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Balanced congruences

Balanced congruences.
Authors: Chajda, Ivan; Eigenthaler, Günther;

Balanced congruences

Abstract

If \(\mathfrak{A}\) is an algebra then \(\text{Con}({\mathfrak A})\) is the set of all congruences on \({\mathfrak A}\). If \(\Phi\in \text{Con}({\mathfrak A})\), \(x\in{\mathfrak A}\), then \([x]\Phi=\{t\in{\mathfrak A}\mid (x;t)\in\Phi\}\). If \(M\subseteq{\mathfrak A}\) then \(\Theta(M)\) is the least congruence on \({\mathfrak A}\) containing the relation \(M\times M\). An algebra \({\mathfrak A}\) with two distinct nullary operations \(0\) and \(1\) is called balanced if for all \(\Phi,\Psi\in \text{Con}({\mathfrak A})\) we have \([0]\Phi=[0]\Psi\) if and only if \([1]\Phi=[1]\Psi\). A variety \({\mathfrak V}\) with two distinct nullary operations \(0\) and \(1\) is called balanced if every \({\mathfrak A}\in{\mathfrak V}\) is balanced. The paper consists of three paragraphs. Balanced congruences on bounded lattices are considered in first paragraph. Let \({\mathfrak L}=\langle L;\vee,\wedge,0,1\rangle\) be a bounded lattice with least element \(0\) and greatest element \(1\). For \(\Phi\in \text{Con}({\mathfrak L})\) put \(I=[0]\Phi\) and \(F=[1]\Phi\). We say that \(\Phi\) is balanced if \([0]\Phi=[0]\Theta(F)\) and \([1]\Phi=[1]\Theta(I)\). We say that \({\mathfrak L}\) is a \(d\)-lattice if for each \(a,b,c,d\in L\) the following holds: \[ (a;0)\in\Theta(\{c;1\})\Rightarrow a\wedge c=0, \] \[ (b;1)\in\Theta(\{d;0\})\Rightarrow b\vee d=1. \] Theorem 1. Let \({\mathfrak L}\) be a \(d\)-lattice and \(\Phi\in \text{Con}({\mathfrak L})\). Then \(\Phi\) is balanced if and only if \[ [0]\Phi=\{a\in L:c\wedge a=0\text{ for some }c\in[1]\Phi\} \] and \[ [1]\Phi=\{b\in L:d\vee b=1\text{ for some }d\in[0]\Phi\}. \] Balanced algebras with two nullary operations are the subject of the second paragraph. Let \(\tau\) be a type containing two distinct nullary operations denoted by \(0\) and \(1\). Theorem 2. Let \({\mathfrak V}\) be a variety of type \(\tau\) containing two distinct nullary operations \(0\) and \(1\). \({\mathfrak V}\) is balanced if for each \({\mathfrak A}\in{\mathfrak V}\) and every \(\Phi\in \text{Con}({\mathfrak V})\) the following property holds: \([0]\Phi\) is a singleton if and only if \([1]\Phi\) is a singleton. A characterization of balanced varieties by a Mal'tsev condition is obtained in the third paragraph.

Keywords

Mal'tsev condition, Lattice ideals, congruence relations, balanced algebra, balanced congruence, Equational logic, Mal'tsev conditions, Subalgebras, congruence relations, balanced variety

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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Published in a Diamond OA journal