Fuzzy radicals and prime fuzzy ideals of ordered semigroups
type: Item
instance of: scholarly article
title: Fuzzy radicals and prime fuzzy ideals of ordered semigroups
zbMATH Open document ID: 1159.06009
author: Jian Tang
author: Xie Xiang-Yun
published in: Information Sciences
publication date: 2008-11-19T00:00:00Z
review text: Let \(S\) be an ordered semigroup, \(f\) a fuzzy subset of \(S\) and \(t\in [0,1]\). Then, the set \(f_t:=\{x\in S\mid f(x)\geq t\}\) is called the level subset of \(f\) (introduced by the same authors in an earlier paper). The authors prove first that a fuzzy subset of \(S\) is a fuzzy ideal of \(S\) if and only if the level subset of \(f\), if it is nonempty, is an ideal of \(S\). For an ordered semigroup \(S\), \(a\in S\) and \(\lambda\in [0,1]\) they define the concept of an ordered fuzzy point \(a_\lambda\) of \(S\) as follows: \(a_\lambda (x)=\lambda\) if \(x\in (a]\) and \(a_\lambda (x)=0\) if \(x\not\in (a]\). Using the concept of ordered fuzzy points they characterize the weakly prime fuzzy ideals of \(S\) in terms of right and left ideals of \(S\). They introduce the notions of weakly prime, completely prime, completely semiprime and weakly completely prime fuzzy ideals of an ordered semigroup and study the relationship between them. They prove, among others, that an ideal \(I\) of an ordered semigroup \(S\) is a prime ideal of \(S\) if and only if the fuzzy subset \(f_I\) is a prime fuzzy ideal of \(S\). If \(P\) is a prime fuzzy ideal of \(S\), then each nonempty level subset \(P_t\) is a prime ideal of \(S\). Moreover, they introduce and investigate the fuzzy radical of a fuzzy subset of \(S\) and prove that the fuzzy radical of a completely semiprime fuzzy ideal of \(S\) can be expressed as the intersection of all weakly completely prime fuzzy ideals of \(S\) containing it. There are further nice results in this paper that are interesting for people working on the subject.
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 5368814
zbMATH Keywords: level subset
zbMATH Keywords: ordered fuzzy point
zbMATH Keywords: fuzzy left (right) ideal
zbMATH Keywords: prime, weakly prime, completely prime, weakly completely prime fuzzy ideal
zbMATH Keywords: fuzzy radical
zbMATH Keywords: continuation of Kehayopulu and Tsingelis' work on ordered semigroups in terms of fuzzy subsets
reviewed by: Niovi Kehayopulu
MaRDI profile type: Publication
full work available at URL: j.ins.2008.07.006
OpenAlex ID: W2046388488
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DOI: 10.1016/J.INS.2008.07.006
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