Duality Theory and Abstract Algebraic Logic
- Autores: María Esteban
- Directores de la Tesis: Ramón Jansana Ferrer (dir. tes.)
- Lectura: En la Universitat de Barcelona ( España ) en 2013
- Idioma: inglés
- Tribunal Calificador de la Tesis: Josep Maria Font i Llobet (presid.), Alessandra Palmigiano (secret.), Mai Gehrke (voc.)
- Materias:
- Enlaces
- Tesis en acceso abierto en: Dipòsit Digital de la UB
- Resumen
In this thesis we present the results of our research on duality theory for non-classical logics under the point of view of Abstract Algebraic Logic (AAL). Firstly, we propose an abstract Spectral-like duality and an abstract Priestley-style duality for every filter distributive finitary congruential logic with theorems. This proposal aims to unify the various dualities for concrete logics that we find in the literature, by showing the abstract template in which all of them fit. Secondly, the dual correspondence of some logical properties is examined. This serves to reveal the connection between our abstract dualities and the concrete dualities related wot concrete logics. We apply those results to get new dualities for suitable expansions of a well-known logic: the implicative fragment of intuitionistic logic. Finally, we develop a new strategy that can be modularly applied to simplify some of the dualities obtained. The first part of the dissertation is devoted to introduce the preliminaries and the basic notation. In Chapter 1 we fix the mathematical concepts that we assume the reader is familiar with. Of particular interest is the section in which we introduce the basic concepts of AAL, such as "S-filter" or "S-algebra". The notion of "closure operator" plays a fundamental role in AAL, as well as in our dissertation. The notions of filter and ideal associated with a closure operator, and the separation lemmas between them are studied in detail in Chapter 2. Moreover, we briefly review the literature on duality theory for non classical logics in Chapter 3. In the second part of the dissertation we present an abstract view of the duality theory for non-classical logics. In Chapter 4 we review previous works on this topic, in which our work relies, and we introduce the notions of "referential algebra", "irreducible and optimal S-filter" and "S-semilattice". This lead us to identify a set of necessary conditions that a logic should satisfy in order to develop a Spectra-like/Priestley-style duality for it. These conditions are: "filter distributivity","congruentiality", "finitarity" and "having theorems". Moreover, we carry out a brief digression in which we argue how those notions can also be used to develop an abstract theory of canonical extensions. The core of the proposed theory consists of the definitions of dual objects and morphisms, for the category of S-algebras and homomorphisms, for any logic S that satisfies the mentioned properties. In Chapter 5 we define a Spectral-like duality and a Priestley-style duality for filter distributive finitary congruential logics with theorems, and we prove the respective duality theorems. Due to the abstraction of our approach, we obtain that the objects of both categories involved in the duality posses algebraic nature. However, through the analysis of the dual correspondence of several well-known logical properties, we can simplify the definitions of the dual categories, provided the logic under consideration satisfies such good logical properties. This analysis is interesting under the point of view of AAL, since our results can be regarded as bridge theorems between logical properties and properties of a Kripke-style semantics. And it is also interesting under the point of view of duality theory, since it confirms the strength of duality theory, that can be developed in a modular way beyond the distributive lattice setting. Moreover, our analysis shows the connection of the general theory proposed with the concrete results that we find in the literature, and lead us to explore the applications of such general theory to obtain new dualities.
