Consistent algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 0990.08001
author: Günther Eigenthaler
author: Ivan Chajda
publication date: 2002-07-18T00:00:00Z
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 08B05
zbMATH DE Number: 1683834
zbMATH Keywords: algebra with \(0\) and \(1\)
zbMATH Keywords: variety
zbMATH Keywords: consistency
zbMATH Keywords: balanced congruences
zbMATH Keywords: allied constant classes
zbMATH Keywords: permutability at \(0\) and \(1\)
zbMATH Keywords: polynomially linked constant classes
MaRDI profile type: Publication
title: Consistent algebras
review text: In the following let \({\mathbf N}\) denote the set of all positive integers, let \({\mathcal A}\) be an algebra with two constants \(0\) and \(1\) and for every \(n\in{\mathbf N}\) let \(\text{P}_n({\mathcal A})\) denote the set of all \(n\)-ary polynomial functions over \({\mathcal A}\). \({\mathcal A}\) is called consistent if for every \(\Theta\in\text{Con}{\mathcal A}\) and every subuniverse \(B\) of \({\mathcal A}\) the inclusion \([0]\Theta\subseteq B\) is equivalent to \([1]\Theta\subseteq B\). \({\mathcal A}\) is said to have balanced congruences (BC) if for every \(\Theta,\Phi\in\text{Con}{\mathcal A}\) the equality \([0]\Theta=[0]\Phi\) is equivalent to \([1]\Theta=[1]\Phi\). \({\mathcal A}\) is said to have allied constant classes (ACC) if for every \(\Theta\in\text{Con}{\mathcal A}\), every subuniverse \(B\) of \({\mathcal A}\), every \(n\in{\mathbf N}\) and every \(p\in\text{P}_n({\mathcal A})\) the following holds: If \([0]\Theta\subseteq B\) and \(p(0,\dots,0)=1\) then \(p(([0]\Theta)^n)\subseteq B\), and if \([1]\Theta\subseteq B\) and \(p(1,\dots,1)=0\) then \(p(([1]\Theta)^n)\subseteq B\). \({\mathcal A}\) is said to be permutable at \(0\) and \(1\) if \([x](\Theta\circ\Phi)=[x](\Phi\circ\Theta)\) for all \(x\in\{0,1\}\) and all \(\Theta,\Phi\in\text{Con}{\mathcal A}\). \({\mathcal A}\) is said to have polynomially linked constant classes (PLCC) if for every \(\Theta\in\text{Con}{\mathcal A},\) every subuniverse \(B\) of \({\mathcal A}\), every \(n\in{\mathbf N}\), every \(p_1,\dots,p_n\in\text{P}_1({\mathcal A})\) and every \(a_1,\dots,a_{n-1}\in[0]\Theta\) the following holds: If \([0]\Theta\subseteq B\), \(p_1(0)=1\) and \(p_i(a_i)=p_{i+1}(0)\) for all \(i=1,\dots,n-1\) then \(p_n(0)\in B\), and if \([1]\Theta\subseteq B\), \(p_1(1)=0\) and \(p_i(a_i)=p_{i+1}(1)\) for all \(i=1,\dots,n-1\) then \(p_n(1)\in B\). A variety with constant unary terms \(0\) and \(1\) is said to have one of the mentioned properties if each of its members has this pro…
reviewed by: Helmut Länger
description: scientific article; zbMATH DE number 1683834