Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

Abstract

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (\({\mathcal{I}}\), N)-algebraic systems, introduced previously by the author as an analog of the \({\mathcal{S}}\) -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated.