Archimedean ordered semigroups as ideal extensions
type: Item
instance of: scholarly article
title: Archimedean ordered semigroups as ideal extensions
zbMATH Open document ID: 1169.06008
published in: Semigroup Forum
publication date: 2009-04-27T00:00:00Z
review text: The main result of the paper is a structure theorem concerning the ideal extensions of Archime\-dean ordered semigroups. Let \((S,\cdot,\leq )\) be an ordered semigroup. The authors prove that if \(S\) is Archimedean and \(e\) is an idempotent element of \(S\), then there exists an ideal \(K\) of \(S\) containing \(e\) such that \(K\) is a simple subsemigroup of \(S\) and the ordered semigroup \(S|K\) is nil. Conversely, let \(K\) be an ideal of \(S\) such that \(K\) is a simple subsemigroup of \(S\) and the ordered semigroup \(S|K\) be nil. Then \(S\) is Archimedean. By the above conclusions, the authors obtain the following results: Let \(S\) be an ordered semigroup. If \(S\) is Archimedean and contains an idempotent, then \(S\) is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if \(S\) is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then \(S\) is Archimedean. Especially, an ordered semigroup \(S\) is Archimedean and contains an idempotent if and only if \(S\) is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 5547495
zbMATH Keywords: ideal extension of an ordered semigroup by an ordered semigroup
zbMATH Keywords: Archimedean ordered semigroup
zbMATH Keywords: idempotent element
zbMATH Keywords: ideal
zbMATH Keywords: simple ordered semigroup
zbMATH Keywords: nil ordered semigroup
zbMATH Keywords: Rees quotient ordered semigroup
author: Niovi Kehayopulu
author: Michael Tsingelis
reviewed by: Xie Xiang-Yun
MaRDI profile type: Publication
full work available at URL: s00233-008-9116-y
OpenAlex ID: W2135247233
cites work: Groupes et anneaux reticules
cites work: Q5532570
cites work: On weak commutativity of po-semigroups and their semilattice decompositions
cites work: On medial semigroups
cites work: Extensions of Semigroups
cites work: Q3848243
cites work: Q5509706
cites work: On weakly commutative poe-semigroups
cites work: Q3204514
cites work: Q3204515
cites work: Q4713301
cites work: Q4713308
cites work: On separative ordered semigroups
cites work: Ideal Extensions of Ordered Semigroups
cites work: Semilattices of Archimedean Ordered Semigroups
cites work: Q5570335
cites work: Special classes of semigroups
cites work: Q4083905
cites work: Semilattice decompositions of semigroups
cites work: Band of t-archimedean semigroups
cites work: Neighbouringly normal archimedean ordered semigroups
cites work: On decompositions of a commutative semigroup
cites work: Bands of weakly \(r\)-Archimedean ordered semigroups
DOI: 10.1007/S00233-008-9116-Y
description: scientific article; zbMATH DE number 5547495