Generalized varieties.
Alg. Universalis 30 (1993), 27-52.
We wish to generalize to arbitrary structures as much as we can of the theory
of varieties from universal algebra. This or something similar has been
attempted several times already; in particular, the production of
category-theoretic generalizations of Birkhoff's theorem that varieties are
equationally definable seems to be a minor industry. But finding a really
satisfactory solution is made difficult by the fact that no important class
of nonalgebraic structures is closed under the most obvious kind of
homomorphism.
The central novelty introduced here is a new type of homomorphism, which,
however, reduces to the classical notion in any purely algebraic setting.
Our approach yields a notion of a variety of structures which enjoys the
following properties: (1) in the context of algebraic structures it is
precisely the usual notion of a variety; (2) it is instantiated by many
important classes of structures, including graphs, partially ordered sets,
ordered groups, ordered rings, metric spaces, normed vector spaces,
valuation rings, discrete valuation rings, topological spaces,
topological groups, and locally convex topological vector spaces.
Return to publication list