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Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties

Published online by Cambridge University Press:  20 November 2018

Michel Hébert
Michel Hébert*
Affiliation:
Département de Mathématiques, Université LavalQuebec, P.Q. Canada G1K 7P4 Department of Mathematics, Mcgill UniversityMontreal, P.Q., Canada H3A 2K6
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Abstract

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Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.

In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.

Keywords

Categories of modelspreservation theoremvarietiesmonadicity03C6008C1018C15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 3 , 01 September 1988 , pp. 287 - 300
Copyright
Copyright © Canadian Mathematical Society 1988

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