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On fuzzy semigroups. (English) Zbl 0714.20052

The author studies properties of fuzzy ideals [bi-ideals] of a semigroup. He obtains characterizations of semigroups which are semilattices of left simple semigroups [left groups, groups]. For instance, 30 years ago the reviewer proved the following: the products \(AB\), \(BA\) of a non-empty subset \(A\) and a \((1,1)\)-ideal \(B\) of a semigroup \(S\) are \((1,1)\)-ideals of \(S\) [see S. Lajos, Magyar Tud. Akad. Mat. Fiz. Tud. Oszt. Közl. 11, 57-66 (1961; Zbl 0136.26501), Theorem 1.11]. The author shows that the products \(f\circ g\) and \(g\circ f\) are fuzzy bi-ideals of a semigroup \(S\) if \(f\) is a fuzzy subset and \(g\) is a fuzzy bi-ideal of \(S\) (Lemma 2.8). A semigroup \(S\) is a semilattice of groups if and only if the set of all [fuzzy] bi-ideals of \(S\) is a semilattice under the multiplication of [fuzzy] subsets. The non-bracketed criterion was proved by S. Lajos [Acta Sci. Math. 33, 315-317 (1972; Zbl 0247.20072)]. Semisimple semigroups are also characterized by fuzzy 2-sided ideals.
Reviewer: S.Lajos

MSC:

20M12 Ideal theory for semigroups
20N25 Fuzzy groups
20M17 Regular semigroups
20M10 General structure theory for semigroups
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References:

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