Every idempotent plain algebra generates a minimal variety
type: Item
instance of: scholarly article
title: Every idempotent plain algebra generates a minimal variety
zbMATH Open document ID: 0618.08002
DOI: 10.1007/BF01229959
author: Ágnes Szendrei
published in: Algebra Universalis
publication date: 1988-01-01T00:00:00Z
review text: An algebra is called plain if it is finite, simple, and has no nontrivial proper subalgebras. It is shown that for every plain idempotent algebra \({\mathcal A}\), either \({\mathcal A}\) generates a congruence distributive variety, or \({\mathcal A}\) is affine, or \({\mathcal A}\) is term equivalent to a 2-element semilattice or left zero semigroup. This implies the claim in the title. The proof is based on the description of idempotent plain algebras established by the author [in Acta Sci. Math. 51, 251-268 (1987)].
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 4002156
zbMATH Keywords: minimal variety
zbMATH Keywords: finite, simple algebras
zbMATH Keywords: congruence distributive variety
zbMATH Keywords: idempotent plain algebras
MaRDI profile type: Publication
cites work: Finite equational bases for finite algebras in a congruence-distributive equational class
cites work: A proof of Lyndon's finite basis theorem
cites work: Para primal algebras
cites work: Varieties generated by para primal algebras
cites work: Q3775604
cites work: Algebras in permutable varieties: Geometrical properties of affine algebras
cites work: A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity
cites work: Nonfinitely based three-element idempotent groupoids
cites work: Algebras Whose Congruence Lattices are Distributive.
cites work: The structure of finite algebras
cites work: Minimal, locally-finite varieties that are not finitely axiomatizable
cites work: The Two-Valued Iterative Systems of Mathematical Logic. (AM-5)
cites work: Algebras with minimal spectrum
cites work: Mal'cev varieties
cites work: Q3770585
full work available at URL: https://doi.org/10.1007/bf01229959
OpenAlex ID: W2109599377
description: scientific article; zbMATH DE number 4002156