Correspondence and Canonicity Theory of Quasi-Inequalities and $\Pi_2$-Statements in Modal Subordination Algebras

TL;DR

This paper extends the correspondence and canonicity theory for modal subordination algebras using quasi-inequalities and $\\Pi_2$-statements, providing algorithms for translating these into first-order logic on dual spaces.

Contribution

It generalizes existing results to include quasi-inequalities and $\\Pi_2$-statements in modal subordination algebras, with an algorithmic approach for their first-order correspondence.

Findings

Developed an algorithm for transforming quasi-inequalities to first-order correspondents.

Extended correspondence results to modal subordination algebras with two relations.

Generalized canonicity theory for a broader class of modal algebras.

Abstract

In the present paper, we study the correspondence and canonicity theory of modal subordination algebras and their dual Stone space with two relations, generalizing correspondence results for subordination algebras in \cite{dR20,dRHaSt20,dRPa21,Sa16}. Due to the fact that the language of modal subordination algebras involves a binary subordination relation, we will find it convenient to use the so-called quasi-inequalities and $\Pi_2$-statements. We use an algorithm to transform (restricted) inductive quasi-inequalities and (restricted) inductive $\Pi_2$-statements to equivalent first-order correspondents on the dual Stone spaces with two relations with respect to arbitrary (resp.\ admissible) valuations.