Annals of Communications in Mathematics

AbstractThe algebraic system Γ-near rings was introduced by Satyanarayana. Tamizh and Ganesan introduced the concept of bi-ideals in near-rings [On bi-ideals of near-rings, Indian J. Pure Appl. Math., 18(11), 1002-1005(1987)]. Tamizh and Meenakumari defined the concept of bi-ideals in Γ-near-rings and characterized Γ-near-fields [Bi-Ideals of Gamma Near-Rings, Southeast Asian Bulletin of Mathematics(2004), 27: 983-988]. Satyanarayana, Yahya, Basar and Kuncham studied abstract affine Γ-nearrings [Some Results on Abstract Affine Gamma-Near-Rings, International Journal of Pure and Applied Mathematical Sciences, 7(1) (2014), 43-49]. Recently, Basar, Satyanarayana, Kuncham, Kumar and Yahya studied some relative ideals in Γ-nearrings [A note on relative Γ-ideals in abstract affine Γ-nearrings, GIS Science Journal, 8(10)(2021), 9-13]. In this paper, we study relative quasi-Γ-ideals and relative bi-Γ-ideals in Γ-near rings. We also proved nice characterizations of Γ-near rings by these basic relative Γ-ideals.

AbstractIn this paper, we introduce the concept of ordered quasi-hyperideals of regular ordered semihypergroups, and study the basic results on ordered quasi-hyperideals of ordered semihypergroups. We also investigate regular ordered semihypergroups in terms of its ordered quasi-hyperideals, ordered right hyperideals and ordered left hyperideals. We prove that: (i) A partially ordered semihypergroup S is regular if and only if for every ordered bi-hyperideal B, every ordered hyperideal I and every ordered quasi-hyperideal Q, we have B ∩ I ∩ Q ⊆ (B ◦ I ◦ Q], and (ii) A partially ordered semihypergroup S is regular if and only if for every ordered quasi-hyperideal Q, every ordered left hyperideal L and every ordered right-hyperideal R, we have R ∩ Q ∩ L ⊆ (R ◦ Q ◦ L].

AbstractIn the present paper, we introduce the relative left, right, lateral, two-sided hyperideal, relative quasi-hyperideal, relative bi-hyperideal, relative sub-idempotent ordered bi-hyperideal, relative generalized quasi-hyperideal, relative generalized bi-hyperideal, relative regularity of ordered ternary semihypergroups and relative left (right, lateral) simple ordered ternary semihypergroups. We characterize relative regular ordered ternary semihypergroups through relative quasi-hyperideals and relative bi-hyperideals. We also obtain some results based on relative simple ordered ternary semihypergroups, and other results connecting these relative hyperideal-theoretic notions.

AbstractIn this paper, we study some properties of hesitant fuzzy ideals and hesitant fuzzy bi-ideals in a semigroup and discuss their characterizations. Also we introduce hesitant fuzzy interior ideals in a semigroup and studied their properties. It is proved that in a semigroup a hesitant fuzzy ideal is a hesitant fuzzy interior ideal but the converse is not true. Moreover we prove that in regular and in intra-regular semigroups the hesitant fuzzy ideals and the hesitant fuzzy interior ideals coincide.