Annals of Communications in Mathematics

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.

AbstractIn this paper, as a further generalization of ideals, we introduce the notion of a bi-quasi-interior ideal as a generalization of ideals, right ideals, left ideals, quasi ideals, bi ideals, interior ideals and quasi interior ideals of a Γ−semigroup and study the properties of bi-quasi-interior ideals of a Γ−semigroup.