Categorical quasivarieties revisited

Abstract

Quasivarieties (and varieties) which are categorical in some power not less than the power of their language have been completely characterized by S. Givant [6], [7] and independently by E. A. Palyutin [11], [12]. These classes fall into two radically different families. A class in the first family is derived from the class of permutational representations of a group. Its members are [n]-th powers of algebras whose operations are unary, for some fixed positive integern. A class in the second family consists of affine algebras. Its members are polynomially equivalent (but not usually definitionally equivalent) to modules over a ring which is isomorphic to the ring ofn-by-n matrices with entries in a division ring.

The general results are faithfully represented in the family ofω-categorical quasivarieties of countable type. Each of them is generated by a finite algebra, and the results can be viewed as very interesting facts about finite algebras and the classes they generate. In this paper, we offer simple new characterizations ofω-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 ofω-categorical quasivarieties. Theorems 3.2 and 4.13 combine to give the characterization ofω-categorical varieties.

The heart of this paper is §2. There we prove that a nontrivial algebra of least cardinality in anω-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 [9] 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).