Abstract
In this paper, some properties of the ( k ∗ , k ) -lower part of ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals are obtained. Then, we characterize regular ordered semigroups in terms of its ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy generalized bi-ideals, ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left ideals and ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy right ideals, and an equivalent condition for ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy left (resp. right) ideals is obtained. Finally, the existence theorems for an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal as well as for the minimality of an ( ∈ , ∈ ∨ ( k ∗ , q k ) ) -fuzzy quasi-ideal of an ordered semigroup are provided.
Highlights
Since the concept of fuzzy sets was introduced by Zadeh in 1965 [1], the theories of fuzzy sets and fuzzy systems developed rapidly
Rosenfeld introduced the notion of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory of fuzzy groups
Interior ideals and a semiprime subset in a semigroup were extended [6] to introduce the concept of fuzzy interior ideals and the fuzzy semiprimality of a fuzzy set in a semigroup, and characterize different classes of a semigroup in terms of fuzzy semiprime interior ideals
Summary
Since the concept of fuzzy sets was introduced by Zadeh in 1965 [1], the theories of fuzzy sets and fuzzy systems developed rapidly. Kehayopulu and Tsingelis [14] were first to initiate the study of fuzzy sets in ordered semigroups They introduced the concept of fuzzy bi-ideals in ordered semigroups [15] and characterized bi-ideals, left and right simple, the completely regular and the strongly regular ordered semigroups in terms of fuzzy bi-ideals. In his study of fuzzy ideals in semigroups, Kuroki [4] investigated some properties of fuzzy quasi-ideals of semigroups [6] and characterized different classes of semigroups in terms of fuzzy semiprime quasi-ideals. Our first aim is to provide some different characterizations of regular ordered semigroups in terms of (∈, ∈ ∨(k∗ , qk ))-fuzzy left ideals, (∈, ∈ ∨(k∗ , qk ))-fuzzy right ideals, (∈, ∈ ∨(k∗ , qk ))-fuzzy generalized bi-ideals, and (∈, ∈ ∨(k∗ , qk ))-fuzzy quasi-ideals. We present the relationships between (∈, ∈ ∨(k∗ , qk ))-fuzzy quasi-ideals and similar types of fuzzy left/right ideals, while the last section offers concluding remarks and some ideas for future work on the topic
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