On regular, strongly regular congruences on ordered semigroups
type: Item
instance of: scholarly article
title: On regular, strongly regular congruences on ordered semigroups
zbMATH Open document ID: 0967.06012
DOI: 10.1007/PL00006018
author: Xie Xiang-Yun
published in: Semigroup Forum
publication date: 2001-02-28T00:00:00Z
review text: If \((S,\cdot,\leq)\) is a partially ordered (p.o.) semigroup and \(\rho\) is a congruence on \((S,\cdot)\), then, in general, the quotient semigroup \((S/\rho,*)\) does not inherit from \((S,\leq)\) a nontrivial partial order \(\preceq\) such that \((S/\rho,*,\preceq)\) forms again a p.o. semigroup. In this paper, \(\rho\) is called regular if there exists a partial order \(\preceq\) on \(S/\rho\) such that \((S/\rho,*,\preceq)\) forms a p.o. semigroup and if the natual homomorphism of \(\rho\) is isotone. First, it is shown that for every ideal \(I\) of \(S\) (defined as a semigroup ideal of \((S,\cdot)\) which is also an order ideal of \((S,\leq))\), the Rees congruence of \(I\) is regular. Using a result on pseudo-orders due to \textit{N. Kahayopulu} and \textit{M. Tsingelis} [Semigroup Forum 50, 389-392 (1995; Zbl 0828.06010)], it is shown that a congruence \(\rho\) on a p.o. semigroup \(S\) is regular if and only if there exists a p.o. semigroup \(T\) and a homomorphism \(\varphi:S\to T\) such that \(a\rho b\Leftrightarrow\varphi(a)=\varphi(b)\). A congruence \(\rho\) on \((S,\cdot,\leq)\) is called strongly regular if \(\rho\circ \leq\) is contained in \(\leq \circ\rho\). Characterizations of regular and strongly regular congruences are given, in particular in the case that \((S,\cdot,\leq)\) is a lattice-ordered semigroup. Finally, it is shown that the p.o. set of all (strongly) regular congruences forms a complete lattice, which is not a sublattice of the congruence lattice of \((S,\cdot)\) in general.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 1523487
zbMATH Keywords: partially ordered semigroup
zbMATH Keywords: quotient semigroup
zbMATH Keywords: ideal
zbMATH Keywords: Rees congruence
zbMATH Keywords: strongly regular congruences
zbMATH Keywords: regular congruences
reviewed by: H. Mitsch
MaRDI profile type: Publication
full work available at URL: pl00006018
OpenAlex ID: W83916435
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description: scientific article; zbMATH DE number 1523487