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Regular ordered semigroups in terms of fuzzy subsets. (English) Zbl 1141.06013

Summary: Given a set \(S\), a fuzzy subset of \(S\) (or a fuzzy set in \(S\)) is, by definition, an arbitrary mapping \(f : S \rightarrow [0, 1]\), where \([0, 1]\) is the usual interval of real numbers. If the set \(S\) bears some structure, one may distinguish some fuzzy subsets of \(S\) in terms of that additional structure. This important concept of a fuzzy set was first introduced by Zadeh. Fuzzy groups have been first considered by Rosenfeld, fuzzy semigroups by Kuroki. A theory of fuzzy sets on ordered groupoids and ordered semigroups can be developed. Some results on ordered groupoids or semigroups have been already given by the same authors [Semigroup Forum 65, No. 1, 128–132 (2002; Zbl 1006.06008); Inf. Sci. 152, 231–236 (2003; Zbl 1039.06006); Inf. Sci. 171, No. 1–3, 13–28 (2005; Zbl 1080.06021)], where \(S\) has been endowed with the structure of an ordered semigroup, and they defined “fuzzy” analogues for several notions that have been proved to be useful in the theory of ordered semigroups. The characterization of regular rings in terms of right and left ideals is well known. The characterization of regular semigroups and regular ordered semigroups in terms of left and right ideals or in terms of left, right ideals and quasi-ideals is well known as well. The characterization of regular \(le\)-semigroups (that is lattice-ordered semigroups having a greatest element) in terms of right ideal elements and left ideal elements or right, left and quasi-ideal elements is also known. In the present paper we first give the main theorem which characterizes the regular ordered semigroups by means of fuzzy right and fuzzy left ideals. Then we characterize the regular ordered semigroups in terms of fuzzy right, fuzzy left ideals and fuzzy quasi-ideals. The paper serves as an example to show that one can pass from the theory of ordered semigroups to the theory of “fuzzy” ordered semigroups. Some of our results are true for ordered groupoids in general.

MSC:

06F05 Ordered semigroups and monoids
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