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Reports on Mathematical Logic

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No. 44

George VOUTSADAKIS

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.

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