International Journal of Analysis and Applications Volume 16, Number 5 (2018), 628-642 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-628 GENERALIZED COMPLEX FUZZY AND ANTI-FUZZY Hv-SUBGROUPS M. AL TAHAN1 AND B. DAVVAZ2,∗ 1Department of Mathematics, Lebanese International University, Lebanon 2Department of Mathematics, Yazd University, Yazd, Iran ∗Corresponding author: davvaz@yazd.ac.ir Abstract. In this paper, we introduce the concept of generalized complex fuzzy subhypergroup (Hv- subgroup) as well as the generalized concept of complex anti-fuzzy subhypergroup (Hv-subgroup). We investigate their properties and their relations with the generalized traditional fuzzy (anti-fuzzy) subhyper- group (Hv-subgroup). 1. Introduction Hyperstructure theory was born in 1934, when Marty [8] gave the definition of hypergroup as a natural generalization of the concept of group based on the notion of hyperoperation at the eighth Congress of Scandinavian Mathematicians. He analyzed their properties and applied them to groups, illustrated some applications and showed its utility in the study of groups, algebraic functions and relational fractions. Recently, the hypergroups are studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics: geometry, topology, cryptography and code theory, graphs and hypergraphs, probability theory, binary relations, theory of fuzzy and rough sets, automata theory, economy, etc. (see [4, 6]). A hypergroup is an algebraic structure similar to a group, but the composition of two elements is a non-empty set. On the other hand, the fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It was introduced in 1965 after the publication of L. A. Zadeh Received 2017-10-30; accepted 2018-01-11; published 2018-09-05. 2010 Mathematics Subject Classification. 20N20, 20N25, 03E72 . Key words and phrases. hypergroup; complex fuzzy set; generalized fuzzy subhypergroup. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 628 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-628 Int. J. Anal. Appl. 16 (5) (2018) 629 (see [15]), who is considered as the pioneer of this theory, as an extension of the classical notion of set, when he proposed the idea of a multi-valued logic, which extends the traditional concept of a bivalent logic, which becomes a particular case of the new theory. The fuzzy set theory is based on the principle called by L. A. Zadeh “the principle of incompatibility”, that is “the closer a phenomenon is studied, the more indistinct its definition becomes”. Fuzzy sets are sets whose elements have degrees of membership. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. Many researchers worked on fuzzy set theory, its applications and its extensions. An important extension of this theory was proposed by Raymot et al. [10, 11]. They introduced the concept of complex fuzzy sets in which the codomain of membership function was the unit disc of the complex plane. Then they defined different fuzzy complex operations and relations such as the complement of a complex fuzzy set, intersection and union of complex fuzzy sets. Davvaz et al. (see [7]) introduced the concept of generalized traditional fuzzy Hv-subgroups and the authors in [1] introduced the concept