Abstract:

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term -institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for -institutions. Necessary and sufficient conditions are given for the quasi-equivalence and the deductive equivalence of two term -institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated -institutions. The -institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory.

Keywords: algebraic logic; institutions; equivalent deductive systems; algebraizable deductive systems; adjunctions; equivalent categories

Language: English

Document Type: Research article

Affiliations: 1: Department of Mathematics, Iowa State University, Ames, IA 50011, USA and Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA