Term minimal algebras
type: Item
instance of: scholarly article
title: Term minimal algebras
zbMATH Open document ID: 0812.08001
DOI: 10.1007/BF01195723
published in: Algebra Universalis
publication date: 1995-01-11T00:00:00Z
review text: The author develops the counterpart of the theory of minimal algebras from tame congruence theory. Instead of polynomial functions, only term functions are considered, like in a paper of \textit{C. Bergman} and \textit{R. McKenzie} [J. Aust. Math. Soc., Ser. A 48, 133-147 (1990; Zbl 0692.08011)]. In general, one cannot expect such strong results as with polynomials, hence the study is restricted here to strictly simple algebras (i.e., simple algebras without nontrivial proper subalgebras). For example, it remains true for these algebras that their minimal (term) algebras are all isomorphic. For the unary term functions of a term minimal algebra there are four possibilities. Three of these cases have been dealt with earlier by the author [Acta Sci. Math. 51, 251-268 (1987; Zbl 0633.08002); J. Aust. Math. Soc., Ser. A 48, 434-454 (1990; Zbl 0702.08003)] and by the reviewer [Algebra Univers. 18, 262-273 (1984; Zbl 0546.08005)]. The study of the fourth case is the main subject of this paper. In this case the unary term functions form a group \(G\) fixing one element, say 0, and acting regularly on the rest of the elements. Such algebras are called \(G^ 0\)-algebras. The main result establishes that a \(G^ 0\)-algebra is either a 1-dimensional vector space or the idempotent operation \(x\wedge y = 0\), for all \(x \neq y\), is a term operation of the algebra. Then these algebras are classified according to their type in the sense of tame congruence theory. As a consequence, it turns out, for example, that every simple \(G^ 0\)-algebra of type 3 or 4 generates a congruence distributive variety, and for type 3 the variety is congruence 3- permutable as well. For a fixed nontrivial group \(G\), continuum many inequivalent \(G^ 0\)-algebras are constructed. At the end of the paper, very handy figures sketch the structure of the lattice of clones of \(G^ 0\)-algebras.
reviewed by: Péter P. Pálfy
Mathematics Subject Classification ID: 08A05
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 710072
zbMATH Keywords: induced algebra
zbMATH Keywords: term functions
zbMATH Keywords: strictly simple algebras
zbMATH Keywords: term minimal algebra
zbMATH Keywords: tame congruence theory
zbMATH Keywords: congruence distributive variety
zbMATH Keywords: lattice of clones
author: Ágnes Szendrei
MaRDI profile type: Publication
cites work: Q3033815
cites work: Plain para primal algebras
cites work: Construction of Large Sets of Clones
cites work: On \(n\)-permutable congruences
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: Unary polynomials in algebras. I
cites work: The Two-Valued Iterative Systems of Mathematical Logic. (AM-5)
cites work: Q3770585
cites work: Q3480114
cites work: Simple Abelian algebras
full work available at URL: https://doi.org/10.1007/bf01195723
OpenAlex ID: W2079906810
description: scientific article; zbMATH DE number 710072