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Categorical Abstract Algebraic Logic: Syntactically Algebraizable pi-Institutions

Voutsadakis G.

Reports on Mathematical Logic

2009

Vol. 44

105-151

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 Wroński 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.