ON ROUGH (m,n) BI-Γ-HYPERIDEALS IN Γ-SEMIHYPERGROUPS

The notion of (m,n)-ideals of semigroups was introduced by Lajos [13, 14]. Later (m,n) quasi-ideals and (m,n) bi-ideals and generalized (m,n) bi-ideals were studied in various algebraic structures. The notion of a rough set was originally proposed by Pawlak [16] as a formal tool for modeling and processing incomplete information in information systems. Some authors have studied the algebraic properties of rough sets. Kuroki, in [12], introduced the notion of a rough ideal in a semigroup. Anvariyeh et al. [3], introduced Pawlak’s approximations in Γ-semihypergroups. Abdullah et al. [1], introduced the notion of M -hypersystem and N -hypersystem in Γ-semihypergroups and Aslam et al. [6], studied rough M -hypersystems and fuzzy M -hypersystems in Γ-semihypergroups, also see [4, 5, 19]. Yaqoob et al. [18], Applied rough set theory to Γ-hyperideals in left almost Γ-semihypergroups. The algebraic hyperstructure notion was introduced in 1934 by a French mathematician Marty [15], at the 8th Congress of Scandinavian Mathematicians. He published some notes on hypergroups, using them in different contexts: algebraic functions, rational fractions, non commutative groups. In 1986, Sen and Saha [17], defined the notion of a Γ-semigroup as a generalization of a semigroup. One can see that Γ-semigroups are generalizations of semigroups. Many classical notions of semigroups have been extended to Γ-semigroups and a lot of results on Γ-semigroups are published by a lot of mathematicians, for instance, Chattopadhyay [7], Chinram and Jirojkul [8], Chinram and Siammai [9], Hila [11]. Then, in [2, 10], Davvaz et al. introduced the notion of Γ-semihypergroup