Correspondence and Canonicity Theory of Quasi-Inequalities and ${\mathrm{\Pi }}_2$-Statements in Modal Subordination Algebras

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 ${\mathrm{\Pi }}_2$-statements. We use an algorithm to transform (restricted) inductive quasi-inequalities and (restricted) inductive ${\mathrm{\Pi }}_2$-statements to equivalent first-order correspondents on the dual Stone spaces with two relations with respect to arbitrary (resp.\ admissible) valuations.