Annals of Communications in Mathematics

ABSTRACT.In this paper, we introduce the notion of a fuzzy soft tri-ideal over \( \Gamma \)-semiring. We characterize the regular \( \Gamma \)-semiring in terms of fuzzy soft tri-ideals, and study some of the properties. \( M \) is a regular \( \Gamma \)-semiring, \( E \) be a parameters set and \( A \subseteq E \). If \( (\mu, A) \) is a fuzzy soft left tri-ideal over \( M \), then \( (\mu, A) \) is a fuzzy soft right ideal over \( M \).

AbstractIn this paper, we introduce the notion of a tri-quasi ideal and a fuzzy tri-quasi ideal as a further generalization of ideals, left ideals, right ideals, bi-ideals, quasi ideals, and interior ideals. We characterize the regular semigroup in terms of tri-quasi ideals, fuzzy tri-quasi ideals and study some of their properties. This generalization enables mathematicians to explore new relationships and enhancing the understanding of these structures. We establish that, a semigroup is a regular semigroup if and only if B ∩ I ∩ L ⊆ BIL, for any tri-quasi ideal B, ideal I and left ideal L of a semigroup, and for a semigroup, if μ is a fuzzy left tri-ideal of a semigroup then μ is a fuzzy tri-quasi ideal.