On left \(\pi\)-regular po-semigroups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0990.06008
author: Xinzhai Xu
publication date: 2002-02-10T00:00:00Z
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 20M17
zbMATH DE Number: 1648874
zbMATH Keywords: partially ordered semigroup
zbMATH Keywords: left \(\pi\)-regular
zbMATH Keywords: left ideals
MaRDI profile type: Publication
title: On left \(\pi\)-regular po-semigroups
published in: Chinese Quarterly Journal of Mathematics
review text: A partially ordered (p.o.) semigroup \((S,\cdot,\leq)\) is called left \(\pi\)-regular if for every element \(a\in S\) there exist \(m>0\), \(x\in S\) such that \(a^m\leq xa^{2m}\). Defining a left ideal \(I\) of \((S,\cdot,\leq)\) as a subset of \(S\) which is at the same time a left ideal of \((S,\cdot)\) and an order-ideal of \((S,\leq)\), three characterizations of such p.o. semigroups in terms of (principal) left ideals are given. In particular, it is shown that a p.o. semigroup \((S,\cdot,\leq)\) is left \(\pi\)-regular iff every left ideal \(I\) of \(S\) has the property that for any \(a\in S\) there exists \(n>0\) such that \(a^{2n}\in I\) implies that \(a^n\in I\). Considering for ``\(\leq\)'' the identity relation this result generalizes the fact that a semigroup \((S,\cdot)\) is left regular (i.e., for every \(a\in S\) there exists \(x\in S\) with \(a= xa^2\)) iff every left ideal \(I\) of \(S\) is semiprime (i.e., if \(a^2\in I\), \(a\in S\), then \(a\in I\)).
reviewed by: H. Mitsch
description: scientific article; zbMATH DE number 1648874