An Application of Tripolar \((m,n)\)-Fuzzy Soft Sets in Ordered Semigroups

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DOI: https://doi.org/10.28924/2291-8639-24-2026-304
Published: Sep 21, 2026
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Maliwan Phattarachaleekul, Somsak Lekkoksung, An Application of Tripolar \((m,n)\)-Fuzzy Soft Sets in Ordered Semigroups, Int. J. Anal. Appl., 24 (2026), 304.

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Maliwan Phattarachaleekul, Somsak Lekkoksung

Abstract

In this paper, we introduce the concept of tripolar \((m,n)\)-fuzzy soft sets over ordered semigroups. First, we define several core structures, including tripolar \((m, n)\)-fuzzy soft subsemigroups, tripolar \((m, n)\)-fuzzy soft left and right ideals, tripolar \((m, n)\)-fuzzy soft quasi-ideals, and tripolar \((m, n)\)-fuzzy soft bi-ideals, and we also provide concrete examples to clearly explain these new concepts. Second, we present our main results concerning regular ordered semigroups, where we prove that the concepts of a tripolar \((m,n)\)-fuzzy soft quasi-ideal and a tripolar \((m,n)\)-fuzzy soft bi-ideal are identical (they coincide) under regularity. Finally, we characterize tripolar \((m,n)\)-fuzzy soft quasi-ideals by proving that a tripolar \((m,n)\)-fuzzy soft set is a quasi-ideal if and only if it is the intersection of a tripolar \((m,n)\)-fuzzy soft left ideal and a tripolar \((m,n)\)-fuzzy soft right ideal.

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