On the ideal extensions in ordered semigroups
type: Item
instance of: scholarly article
title: On the ideal extensions in ordered semigroups
zbMATH Open document ID: 0848.06012
DOI: 10.1007/BF02574121
author: Ming Fen Wu
author: Xie Xiang-Yun
published in: Semigroup Forum
publication date: 1996-10-27T00:00:00Z
full work available at URL: https://eudml.org/doc/135470
review text: The concepts of \(n\)-prime ideal and extension of an ideal in semilattices (introduced by K. P. Shum) are investigated in p.o. commutative semigroups \((S, \cdot, \leq)\). For any ideal \(I\) of \(S\) (i.e., semigroup- and order-ideal) and \(a \in S\), the set \(a : I = \{x \in S \mid ax \in I\}\) is called the ``extension of \(I\) by \(a\)''. \(I\) is called ``\(n\)-prime'' if the condition \(a_1 \dots a_n \in I\) \((a_1, \dots, a_n \in S)\) implies that at least \(n - 1\) of the products of \(a_1, \dots, a_n\) which contain only \(n - 1\) factors, belong to \(I\) (thus, the 2-prime ideals are exactly the prime ideals of \(S)\). In particular, it is shown that (1) if \(I\) is semiprime then the relation: \(a \rho_I b \Leftrightarrow a : I = b : I\), is a semilattice congruence on \(S\) satisfying \(x \leq y \Rightarrow x \rho_I xy\); (2) if \(I\) is prime then: \(a \rho_I b \Leftrightarrow a\), \(b \in I\) or \(a,b \notin I\), and \({\mathcal N} \subseteq \rho_I\), where \(a {\mathcal N} b \Leftrightarrow N(a) = N(b)\), \(N(x)\) denoting the least filter of \(S\) containing \(x \in S\); (3) every \((n - 1)\)-prime ideal is \(n\)-prime; (4) \(I\) is \(n\)-prime iff \(a:I\) is \((n - 1)\)-prime for every \(a \in S\) \((n \geq 3)\); (5) if \(I\) is semiprime and \(n\)-prime \((n \geq 3)\) then \(I\) equals the intersection of all \((n - 1)\)-prime ideals containing \(I\).
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 895851
zbMATH Keywords: partially ordered semigroup
zbMATH Keywords: \(n\)-prime ideal
zbMATH Keywords: extension of an ideal
reviewed by: H. Mitsch
MaRDI profile type: Publication
cites work: Q3204514
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cites work: On prime, weakly prime ideals in ordered semigroups
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description: scientific article; zbMATH DE number 895851