Categorical quasivarieties revisited
We offer simple new characterizations of \(\omega\)-categorical quasivarieties and varieties of countable type. Our arguments are distinguished by the absence of any sophisticated model theory. In the beginning we use some very basic model theory, but after that we find that combinatorial reasoning about finite sets and elementary algebraic arguments, combined with two classical theorems describing the structure of finite simple rings and their modules, suffice to derive the results. Theorems 3.1 and 4.12 combine to give the characterization of \(\omega\)- categorical quasivarieties. Theorems 3.2 and 4.13 combine to give the characterization of \(\omega\)-categorical varieties. The heart of this paper is {\S}2. There we prove that a nontrivial algebra of least cardinality in an \(\omega\)-categorical quasivariety (which must generate the class) is a finite ''tame'' algebra. Tameness is the principal tool used in a relatively quick and painless proof that the generating algebra must be affine or an [n]-th power of a unary algebra. The concept of a tame algebra was introduced [in Lect. Notes Math. 1004, 176-205 (1983; Zbl 0523.06012)] where we proved, among other things, that finite simple algebras are tame. When we had gained some experience with this concept, it became clear to us that the arguments in this present paper should exist (and it didn't take long to find them).
scientific article; zbMATH DE number 3966167
scientific article