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.

ABSTRACT.In 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 \[B \cap I \cap Q \subseteq (B \circ I \circ Q),\]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 \[R \cap Q \cap L \subseteq (R \circ Q \circ L).\]

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.

AbstractIn this paper, we introduce the concept of tri-quasi hyperideal in Γ-semihyperring generalizing the classical ideal, left ideal, right ideal, bi-ideal, quasi ideal, interior ideal, bi-interior ideal, weak interior ideal, bi-quasi ideal, tri-ideal, quasi-interior ideal and bi- quasi-interior ideal of Γ-semihyperring and semiring. Furthermore, charecterizations of Γ-semihyperring, regular Γ-semihyperring and simple Γ-semihyperring with relative tri- quasi hyperideals are provided discussing the characteristics of Γ-semihyperring of relative tri-quasi hyperideals.