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# The Semantic Isomorphism Theorem in Abstract Algebraic Logic

- 1.0465843 - ÚI 2017 RIV NL eng J - Journal Article
**Moraschini, Tommaso**

The Semantic Isomorphism Theorem in Abstract Algebraic Logic.*Annals of Pure and Applied Logic*. Roč. 167, č. 12 (2016), s. 1298-1331. ISSN 0168-0072. E-ISSN 1873-2461**R&D Projects**: GA ČR GA13-14654S**Institutional support**: RVO:67985807**Keywords**: algebraizable logics * abstract algebraic logic * structural closure operators * semantic isomorphism theorem * evaluational frames * compositional lattice**Subject RIV**: BA - General Mathematics**Impact factor**: 0.647, year: 2016

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.

**Permanent Link:**http://hdl.handle.net/11104/0264288

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