Joins of subalgebras and normals in 0-regular varieties
type: Item
instance of: scholarly article
title: Joins of subalgebras and normals in 0-regular varieties
zbMATH Open document ID: 1351.08004
author: N. R. McConnell
author: T. E. Stokes
published in: Algebra Universalis
publication date: 2015-10-14T00:00:00Z
full work available at URL: https://hdl.handle.net/10289/9775
review text: Let \(\mathcal V\) be a variety of algebras with a nullary operation \(0\). If for each congruence \(\theta\) on each algebra \(A\in\mathcal V\), the \(\theta\)-class containing \(0\) determines \(\theta\), then \(\mathcal V\) is called \textit{\(0\)-regular}. If additionally, for each algebra \(A\in\mathcal V\), the singleton \(\{0\}\) is a subalgebra of \(A\), then \(\mathcal V\) is called \textit{\(0\)-normal}. Evidently, a \(0\)-regular variety is \(0\)-normal if and only if for any congruence \(\theta\) on any \(A\in\mathcal V\), the \(\theta\)-class containing \(0\) is a subalgebra of \(A\). Examples of \(0\)-normal varieties are groups or rings or -- more generally -- multioperator groups. If \(\mathcal V\) is a \(0\)-regular variety, then for any \(A\in\mathcal V\) the congruence classes containing \(0\) are called its \textit{normals}. If \(I\) is a normal of \(A\) and \(a\in A\), then \(aI\) denotes the \(\theta\)-class containing \(a\), where \(\theta\) is the congruence corresponding to \(I\). Evidently, the lattice \((\mathcal I(A),\cap,+)\) of normals of \(A\) is isomorphic to the congruence lattice of \(A\). A \(0\)-normal variety \(\mathcal V\) is called \textit{join invariant} if for any \(A\in\mathcal V\) the set \(\mathcal I(A)\) forms a sublattice of the lattice \((\mathcal S(A),\cap,\vee)\) of subuniverses of \(A\), i.e., if \(I+J=I\vee J\) for all \(I,J\in\mathcal I(A)\). Let \(\mathcal V\) be a \(0\)-normal variety and \(A\in\mathcal V\). Two congruences \(\rho\), \(\theta\) on \(A\) are said to \textit{permute at \(0\)} if for any \(a\in A\), \((a,0)\in\rho\circ\theta\) implies \((a,0)\in\theta\circ\rho\) and vice versa. The variety \(\mathcal V\) is called \textit{ideal determined} if for any \(A\in\mathcal V\) any two congruences on \(A\) permute at \(0\) (for this notion, see in particular [\textit{H. P. Gumm} and \textit{A. Ursini}, ibid. 19, 45--54 (1984; Zbl 0547.08001)]). The variety is called \textit{\(0\)-coherent} if for any \(A\in\…
reviewed by: Günther Eigenthaler
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 03C05
zbMATH DE Number: 6494312
zbMATH Keywords: \(0\)-regular variety
zbMATH Keywords: ideal determined variety
zbMATH Keywords: Mal'cev condition
zbMATH Keywords: EQ-monoid
MaRDI profile type: Publication
OpenAlex ID: W1127132200
cites work: An idealtheoretic characterization of varieties of abelian \(\Omega\)- groups
cites work: Q4707050
cites work: Q3934450
cites work: VARIETIES OF EQUALITY STRUCTURES
cites work: Q5595183
cites work: Ideals in universal algebras
cites work: Semi-abelian categories
cites work: Q4931926
cites work: Q4663636
cites work: Q3659833
cites work: Radicals of 0-regular algebras
cites work: Q5181407
cites work: On subtractive varieties. I
DOI: 10.1007/S00012-015-0344-1
description: scientific article; zbMATH DE number 6494312