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The Semantic Isomorphism Theorem in Abstract Algebraic Logic
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SYSNO ASEP 0465843 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title The Semantic Isomorphism Theorem in Abstract Algebraic Logic Author(s) Moraschini, Tommaso (UIVT-O) SAI, RID Source Title Annals of Pure and Applied Logic. - : Elsevier - ISSN 0168-0072
Roč. 167, č. 12 (2016), s. 1298-1331Number of pages 34 s. Language eng - English Country NL - Netherlands Keywords algebraizable logics ; abstract algebraic logic ; structural closure operators ; semantic isomorphism theorem ; evaluational frames ; compositional lattice Subject RIV BA - General Mathematics R&D Projects GA13-14654S GA ČR - Czech Science Foundation (CSF) Institutional support UIVT-O - RVO:67985807 UT WOS 000385604800006 EID SCOPUS 84989871444 DOI 10.1016/j.apal.2016.08.001 Annotation One of the most interesting aspects of Blok and Pigozzi's algebraizability theory is that the notion of algebraizable logic L can be characterised by means of Syntactic and Semantic Isomorphism Theorems. While the Syntactic Isomorphism Theorem concerns the relation between the theories of the algebraizable logic L and those of the equational consequence relative to its equivalent algebraic semantics K, the Semantic Isomorphism Theorem describes the interplay between the filters of L on an arbitrary algebra A and the congruences of A relative to K. The pioneering insight of Blok and Jónsson, and the further generalizations by Galatos, Tsinakis, Gil-Férez and Russo, showed that the concept of algebraizability was not intrinsic to the connection between a logic and an equational consequence, thus inaugurating the abstract theory of equivalence between structural closure operators. However all these works focus only on the Syntactic Isomorphism Theorem, disregarding the semantic aspects present in the original theory. In this paper we fill this gap by introducing the notion of compositional lattice, which acts on a category of evaluational frames. In this new framework the non-linguistic flavour of the Semantic Isomorphism Theorem can be naturally recovered. In particular, we solve the problem of finding sufficient and necessary conditions for transferring a purely syntactic equivalence to the semantic level as in the Semantic Isomorphism Theorem. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2017
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