\(*\)-congruences on abundant semigroups with \(SQ\)-adequate transversals.
type: Item
instance of: scholarly article
title: \(*\)-congruences on abundant semigroups with \(SQ\)-adequate transversals.
zbMATH Open document ID: 1307.20052
author: Aifa Wang
author: Lili Wang
published in: Acta Mathematica Hungarica
publication date: 2015-01-09T00:00:00Z
review text: A congruence \(\rho\) on an abundant semigroup \(S\) is called a \(*\)-congruence if for all \(a,b\in S\), \(a\rho b\) implies \(a^*\rho b^*\) and \(a^\dag\rho b^\dag\). Let \(T\) be an adequate semigroup (that is, an abundant semigroup in which idempotents commute) with semilattice \(Y\), \(L=[Y;L_\alpha,\varphi_{\alpha,\beta}]\) be a strong semilattice of left zero bands \(L_\alpha\) and \(R=[Y;R_\alpha,\phi_{\alpha,\beta}]\) be a strong semilattice of right zero bands \(R_\alpha\). Then the set \(SQ=\{(x,s,y)\in L\times T\times R,\;x\in L_{s^\dag},\;y\in R_{s^*}\}\) with a suitable multiplication, becomes an abundant semigroup. A triple \((\rho^L,\pi^T,\rho^R)\), where \(\rho^L\) and \(\rho^R\) are congruences on \(L\) and \(R\), respectively, and \(\pi^T\) is a \(*\)-congruence on \(T\), satisfying certain conditions, is called a \(*\)-congruence triple on \(SQ\). The sets of all \(*\)-congruences and \(*\)-congruence triples on \(SQ\) are denoted by \(C^*(SQ)\) and \(CT^*(SQ)\), respectively. Defining an order relation \(\leq\) on \(CT^*(SQ)\) by \((\rho_1^L,\pi_1^T,\rho_1^R)\leq(\rho_2^L,\pi_2^T,\rho_2^R)\Leftrightarrow\rho_1^L\subseteq\rho_2^L,\;\pi_1^T\subseteq\pi_2^T,\;\rho_1^R\subseteq\rho_2^R\), it is proved that \(CT^*(SQ)\) forms a complete lattice with respect to \(\leq\) and \(C^*(SQ)\) is isomorphic to \(CT^*(SQ)\) as complete lattice.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M17
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 20M10
zbMATH DE Number: 6385398
zbMATH Keywords: abundant semigroups
zbMATH Keywords: adequate transversals
zbMATH Keywords: quasi-ideal transversals
zbMATH Keywords: congruences
zbMATH Keywords: congruence triples
zbMATH Keywords: congruence lattices
zbMATH Keywords: complete lattices
MaRDI profile type: Publication
full work available at URL: s10474-013-0345-5
OpenAlex ID: W2028366213
cites work: Regular semigroups with a multiplicative inverse transversal
cites work: Abundant semigroups with a multiplicative type \(A\) transversal
cites work: On definability of hypergraphs by their semigroups of homomorphisms
cites work: Abundant semigroups with a multiplicative adequate transversal
cites work: IDEMPOTENT-CONNECTED ABUNDANT SEMIGROUPS WHICH ARE DISJOINT UNIONS OF QUASI-IDEAL ADEQUATE TRANSVERSALS
cites work: Abundant Semigroups with Q-adequate Transversals and Some of Their Special Cases
cites work: Abundant Semigroups
cites work: Adequate Semigroups
cites work: On congruence lattices of regular semigroups with \(Q\)-inverse transversals
cites work: Congruences on regular semigroups with inverse transversals*
cites work: Congruence lattices of regular semigroups with inverse transversals1
DOI: 10.1007/S10474-013-0345-5
Recommended article: CONGRUENCES ON ABUNDANT SEMIGROUPS WITH QUASI-IDEAL S-ADEQUATE TRANSVERSALS
Recommended article: Good congruences on abundant semigroups with \(PSQ\)-adequate transversals.
Recommended article: Good congruences on abundant semigroups with multiplicative quasi-adequate transversals.
Recommended article: Good congruences on abundant semigroups with multiplicative adequate transversals.
Recommended article: Abundant Semigroups with Q-adequate Transversals and Some of Their Special Cases
Recommended article: ABUNDANT SEMIGROUPS WITH QUASI-IDEAL S-ADEQUATE TRANSVERSALS
Recommended article: Good congruences on abundant semigroups with quasi-ideal adequate transversals
Recommended article: Q3014748
Recommended article: Q5381450
Recommended article: On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup
description: scientific article; zbMATH DE number 6385398