Structures of N-∧-Hyperideals in Left Almost ∧-Semihypergroups

Hyperstructure theory was born in 1934 when Marty [9] defined hypergroups, began to analyze their properties and applied them to groups, rational algebraic functions. Now they are widely studied from theoretical point of view and for their applications to many subjects of pure and applied properties. In 1986, Sen and Saha [12] introduced the concept of Γsemigroup as a generalization of semigroup and ternary semigroup. Many classical notions and results of the theory of semigroups have been extended and generalized to Γ-semigroups. Recently, Yaqoob and Aslam [15] introduced the notion of LA -Γsemihypergroups as a generalization of commutative semigroups, commutative semihypergroups and of commutative Γ-semigroups. They proved some results in this respect and presented some examples of LA -Γsemihypergroups. Yaqoob et al. [16, 17] applied rough set theory and soft set theory to LA -Γsemihypergroups. In 1965, Zadeh [19] introduced the notion of a fuzzy subset of a non-empty set X, as a function from X to [0,1]. After the introduction of the concept of fuzzy sets by Zadeh, several researches conducted the researches on the generalizations of the notion of fuzzy set with huge applications in computer, logics, automata and many branches of pure and applied mathematics. Rosenfeld [10] defined the concept of fuzzy group. Since then many papers have been published in the field of fuzzy algebra, see [1, 2, 13, 14]. Recently, fuzzy set theory has been well developed in the context of hyperalgebraic structure theory. A recent book [3] contains a wealth of applications. In [4], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup. Recently in [5], Davvaz and Leoreanu-Fotea studied the structure of fuzzy Γ-hyperideals in Γ-semihypergroups. In fuzzy set theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees. The generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point {1} into the interval [0,1]. Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. [6] introduced and used a new function which is called negative-valued function and constructed N-structures. They discussed Nsubalgebras and N-ideals in BCK/BCI-algebras. Jun et al. discussed the N-structure in hyper BCK-algebras and subtraction algebras [7, 8]. In [11], Borumand Saeid introduced the concept of bipolar-valued fuzzy BCK/BCI-algebras. The concept of bipolar-valued fuzzification in an LA -semigroup was introduced by Yaqoob [18]. In this paper, the concept of N-structures is applied to LA -Γ -semihypergroups and introduced some results on sub N-LA-Γ-semihypergroup, N-Γ-hyperideals and N-bi-Γ-hyperideals in left almost Γ-semihypergroups.