Basic notions of algebraic logic: formulae, proofs, logical matrices, filters, closure operators, closure systems, Schmidt Theorem, abstract Lindenbaum Lemma. Completeness theorem w.r.t.\ the class of all models. (Weakly) implicative logics. Examples on substructural and fuzzy logics.
Lindenbaum--Tarski method for weakly implicative logics: Leibniz congruence, reduced matrices, and completeness theorem w.r.t.\ the class of reduced models. Operators on classes of matrices. Relatively (finitely) subdirectly irreducible matrices (RFSI). Completeness theorem w.r.t.\ RFSI reduced models. Algebraizability and order algebraizability. Examples on substructural and fuzzy logics.
Leibniz operator on arbitrary logics. Leibniz hierarchy: protoalgebraic, equivalential and (weakly) algebraizable logics. Regularity and finiteness conditions. Alternative characterizations of the classes in the hierarchy.
Bridge theorems (deduction theorems, Craig interpolation, Beth definability). Generalized disjunctions and proof by cases properties and their role in AAL.