International Journal of Mathematical Archive-12(5), 2021, 23-30 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 12(5), May-2021 23 SOME CHARACTERIZATIONS OF ORDERED INVOLUTION Γ-SEMIHYPERGROUPS BY WEAKLY PRIME Γ-HYPERIDEALS ABUL BASAR 1* , NAVEED YAQOOB 2 , M. YAHYA ABBASI 3 AND S. ALI KHAN 4 1 Department of Natural and Applied Sciences, Mirzapur, Saharanpur, Uttar Pradesh-247 121, India. 2 Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad, Pakistan. 3,4 Department of Mathematics, Jamia Millia Islamia, New Delhi-110 025, India. (Received On: 17-04-21; Revised & Accepted On: 02-05-21) ABSTRACT In this paper, we introduce ordered Γ-semihypergroups with involution and weakly prime Γ- hyperideal, then we investigate some properties of prime, semiprime and weakly prime Γ-hyperideals in ordered Γ-semihypergroup with involution. Also, we study intra-regular ordered Γ-semihypergroups with involution. 2010 AMS Classification: 06F05, 16D25. Keywords: Γ-semigroup, ordered Γ-semihypergroup, Γ-hyperideal, involution, weakly prime Γ-hyperideal. 1. INTRODUCTION AND PRELIMINARIES The notion of Γ-semigroup was introduced by Sen [19]. The concept of prime and weakly prime ideal in semigroups has been given by Szasz [21], and then Petrich [18] studied these notions for semigroups. Fur- thermore, Kehayopulu [10], [11], [12] introduced prime, weakly prime ideals in ordered semigroups (partially ordered semigroups) by extending the analogous concepts of ring theory that was given by McCoy [16] and Steinfeld [20]. Khan et al [24] studied derivations of -prime rings. The concept of algebraic hyperstructures was given by Marty [15]. Algebraic hyperstructures are a standard generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The first association between binary relations and hyperstructures appeared in Nieminem [17]. For comprehensive study on semihypergroup by different algebraists, we refer [8], [4], [6], [3] and [1]. Kondo and Lekkoksung [13] studied intra-regular ordered Γ-semihypergroups. Later, Tang et al. [22] studied (fuzzy) quasi-Γ-hyperideals in ordered Γ-semihypergroups. Foulis [7] introduced the concept of involution semigroups. Later, Baxter [2] studied rings with proper involution, an d Drazin [5] studied regular semigroups with involution. Herstein [9] studied ring with involution, and Wu [23] studied intra-regular ordered semigroups with involution. In this paper, the notion of a weakly prime Γ-hyperideal of a Γ-semihypergroup with involution is introduced. A weakly prime Γ-hyperideal of a Γ-semihypergroup is a generalization of a weakly prime ideal of a semigroup, a generalization of a weakly prime hyperideal of a semihypergroup and a generalization of a Γ-ideal of a Γ- semigroup. Corresponding Author: Abul Basar 1 *, 1 Department of Natural and Applied Sciences, Mirzapur, Saharanpur, Uttar Pradesh-247 121, India.