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COMPATIBILITY OPERATORS IN ABSTRACT ALGEBRAIC LOGIC

Published online by Cambridge University Press:  29 June 2016

HUGO ALBUQUERQUE ,
JOSEP MARIA FONT  and
RAMON JANSANA
HUGO ALBUQUERQUE
Affiliation:
DEPARTAMENT DE LÒGICA, HISTÒRIA I FILOSOFIA DE LA CIÈNCIA UNIVERSITAT DE BARCELONA (UB) MONTALEGRE 7, E-08001 BARCELONA, SPAINE-mail:hugo.albuquerque@ua.pt
JOSEP MARIA FONT
Affiliation:
DEPARTAMENT DE PROBABILITAT, LÒGICA I ESTADÍSTICA UNIVERSITAT DE BARCELONA (UB) GRAN VIA DE LES CORTS CATALANES 585 E-08007 BARCELONA, SPAINE-mail:jmfont@ub.edu
RAMON JANSANA
Affiliation:
DEPARTAMENT DE LÒGICA, HISTÒRIA I FILOSOFIA DE LA CIÈNCIA UNIVERSITAT DE BARCELONA (UB) MONTALEGRE 7, E-08001 BARCELONA, SPAINE-mail:jansana@ub.edu
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Abstract

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

Keywords

Leibniz operatorSuszko operatorTarski operatorcompatibility operatorcoherent familyprotoalgebraic logictruth-equational logicalgebraic logic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 2 , June 2016 , pp. 417 - 462
Copyright
Copyright © The Association for Symbolic Logic 2016 

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