Strong amalgamation, Beck-Chevalley for equivalence relations and interpolation in algebraic logic

We extend Makkai’s proof of strong amalgamation (push-outs of monos along arbitrary maps are monos) from the category of Heyting algebras to a class which includes the categories of symmetric bounded distributive lattices, symmetric Heyting algebras, Heyting modal S4-algebras, Heyting modal bi-S4-albegras, and Lukasiewicz n-valued algebras. We also extend and improve Pitt’s proof that strong amalgamation implies Beck-Chevalley for filters of Heyting algebras to exact categories with certain push-outs. As a consequence, a form of the Interpolation Lemma for some non-classical calculi is proved. Amalgamation properties for certain categories of algebras arising in Algebraic Logic have been used to show Craig’s Interpolation Lemma for the corresponding logics. It seems that the first example is due to A. Daigneault who, in the earlier 60’s showed that Craig’s Interpolation Lemma for classical propositional logic is a consequence of the Amalgamation Property (A) for the category of Boolean Algebras (see [3]). More recently, several authors have shown that Craig’ Interpolation Lemma for propositional intuitionistic logic (and some extensions) as well as for (classical) propositional S4 logic (and some extensions) are consequences of (A) in the corresponding categories of algebras (see e.g. the exhausting studies of Maksimova in [10], [11], [12] and the references therein indicated).