Categorical Abstract Algebraic Logic Local Deduction Theorems for π-Institutions

In this paper, some of the results of Blok and Pigozzi on the local deductiondetachment theorems (LDDT) in Abstract Algebraic Logic, which followed pioneering work of Czelakowski on the same topic, are abstracted to cover logics formalized as π-institutions. The relationship between the LDDT and the property of various classes of I-matrices having locally definable principal I-filters is investigated. Moreover, it is shown that the LDDT implies various forms of the principal and the local filter extension properties that are mutually equivalent. Finally, the notion of algebraic equivalence for π-institutions, a strengthening of the notion of deductive equivalence, previously introduced by the author, is formulated and it is shown that the property of having the LDDT is preserved under both bilogical morphisms and algebraic equivalence.