Implicit definition of the quaternary discriminator
type: Item
instance of: scholarly article
title: Implicit definition of the quaternary discriminator
zbMATH Open document ID: 1261.08003
author: Miguel Campercholi
author: Diego J. Vaggione
published in: Algebra Universalis
publication date: 2012-11-27T00:00:00Z
review text: The quaternary discriminator on a set \(A\) is the function \(d:A^4\rightarrow A\) defined by \(d(x,y,z,t)=z\) if \(x=y\), otherwise \(d(x,y,z,t)=t\). Let \({\mathbf A}\) be an algebra. The quaternary discriminator is called implicitly definable by a system of term equations \(\bigwedge t_i(x_1,x_2,x_3,x_4,x_5)=s_i(x_1,x_2,x_3,x_4,x_5)\) if \(d\) is the only quaternary operation on \(A\) making the identities \(t_i(x_1,x_2,x_3,x_4,d(x_1,x_2,x_3,x_4))\approx s_i(x_1,x_2,x_3,x_4,d(x_1,x_2,x_3,x_4))\) hold in \({\mathbf A}\). The main theorem gives five equivalent conditions for a class \(K\) of non-trivial algebras to have the quaternary discriminator implicitly definable for each \({\mathbf A}\) in \( K\). As a corollary the authors obtain the description of a quasivariety generated by \(K\). Each member \({\mathbf A}\) of such a quasivariety is shown to have a unique special extension \({\mathbf E}\) in a discriminator variety such that the congruence lattice of \({\mathbf E}\) and the lattice of relative congruences of \({\mathbf A}\) are isomorphic.
reviewed by: Anna Zamojska-Dzienio
Mathematics Subject Classification ID: 08C15
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 6110487
zbMATH Keywords: quaternary discriminator
zbMATH Keywords: implicit equational definition
zbMATH Keywords: equationally definable principal congruences
MaRDI profile type: Publication
full work available at URL: s00012-012-0189-9
OpenAlex ID: W2171355054
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DOI: 10.1007/S00012-012-0189-9
description: scientific article; zbMATH DE number 6110487