Categorical Abstract Algebraic Logic: Tarski Congruence Systems, Logical Morphisms and Logical Quotients

Abstract

A general notion of a congruence system is introduced for π-institutions. Congruence systems in this sense are collections of equivalence relations on the sets of sentences of the π-institution that are preserved both by signature morphisms and by fixed collections of natural transformations from finite tuples of sentences to sentences. Based on this notion of a congruence system, the notion of a Tarski congruence system, generalizing the notion of a Tarski congruence from sentential logics, is considered. Logical and bilogical morphisms are introduced for π-institutions, also generalizing similar concepts from the theory of sentential logics, and their relationship with the familiar translations and interpretations of institutions is discussed. Finally, the interplay between these logical maps and the formation of logical quotients of π-institutions and the way they transform the Tarski congruence systems is investigated.