## $$(\lambda, \mu)$$-fuzzy version of ideals, interior ideals, quasi-ideals, and bi-ideals.(English)Zbl 1241.06012

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 [N. Kehayopulu and 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 $$b\in Q$$.

### MSC:

 06F05 Ordered semigroups and monoids

Zbl 1142.06004
Full Text:

### References:

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