COMPLEX FUZZY HYPERGROUPS BASED ON COMPLEX FUZZY SPACES

The aim of this study is to define and elaborate the concept of a complex fuzzy hypergroup, which is based on the concept of a complex fuzzy space.


Introduction
Fuzzy algebraic structures started in early 70's when Rosenfeld (1971) introduced the concept of fuzzy subgroup of a group. Dib (1994) stated the absence of the fuzzy universal set and explained some problems in Rosenfeld's approach. The absence of the fuzzy universal set has direct effect on the mentioned structure of fuzzy theory. A new approach of how to define and study fuzzy groups and fuzzy subgroups is derived in Dib (1994). Davvaz et al (2013) defined the notion of fuzzy hyperstructure and introduced a correspondence relation between these fuzzy notion and the classical ordinary notion. After introducing fuzzy hyperstructure, they defined the notion of a fuzzy hypergroup (fuzzy H v -group) and fuzzy sub-hypergroup (fuzzy H v -subgroup). Buckley (1989) incorporated the concepts of fuzzy numbers and complex numbers under the name fuzzy complex numbers. This concept has become a famous research topic and a goal for many researchers (Buckley 1989;Ramot et al 2002Ramot et al , 2003. However, the concept given by Buckley  The purpose of this study is to use the notion of complex fuzzy space to define a complex fuzzy hyperstructure and complex fuzzy hyperoperation as a generalization of fuzzy hypergroup in the sense of Davvaz et al (2013) and to introduce and discuss the concept of complex fuzzy hypergroup.

Preliminaries
In this section, we summarize the preliminary definitions and results of Marty (1934) Vougiouklis (1994)required in the sequel.
Let H be a non-empty set and let P * (H) be the set of all non-empty subsets of H . A hyperoperation on H is a map • : H × H → P * (H) and the couple (H, •) is called a hypergroupoid.
If A and B are non-empty subsets of H then we denote We say that a semihypergroup (H, •) is a hypergroup if for all x ∈ H, we have The hypergroupoid (H, •) is called an H v group, if for all x, y, z ∈ H the following conditions hold: An H v -subgroup (K, •) of (H, •) is a non-empty set K, such that for all k ∈ K, we have k • K = K • k = K.

Complex Fuzzy Spaces
In this section, we present some basic definitions of the complex fuzzy space, complex fuzzy subspace and complex fuzzy binary operation.
Definition 3.2. (Al-Husban, Salleh 2016) The complex fuzzy subspace U of the complex fuzzy space (X, E 2 ) is the collection of all ordered pairs (x, r x e iθx ), where x ∈ U 0 for some U 0 ⊂ X and r x e iθx is a subset of E 2 , which contains at least one element beside the zero element. If it happens that x / ∈ U 0 , then r x = 0 and θ x = 0. The complex fuzzy subspace U is denoted by . and denoted by SU = U 0 . Any empty complex fuzzy subspace is defined as i.e. S∅ = ∅. (i) f xy (re iθr , se iθs ) = 0 if r = 0 , s = 0 , θ s = 0 and θ r = 0 The complex fuzzy binary operation F = (F, f xy ) on a set X is a complex fuzzy function from X × X to X and is said to be uniform if F is a uniform complex fuzzy function.

Complex Fuzzy Hypergroup
In this section, we use the notion of complex fuzzy space to define complex fuzzy hyperstructure, complex fuzzy hyperoperation and complex fuzzy hypergroup. where P * (H, E 2 ) denotes the set of all non-empty complex fuzzy subspaces of (H, E 2 ) and ♦ = (∆, ∇ xy ) with ∆ : H×H → P * (H) and ∆ xy : A complex fuzzy hyperoperation ♦ = (∆, ∇ xy ) on (H, E 2 ) is said to be uniform if the associated co-membership functions ∇ xy are identical, i.e. ∇ xy = ∇ for all x, y ∈ H.

Definition 4.2.
A complex fuzzy hypergroup is a complex fuzzy hyperstrucure (H, E 2 ), ♦ satisfying the following axioms:

Definition 4.3.
A complex fuzzy H v -group is a complex fuzzy hyperstructure (H, E 2 ); ♦ satisfying the following conditions: If (H, E 2 ); ♦ satisfies only the first condition of Definition 4.2 (Definition 4.3) then it is called a complex fuzzy semihypergroup (complex fuzzy H vsemigroup) A uniform complex fuzzy hypergroup (complex fuzzy H v -group) is a complex fuzzy hypergroup (complex fuzzy H v -group) having uniform co-membership functions, i.e. ♦ = (∆, ∇ x = ∇) for x ∈ H.
A complex fuzzy hypergroup (complex fuzzy H v -group) (H, E 2 ); ♦ is called a commutative (or abelian) fuzzy hypergroup if  Proof. Assume that conditions (i) and (ii) are satisfied We want to show that U ; ♦ is a complex fuzzy sub-hypergroup of the complex fuzzy hypergroup (H, E 2 ); ♦ .
Also for any (x, r x e iθx ) ∈ U we have Similarly Therefore, by (1) and (2), U ; ♦ is a complex fuzzy sub-hypergroup.
Thus ∇ xy (r x e iθx , r y e iθy ) is the corresponding possible membership values for x∆y. That is ∇ xy (r x e iθx , r y e iθy ) = r x ∆ y e iθ x ∆ y .
In order to prove that (H, E 2 ); ♦ is a complex fuzzy H v -subgroup the same method can be applied. By using the above theorem and Theorem 4.10 of Davvaz et al (2013) we get the following corollary. (ii) ∇ xy (r x e iθx , r y e iθy ) = r x∆y e iθ x ∆ y .

Conclusion
In this paper, we continue the study of fuzzy hypergroup to the context of complex fuzzy hypergroup. We have defined the notion of a complex fuzzy hypergroup and its sub-hypergroups using the notion of a complex fuzzy space.