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Duality Theory, Canonical Extensions and Abstract Algebraic Logic

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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 L´ogicaAlgebraica Abstracta nos proporciona el marco te´oricoadecuado para desarrollar una teor´ıauniforme de la dualidad y de las extensiones can´onicaspara l´ogicasno clsicas. Una teor´ıatal sirve, en esencia, para definir de manera uniforme sem´anticas referenciales (e.g. relacionales, al estilo Kripke) de un amplio rango de l´ogicaspara las cuales se conoce una sem´antica 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 MAR´IA ESTEBAN GARC´IA Conditions (C1)-(C3) are known as extensive, idempotent and isotone re- spectively. C is finitary or algebraic (cf. Definition 5.4 in [5]), when moreover satisfies: (C4) for all Y ⊆ X and all x 2 X, if x 2 C(Y ), then there is a finite Y 0 ⊆! Y such that x 2 C(Y 0). When X is the carrier of an algebra X, C is called structural when it satisfies: (C5) for all Y [ fxg ⊆ X and all h endomorphism on X, if x 2 C(Y ) then h(x) 2 C(h[Y ]). Any closure operator C on X can be transformed in a relation `C on X as follows: for all Y [ fxg ⊆ X Y `C x iff x 2 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 2 Y , then Y `C x, (C2') if Y `C z for all z 2 Z and Z `C x, then Y `C x. Clearly, any consequence relation `C on X defines a closure operator C` by x 2 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 ' 2 Γ, then Γ `S ', (C2') if ∆ `S γ for all γ 2 Γ and Γ `S ', then ∆ `S ', (C3') (structurality) if Γ `S ', then e[Γ] `S e(') 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 2 U and b2 = 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 G¨odel'scompleteness the- orem for first-order logic; J´onssonand Tarski [23,24] proved relational com- pleteness 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 MAR´IA ESTEBAN GARC´IA 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 J´onssonand 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 gen- eralize this pioneering work on representation of boolean algebras with op- erators, 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.
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