Ordered semigroups whose prime ideals form a chain
type: Item
instance of: scholarly article
title: Ordered semigroups whose prime ideals form a chain
zbMATH Open document ID: 1064.06010
author: Xinzhai Xu
author: Jun-Ying Ma
published in: Southeast Asian Bulletin of Mathematics
publication date: 2005-03-14T00:00:00Z
review text: For an ordered semigroup \((S,\cdot,\leq)\), we consider the ordered semigroup \((S/{\mathcal N},\cdot,\preceq)\), where \(\mathcal N\) is the complete semilattice congruence on \(S\) defined by \(\mathcal N\):= \(\{(x,y)\mid N(x)=N(y)\}\), \(N(x)\) is the filter of \(S\) generated by \(x\) \((x\in S)\), \((x)_{\mathcal N}(y)_{\mathcal N}:=(xy)_{\mathcal N}\), \((x)_{\mathcal N}\preccurlyeq (y)_{\mathcal N}\Leftrightarrow (x)_{\mathcal N}=(xy)_{\mathcal N}\). In this paper the authors first show that there exists an order-preserving bijection between the set of prime ideals of \(S\) and the set of prime ideals of \(S/{\mathcal N}\). Then, inspired by the result of \textit{M. Satyanarayana} that the prime ideals in commutative primary semigroups form a chain [Czech. Math. J. 22 (97), 509--516 (1972; Zbl 0248.20072)], they give necessary and sufficient conditions for \(S/{\mathcal N}\) to be a chain. They prove the following Theorem. Let \(S\) be an ordered semigroup. The following are equivalent: 1) \((S/{\mathcal N},\cdot, \preceq)\) is a chain. 2) The set of prime ideals of \(S\) is a chain (under set inclusion). 3) Every non-empty intersection of prime ideals of \(S\) is a prime ideal of \(S\). 4) For each \(x,y\in S\), we have \(\{x,y\}\cap N(x)\cap N(y)\neq \emptyset\).
reviewed by: Niovi Kehayopulu
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 2145566
zbMATH Keywords: ordered semigroup
zbMATH Keywords: complete semilattice congruence
zbMATH Keywords: filter
zbMATH Keywords: prime ideal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2145566