Ordered semigroups whose elements are separated by prime ideals
type: Item
instance of: scholarly article
title: Ordered semigroups whose elements are separated by prime ideals
zbMATH Open document ID: 1314.06018
author: Niovi Kehayopulu
published in: Mathematica Slovaca
publication date: 2012-05-14T00:00:00Z
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 6034817
zbMATH Keywords: ideal in an ordered semigroup
zbMATH Keywords: prime (semiprime) subset
zbMATH Keywords: intra-regular (regular) semigroup
zbMATH Keywords: intra-regular (regular) ordered semigroup
zbMATH Keywords: elements separated by prime ideals
zbMATH Keywords: idempotent element
zbMATH Keywords: idempotent ordered semigroup
MaRDI profile type: Publication
full work available at URL: s12175-012-0018-9
OpenAlex ID: W2064003389
cites work: On intra-regular ve-semigroups
cites work: On intra-regular ordered semigroups
DOI: 10.2478/S12175-012-0018-9
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review text: Let \(S\) be an ordered semigroup. \(S\) is called intra-regular if for every \(a\in S\) there exist \(x, y\in S\) such that \(a\leq xa^2y\). A subset \(T\) of \(S\) is called prime if the complement \(S\setminus T\) of \(T\) to \(S\) is either empty or a subsemigroup of \(S\). A nonempty subset \(T\) of \(S\) is called an ideal of \(S\) if (1) \(TS\subseteq T\), \(ST\subseteq T\) and (2) if for \(a\in T\), \(b\in S\) and \(b\leq a\), \(b\in T\). The ideal of \(S\) generated by \(a\in S\) is denoted by \(I(a)\). We say that the elements of \(S\) are separated by prime ideals of \(S\) if for all \(a,b\in S\) such that \(b\notin I(a)\) there exists a prime ideal \(P\) of \(S\) such that \(a\in P\) and \(b\notin P\). The main result of the paper is as follows: The elements of an ordered semigroup \(S\) are separated by prime ideals of \(S\) if and only if \(S\) is intra-regular. This result generalizes the corresponding result for semigroups (without order).
description: scientific article; zbMATH DE number 6034817