\((\lambda, \mu)\)-fuzzy version of ideals, interior ideals, quasi-ideals, and bi-ideals
If \(S\) is an ordered semigroup, then every fuzzy right (or left) ideal of \(S\) is a fuzzy quasi-ideal of \(S\); every fuzzy quasi-ideal of \(S\) is a fuzzy bi-ideal of \(S\); in regular ordered semigroups the fuzzy quasi-ideals and the fuzzy bi-ideals coincide [\textit{N. Kehayopulu} and \textit{M. Tsingelis}, ``Fuzzy ideals in ordered semigroups'', Quasigroups Relat. Syst. 15, No. 2, 279--289 (2007; Zbl 1142.06004)]. In the present paper the authors consider the following definition: Let \(S\) be an ordered semigroup and \(0\leq \lambda< \mu\leq 1\). A fuzzy subset \(f\) of \(S\) is called a \((\lambda,\mu)\)-fuzzy right (resp. left) ideal of \(S\) if (1) \(\max\{f(xy),\lambda\}\geq \min\{f(x),\mu\}\) (resp. \(\max\{f(xy),\lambda\}\geq \min\{f(y),\mu\})\) and (2) if \(x\leq y\), then \(\max\{f(x),\lambda\}\geq \min\{f(y),\mu\}\) for all \(x,y\in S\). Considering the above definition for any real numbers \(\lambda\), \(\mu\) (and not only for \(0\leq \lambda< \mu\leq 1\)), any fuzzy right (resp. left) ideal of \(S\) is a \((\lambda,\mu)\)-fuzzy right (resp. left) ideal of \(S\), but the converse statement does not hold in general. The class of such \((\lambda,\mu)\)-ideals is larger than the corresponding class of fuzzy ideals. The authors examine the results in [loc. cit.] for \((\lambda,\mu)\)-fuzzy right (left), \((\lambda,\mu)\)-fuzzy quasi- and bi-ideals. They also prove that if \(f\) is a \((\lambda,\mu)\)-fuzzy ideal of \(S\), then \(f\) is a \((\lambda,\mu)\)-fuzzy interior ideal of \(S\). For \(\lambda=0\), \(\mu=1\), the corresponding results of fuzzy ideals in [loc. cit.] can be obtained. In an ordered semigroup \(S\), the definition of a quasi-ideal \(Q\) is as follows: (1) \((QS]\cap (SQ]\subseteq Q\) and (2) if \(a\in Q\) and \(S\ni b\leq a\), then \(b\in Q\).
scientific article; zbMATH DE number 6021687