of complex fuzzy and anti-fuzzy Hv-subgroups. Our paper extends their results to complex fuzzy sets, and it is constructed as follows: after an Introduction, in Section 2 we present some definitions and results about hyperstructures and traditional fuzzy subhyperstructures. In Section 3, we present the results of generalized fuzzy Hv-subgroups and introduce the concept of generalized anti-fuzzy Hv-subgroups. In Section 4, we extend the definitions of generalized fuzzy and anti-fuzzy Hv- subgroups and define generalized complex fuzzy and anti-fuzzy Hv-subgroups. We investigate their properties and present different examples on them. 2. Preliminaries In this section, we present some definitions and theorems related to hyperstructures and fuzzy subhy- perstructures that are used throughout the paper. Definition 2.1. Let H be a non-empty set. Then, a mapping ◦ : H × H → P∗(H) is called a binary hyperoperation on H, where P∗(H) is the family of all non-empty subsets of H. The couple (H,◦) is called a hypergroupoid. In the above definition, if A and B are two non-empty subsets of H and x ∈ H, then we define: A◦B = ⋃ a∈A b∈B a◦ b, x◦A = {x}◦A and A◦x = A◦{x}. Int. J. Anal. Appl. 16 (5) (2018) 630 Definition 2.2. A hypergroupoid (H,◦) is called a: • semihypergroup if for every x,y,z ∈ H, we have x◦ (y ◦z) = (x◦y) ◦z; • quasihypergroup if for every x ∈ H, x ◦ H = H = H ◦ x (This condition is called the reproduction axiom); • hypergroup if it is a semihypergroup and a quasihypergroup; • Hv-group if it is a quasihypergroup and for every x,y,z ∈ H, we have x◦ (y ◦z) ∩ (x◦y) ◦z 6= ∅. Definition 2.3. Let (H,◦) be a hypergroup (or Hv-group) and K ⊆ H. Then (K,◦) is a subhypergroup (or Hv-subgroup) of (H,◦) if for all a ∈ K, we have that a◦K = K ◦a = K. Definition 2.4. [15] A fuzzy set, defined on a universe of discourse U is characterized by a membership function µA(x) that assigns any element a grade of membership in A. The fuzzy set may be represented by the set of ordered pairs A = {(x,µA(x)) : x ∈ U}, where µA(x) ∈ [0, 1]. Definition 2.5. [7] Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x) ∈ [0, 1]. Then A is a fuzzy subhypergroup (or Hv-subgroup) of H if the following conditions hold: (1) inf{µA(z) : z ∈ x◦y}≥ min{µA(x),µA(y)} for all x,y ∈ H; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and min{µA(x),µA(a)}≤ µA(y); (3) For all x,a ∈ H, there exists z ∈ H such that x ∈ z ◦a and min{µA(x),µA(a)}≤ µA(z). Definition 2.6. [7] Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an anti-fuzzy subhypergroup (or Hv-subgroup) of H if the following conditions hold: (1) sup{µA(z) : z ∈ x◦y}≤ max{µA(x),µA(y)} for all x,y ∈ H; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(y) ≤ max{µA(x),µA(a)}; (3) For all x,a ∈ H, there exists z ∈ H such that x ∈ z ◦a and µA(z) ≤ max{µA(x),µA(a)}. Theorem 2.1. [7] Let (H,◦) be a hypergroup (or Hv-group) and µ be a fuzzy subset of H. Then µ is a fuzzy subhypergroup (or Hv-subgroup) of H if and only if its complement µ c is an anti-fuzzy subhypergroup (or Hv-subgroup) of H. Here, µ c(x) = 1 −µ(x) for all x ∈ H. 3. Generalized traditional fuzzy subhyperstructures Davvaz et al. (see [7]) introduced the concept of generalized traditional fuzzy Hv-subgroup. In this section, we present their results. And we introduce the concept generalized traditional anti-fuzzy Hv- subgroup. Int. J. Anal. Appl. 16 (5) (2018) 631 Notation 3.1. