Polyatomic Logics and Generalised Blok-Esakia Theory

TL;DR

This paper introduces Polyatomic Logics inspired by Inquisitive semantics, develops their algebraic semantics, and connects them to a generalized Blok-Esakia theory through translations, advancing the algebraic understanding of logical fragments.

Contribution

It systematically studies Polyatomic Logics, introduces their algebraic semantics, and links them to a generalized Blok-Esakia theory via translations, providing new insights into algebraic logic.

Findings

Established algebraic completeness for Polyatomic Logics

Developed a generalized Blok-Esakia theory for classes of translations

Connected Polyatomic Logics with algebraic and semantic frameworks

Abstract

This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.