Correspondence and Canonicity Theory of Quasi-Inequalities and $Π_2$-Statements in Modal Subordination 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.