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). We say that: (1) xt ∈ µA if µA(x) ≥ t; (2) xt ∈ qµA if µA(x) + t > 1; (3) xt ∈∨qµA if xt ∈ µA or xt ∈ qµA. Otherwise, we say that xt ∈∨qµA. Definition 3.1. [7] Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup) of H if for all t,s ∈]0, 1] and x,y ∈ H, the following conditions hold: (1) xt,ys ∈ µ implies zt∧s ∈∨qµ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ H such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ H such that x ∈ y ◦a. Theorem 3.1. [7] Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈ ∨q) fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all x,y ∈ H, the following conditions hold: (1) µA(x) ∧µA(y) ∧ 0.5 ≤ µA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∧µA(a) ∧ 0.5 ≤ µA(y); (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∧µA(a) ∧ 0.5 ≤ µA(y). Definition 3.2. Let A = {(x,µA(x)) : x ∈ U} be a fuzzy set. Then the set Aπ = {(x, 2πµA(x)) : x ∈ U} is said to be a π-fuzzy set. Proposition 3.1. Let (H,◦) be a hypergroup (or Hv-group). A π-fuzzy set Aπ is an (∈,∈ ∨q) π-fuzzy subhypergroup (or Hv-subgroup) of H if and only if A is an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup) of H. Proof. The proof is straightforward. � Notation 3.2. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). We say that: (1) xt ∈ µA if µA(x) ≤ t; (2) xt ∈ q̂µA if µA(x) + t < 1; (3) xt ∈∨q̂µA if xt ∈ µA or xt ∈ q̂µA. Otherwise, we say that xt ∈∨q̂µA. Definition 3.3. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H if for all t,s ∈]0, 1] and x,y ∈ H, the following conditions hold: Int. J. Anal. Appl. 16 (5) (2018) 632 (1) xt,ys ∈ µ implies zt∨s ∈∨q̂µ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ H such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ H such that x ∈ y ◦a. Theorem 3.2. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all x,y ∈ H, the following conditions hold: (1′) µA(x) ∨µA(y) ∨ 0.5 ≥∨µA(z) for all z ∈ x◦y, (2′) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∨µA(a) ∨ 0.5 ≥ µA(y), (3′) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∨µA(a) ∨ 0.5 ≥ µA(y). Proof. (1 ⇒ 1′): Suppose that x,y ∈ H. We consider the following cases: • Case µ(x)∨µ(y) < 0.5. Assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) > 0.5. It is clear that x0.5,y0.5 ∈ µ and that z0.5 is not in µ. Having that µ(z) + 0.5 > 1 implies that z0.5 is not in q̂µ. We get that z0.5 ∈∨q̂µ. • Case µ(x) ∨µ(y) ≥ 0.5. Assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) > µ(x) ∨µ(y) ∨ 0.5 ≥ 0.5. Choose a real number t such that 0.5 ≤ µ(x) ∨µ(y) < t < µ(z). It is clear that xt,yt ∈ µ and that zt is not in µ. Having that µ(z) + t > 1 implies that zt is not in q̂µ. We get that zt ∈∨q̂µ. (2 ⇒ 2′): Suppose that x,a ∈ H. We consider the following cases: • Case µ(x) ∨µ(a) < 0.5. Assume, to get contradiction, that for every y ∈ H such that x ∈ a◦y we have µ(y) > µ(a) ∨µ(x) ∨ 0.5 = 0.5. It is clear that x0.5,a0.5 ∈ µ and that y0.5 is not in µ. Having that µ(y) + 0.5 > 1 implies that y0.5 is not in q̂µ. We get that y0.5 ∈∨q̂µ. • Case µ(x)∨µ(a) ≥ 0.5. Assume, to get contradiction, that for every y ∈ H such that x ∈ a◦y we have µ(y) > µ(a)∨µ(x)∨0.5 = µ(a)∨µ(x). Choose a real number t such that 0.5 ≤ µ(x)∨µ(a) < t < µ(y). It is clear that xt,at ∈ µ and that yt is not in µ. Having that µ(y) + t > 1 implies that yt is not in q̂µ. We get that yt ∈∨q̂µ. (3 ⇒ 3′): Can be done in a similar manner to (2 ⇒ 2′). (1′ ⇒ 1): Let xt,ys ∈ µ. Then µ(x) ≤ t,µ(y) ≤ t. Let z ∈ x◦y. We consider the following cases: • Case t∨s < 0.5. We obtain that µ(z) ≤ µ(x)∨µ(y)∨0.5 ≤ t∨s∨0.5 ≤ 0.5. Then µ(z) + t∨s < 1. Then zt∨s ∈ q̂µ. • Case t∨s ≥ 0.5. We obtain that µ(z) ≤ µ(x) ∨µ(y) ∨ 0.5 ≤ t∨s∨ 0.5 ≤ t∨s. Then zt∨s ∈ µ. (2′ ⇒ 2): Let xt,as ∈ µ. Then µ(x) ≤ t,µ(y) ≤ t. Let y ∈ H such that x ∈ a ◦ y. We consider the following cases: Int. J. Anal. Appl. 16 (5) (2018) 633 • Case t∨s < 0.5. We get that µ(y) ≤ µ(x)∨µ(a)∨0.5 ≤ t∨s∨0.5 ≤ 0.5. Then µ(y) + t∨s ∈ µ < 1. • Case t∨s ≥ 0.5. We get that µ(y) ≤ µ(x) ∨µ(a) ∨ 0.5 ≤ t∨s∨ 0.5 ≤ t∨s. Then yt∨s ∈ µ. (3′ ⇒ 3): Can be done in a similar manner to (2′ ⇒ 2). � Proposition 3.2. Let (H,◦) be a hypergroup (or Hv-group). A π-fuzzy set Aπ is an (∈,∈ ∨q̂) π-anti- fuzzy subhypergroup (or Hv-subgroup) of H if and only if A is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H. Proof. The proof is straightforward. � Theorem 3.3. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈ ∨q) fuzzy subhypergroup (or Hv-subgroup) of H if and only if its complement, Ac, is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H. Proof. Let µA be an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup). Then the conditions of Theorem 3.1 are satisfied and we need to show that the conditions of Theorem 3.2 are satisfied. To prove (1′), let x,y ∈ H. For all z ∈ x◦y, we have that µA(x) ∧µA(y) ∧ 0.5 ≤ µA(z). We get now that 1 − min{µA(x),µA(y), 0.5}≥ 1 −µA(z). The latter implies that max{µcA(x),µ c A(y), 0.5}≥ µ c A(z). To prove (2′), let x,a ∈ H. Then there exists y ∈ H such that x ∈ a◦y and µA(x)∧µA(a)∧0.5 ≤ µA(y). We get now that 1−min{µA(x),µA(a), 0.5}≥ 1−µA(y). The latter implies that max{µcA(x),µ c A(a), 0.5}≥ µ c A(y). In a similar manner, we can prove the validity of condition (3′). Thus, µcA, is an (∈,∈ ∨q) anti-fuzzy subhypergroup (or Hv-subgroup) of H. Let µcA be an (∈,∈∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup). Then the conditions of Theorem 3.2 are satisfied and we need to show that the conditions of Theorem 3.1 are satisfied.. To prove (1), let x,y ∈ H. For all z ∈ x◦y, we have that µcA(x)∨µ c A(y)∨0.5 ≥ µ c A(z). We get now that 1 −max{µ c A(x),µ c A(y), 0.5}≤ 1 −µcA(z). The latter implies that min{µA(x),µA(y), 0.5}≤ µA(z). In order to prove (2), let x,a ∈ H. Then there exists s y ∈ H such that x ∈ a ◦ y and µcA(x) ∨ µcA(a) ∨ 0.5 ≥ µ c A(y). We get now that 1 − max{µ c A(x),µ c A(a), 0.5} ≤ 1 − µ c A(y). The latter implies that min{µA(x),µA(a), 0.5}≤ µA(y). In a similar manner, we can prove the validity of condition (3). Thus, µA, is an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup) of H. � 4. Generalized complex fuzzy and anti-fuzzy subhyperstructures In this section, we use the concept of generalized fuzzy subhypergroups, discussed in Section 3, to define complex fuzzy (anti-fuzzy) subhypergroups. And we investigate their properties. 