font and R. Jansana. A General Algebraic Semantics for Sentential Logics, volume 7 of Lectures Notes in Logic. The Association for Symbolic Logic, Ithaca, N.Y., second edition, 2009. [17] M. Gehrke. Generalized Kripke frames. Studia Logica, 84(2):241–275, 2006. DUALITY THEORY, CANONICAL EXTENSIONS AND AAL 11

[18] M. Gehrke, Serge Grigorieff, and Jean-Eric´ Pin. Stone duality, topological algebra and recognition. manuscript, 2012. [19] M. Gehrke, R. Jansana, and A. Palmigiano. Canonical extensions for congruential logics with the deduction theorem. Annals of Pure and Applied Logic, 161:1502–1519, 2010. [20] M. Gehrke and H. A. Priestley. Duality for double quasioperator algebras via their canonical extensions. Studia Logica, 68(1):31–68, 2007. [21] R. Goldblatt. Varieties of complex algebras. Annals of Pure and Applied Logic, 44:173–242, 1989. [22] R. Jansana and A. Palmigiano. Referential semantics: duality and applications. Re- ports on Mathematical Logic, 41:63–93, 2006. [23] B. Jónssonand A. Tarski. Boolean algebras with operators. I. Amer. J. Math., 73:891– 939, 1951. [24] B. Jónssonand A. Tarski. Boolean algebras with operators. II. Amer. J. Math., 74:127–162, 1952. [25] K. Pa lasińska. Finite basis theorem for filter-distributive protoalgebraic deductive systems and strict universal horn classes. Studia Logica, 74(1-2):233–273, 2003. [26] H. A. Priestley. Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc., 2:186–190, 1970. [27] H. Rasiowa and R. Sikorski. A proof of the completeness theorem of gödel. Funda- menta Mathematicae, 37(193-200), 1950. [28] M. H. Stone. The theory of representations for Boolean algebras. Transactions of the American Mathematical Society, 39:37–111, 1936. [29] L. van Rooijen, A. Chernilovskaya, and M. Gehrke. Generalised kripke semantics for the lambek-grishin calculus. to appear in Logic Journal of IGPL, 2012. [30] L. van Rooijen, D. Coumans, and M. Gehrke. Relational semantics for full linear logic. to appear in Journal of Applied Logic, 2012. [31] R. Wójcicki. Theory of Logical Calculi. Basic Theory of Consequence Operations. Kluwer, Dordrecht, 1988. [32] R. Wójcicki. A logic is referential iff it is selfextensional. Studia Logica, 73(3):323–335, 2003.

Departament de Logica,` Historia` i Filosofia de la Ciencia,` Facultat de Filosofia, Universitat de Barcelona (UB). Montalegre 6, 08001 Barcelona, Spain E-mail address: [email protected]