Applying Algebraic Logic; a General Methodology

Connections between Algebraic Logic and (ordinary) Logic. Algebraic counterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbertstyle inference system iff Alg(L) is a finitely axiomatizable quasi-variety. Similarly, L is compact iff Alg(L) is closed under taking ultraproducts; L has the Craig interpolation property iff Alg(L) has the amalgamation property, etc.)