Fuzzy radicals and prime fuzzy ideals of ordered semigroups
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.
prime, weakly prime, completely prime, weakly completely prime fuzzy ideal
continuation of Kehayopulu and Tsingelis' work on ordered semigroups in terms of fuzzy subsets
scientific article; zbMATH DE number 5368814
scientific article