## Reports on Mathematical Logic

### No. 44

Categorical Abstract Algebraic Logic: Syntactically Algebraizable $$\pi$$-Institutions
A b s t r a c t. This paper has a two-fold purpose. On the one hand, it introduces the concept of a syntactically $$N$$-algebraizable $$\pi$$-institution, which generalizes in the context of categorical abstract algebraic logic the notion of an algebraizable logic of Blok and Pigozzi. On the other hand, it has the purpose of comparing this important notion with the weaker ones of an $$N$$-protoalgebraic and of a syntactically $$N$$-equivalential $$\pi$$-institution and with the stronger one of a regularly $$N$$-algebraizable $$\pi$$-institution. $$N$$-protoalgebraic $$\pi$$-institutions and syntactically $$N$$-equivalential $$\pi$$-institutions were previously introduced by the author and abstract in the categorical framework the protoalgebraic logics of Blok and Pigozzi and the equivalential logics of Prucnal and Wro\'{n}ski and of Czelakowski. Regularly $$N$$-algebraizable $$\pi$$-institutions are introduced in the present paper taking after work of Czelakowski and of Blok and Pigozzi in the sentential logic framework. On the way to defining syntactically $$N$$-algebraizable $$\pi$$-institutions, the important notion of an equational $$\pi$$-institution associated with a given quasivariety of $$N$$-algebraic systems is also introduced. It is based on the notion of an $$N$$-quasivariety imported recently from the theory of Universal Algebra to the categorical level by the author.