Categorical quasivarieties revisited
type: Item
instance of: scholarly article
title: Categorical quasivarieties revisited
zbMATH Open document ID: 0599.08013
DOI: 10.1007/BF01201096
published in: Algebra Universalis
publication date: 1984-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08C10
Mathematics Subject Classification ID: 03C35
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3966167
zbMATH Keywords: \(\omega \) -categorical quasivarieties
zbMATH Keywords: varieties of countable type
zbMATH Keywords: finite simple rings
zbMATH Keywords: \(\omega \) -categorical varieties
zbMATH Keywords: tame algebra
zbMATH Keywords: finite simple algebras
author: Ralph McKenzie
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01201096
OpenAlex ID: W1977943016
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description: scientific article; zbMATH DE number 3966167