An approach to tame congruence theory via subtraces
type: Item
instance of: scholarly article
title: An approach to tame congruence theory via subtraces
zbMATH Open document ID: 0791.08001
DOI: 10.1007/BF01195379
author: Steven Seif
author: Joel Berman
published in: Algebra Universalis
publication date: 1994-07-04T00:00:00Z
review text: In the present paper the ideas and results of the so-called ``tame congruence theory'', which was developed by \textit{D. Hobby} and \textit{R. McKenzie} in their book ``The structure of finite algebra'' (1988; Zbl 0721.08001), are generalized. The basic ideas of this theory can be outlined as follows: Let \(A\) be a finite algebra. A prime quotient of congruences is a pair \((\alpha,\beta)\) of congruences such that \(\beta\) covers \(\alpha\) in \(\text{Con }A\). So-called tame quotients are considered which are generalizations of prime quotients. There are five types of tame quotients. The type set of \(A\) which is defined as the set of all types of all prime quotients of \(A\) then determines the algebraic properties of \(A\) and of the variety generated by \(A\). In the paper under review so-called subtraces of \(A\) are considered. These are special two- element subsets of \(A\) which have certain properties with respect to the unary polynomials of \(A\). There are five types of subtraces according to the action of the binary polynomials of \(A\) on these subtraces. Now to each prime quotient a certain non-empty set of subtraces can be assigned. It turns out that all subtraces corresponding to the same prime quotient have the same type as this prime quotient. The advantage of the approach via subtraces is that the theory can be generalized to certain infinite, so-called core-finite algebras. Moreover, many of the proofs become simpler. So-called pretame quotients (which are generalizations of tame quotients) are also considered. Finally, a lot of varieties are described all of whose members are core-finite.
reviewed by: Helmut Länger
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 488264
zbMATH Keywords: type set
zbMATH Keywords: variety generated by a single algebra
zbMATH Keywords: tame congruence
zbMATH Keywords: prime quotient of congruences
zbMATH Keywords: tame quotients
zbMATH Keywords: subtraces
zbMATH Keywords: unary polynomials
zbMATH Keywords: binary polynomials
zbMATH Keywords: core-finite algebras
zbMATH Keywords: pretame quotients
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 488264