Semiconic idempotent logic II: Beth definability and deductive interpolation

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F25%3A00604110" target="_blank" >RIV/67985807:_____/25:00604110 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1016/j.apal.2024.103528" target="_blank" >https://doi.org/10.1016/j.apal.2024.103528</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.apal.2024.103528" target="_blank" >10.1016/j.apal.2024.103528</a>

Result language

angličtina

Original language name

Semiconic idempotent logic II: Beth definability and deductive interpolation

Original language description

Semiconic idempotent logic sCI is a common generalization of intuitionistic logic, semilinear idempotent logic sLI, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of sCI, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of stratified and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by stratified and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of sCI, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending sCI.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

—

OECD FORD branch

10102 - Applied mathematics

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2025

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

1873-2461

Volume of the periodical

176

Issue of the periodical within the volume

3

Country of publishing house

NL - THE KINGDOM OF THE NETHERLANDS

Number of pages

40

Pages from-to

103528

UT code for WoS article

001373204800001

EID of the result in the Scopus database

2-s2.0-85210531587