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Research Paper | Mathematics | Volume 15 Issue 9, September 2026 | Pages: 1342 - 1346 | India
A Unified Framework for Fuzzy Semigroup Subsets Based on Inverse and Soft Set Methods
Abstract: Let S be a semigroup and let ℱ(S) denote the family of fuzzy subsets of S. In this paper we introduce a unified framework for fuzzy semigroup subsets generated simultaneously by inverse-image constructions and soft set?based parameterization. More precisely, given a parameter set E, a family of semigroup homomorphisms {φe: S → Te}e∈E, and fuzzy subsets {µe: Te → [0,1]}e∈E, we define a generalized inverse soft fuzzy subset µ: S → [0,1] via suitable inverse-image and aggregation operators, and we isolate conditions under which µ induces unified inverse soft fuzzy ideals, bi-ideals, quasi-ideals and interior ideals of S. The new framework accommodates classical fuzzy ideals and fuzzy bi-ideals in semigroups, fuzzy soft semigroups and fuzzy soft ideals, as well as int-soft semigroups, int-soft generalized bi-ideals, union-soft ideals and inverse int-fuzzy soft bi-ideals, as particular instances. We establish closure and homomorphism-invariance properties in the unified setting and provide representation theorems showing that existing constructions can be embedded into the generalized inverse soft fuzzy scheme. Several examples on finite and ordered semigroups illustrate how different choices of parameters, homomorphisms and aggregation operators recover known structures and yield new characterizations of regular and intra-regular semigroups.
Keywords: Fuzzy semigroup subsets, Unified inverse soft fuzzy ideals, Soft set parameterization, Generalized inverse soft fuzzy subset, Fuzzy bi?ideals, quasi?ideals, interior ideals
How to Cite?: S. Elakkiya, Dr. J. Jayaraj, "A Unified Framework for Fuzzy Semigroup Subsets Based on Inverse and Soft Set Methods", Volume 15 Issue 9, September 2026, International Journal of Science and Research (IJSR), Pages: 1342-1346, https://www.ijsr.net/getabstract.php?paperid=SR26921191904, DOI: https://dx.doi.org/10.21275/SR26921191904