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

Abstract

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)”.

Syntactic characterization of closure under connected limits

“…In [8], they are shown to be precisely those which are invariant under limits and factorizations [i.e. f: 9.1~ in M(T)=~the substructure f(9.I) of ~ on the image of f is a model of T], and a syntactic characterization of these theories is given [like (d) of Theorem 1, but with fl(X)=-l-in the sentences where Y is non-empty].…”