The Semantic Isomorphism Theorem in Abstract Algebraic Logic

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F16%3A00465843" target="_blank" >RIV/67985807:_____/16:00465843 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1016/j.apal.2016.08.001" target="_blank" >http://dx.doi.org/10.1016/j.apal.2016.08.001</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.apal.2016.08.001" target="_blank" >10.1016/j.apal.2016.08.001</a>

Result language

angličtina

Original language name

The Semantic Isomorphism Theorem in Abstract Algebraic Logic

Original language description

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.

Czech name

—

Czech description

—

Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

—

Project

<a href="/en/project/GA13-14654S" target="_blank" >GA13-14654S: An Order-Based Approach to Non-Classical Propositional and Predicate Logics</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2016

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Annals of Pure and Applied Logic

ISSN

0168-0072

e-ISSN

—

Volume of the periodical

167

Issue of the periodical within the volume

12

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

34

Pages from-to

1298-1331

UT code for WoS article

000385604800006

EID of the result in the Scopus database

2-s2.0-84989871444