Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

Two classes of 7r-institutions are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of iV-protoalgebraic 7r-institutions and the second is the wider class of iV-prealgebraic 7r-institutions. Several characterizations are provided. For instance, iV-prealgebraic ttinstitutions are exactly those ttinstitutions that satisfy monotonicity of the TV-Leibniz operator on theory systems and A/-protoalgebraic 7r-institutions those that satisfy monotonicity of the ATLeibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for 7r-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (J, N)algebraic systems, introduced previously by the author as an analog of the «S-algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for ninstitutions are also investigated.