4.1. Generalized complex fuzzy Hv-subgroups. Int. J. Anal. Appl. 16 (5) (2018) 634 Definition 4.1. [10] A complex fuzzy set, defined on a universe of discourse U is characterized by a membership function µA(x) that assigns any element a complex-valued grade of membership in A. The complex fuzzy set may be represented by the set of ordered pairs A = {(x,µA(x)) : x ∈ U}, where µA(x) = r(x)e iw(x), i = √ −1, r(x) ∈ [0, 1] and w(x) ∈ [0, 2π]. Remark 4.1. By setting w(x) = 0 in the above definition, we return back to the traditional fuzzy set. Definition 4.2. Let A = {(x,µA(x)) : x ∈ H} be complex fuzzy subset of a non-void set H with membership function µA(x) = rA(x)e iwA(x). Then A is said to be homogeneous if for all x,y ∈ H, we have rA(x) ≤ rA(y) if and only if wA(x) ≤ wA(y). Notation 4.1. Let A = {(x,µA(x)) : x ∈ H} and B = {(x,µB(x)) : x ∈ H} be complex fuzzy subsets of a non-void set H with membership functions µA(x) = rA(x)e iwA(x) and µB(x) = rB(x)e iwB (x) respectively. By µA(x) ≤ µB(x), we mean that rA(x) ≤ rB(x) and wA(x) ≤ wB(x). Throughout this paper, all complex fuzzy sets are considered homogeneous. Notation 4.2. Let (H,◦) be a hypergroup (or Hv-group) and A be a complex fuzzy subset of H with mem- bership function µA(x) = rA(x)e iwA(x). We say, for all 0e0i < t = seiθ ≤ 1e2πi, that: (1) xt ∈ µA if rA(x) ≥ s and wA(x) ≥ θ; (2) xt ∈ qµA if rA(x) + s > 1 and wA(x) + θ > 2π; (3) xt ∈∨qµA if xt ∈ µA or xt ∈ qµA. Otherwise, we say that xt ∈∨qµA. Definition 4.3. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x) = rA(x)e iwA(x). Then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H if for all t,s ∈]0e0i, 1e2πi] and x,y ∈ H, the following conditions hold: (1) xt,ys ∈ µ implies zt∧s ∈∨qµ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ H such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∧s ∈∨qµ for some y ∈ H such that x ∈ y ◦a. Theorem 4.1. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all x,y ∈ H, the following conditions hold: (1′): µA(x) ∧µA(y) ∧ 0.5eiπ ≤ µA(z) for all z ∈ x◦y, (2′): For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∧µA(a) ∧ 0.5eiπ ≤ µA(y), Int. J. Anal. Appl. 16 (5) (2018) 635 (3′): For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∧µA(a) ∧ 0.5eiπ ≤ µA(y). Proof. (1 ⇒ 1′): Suppose that x,y ∈ H. We consider the following cases: • Case µ(x) ∧ µ(y) < 0.5eiπ. Assume, to get contradiction, that there exists z ∈ x ◦ y such that µ(z) < µ(x)∧µ(y)∧0.5eiπ ≤ 0.5eiπ. Choose a real number t such that µ(z) < t ≤ µ(x)∧µ(y) < 0.5eiπ. It is clear that xt,yt ∈ µ and that zt is not in µ. Having that µ(z) + t < 1e2iπ implies that zt is not in qµ. We get that zt ∈∨qµ. • Case µ(x)∧µ(y) ≥ 0.5eiπ. Assume, to get contradiction, that there exists z ∈ x◦y such that µ(z) < 0.5eiπ. It is clear that x0.5eiπ,y0.5eiπ ∈ µ and that z0.5eiπ is not in µ. Having that µ(z)+0.5eiπ < 1e2iπ implies that z0.5eiπ is not in qµ. We get that z0.5eiπ ∈∨qµ. (2 ⇒ 2′): Suppose that x,a ∈ H. We consider the following cases: • Case µ(x) ∧µ(a) ≤ 0.5eiπ. Assume, to get contradiction, that for every y ∈ H such that x ∈ a◦y we have µ(y) < µ(a) ∧ µ(x) ∧ 0.5eiπ = µ(a) ∧ µ(x). Choose a real number t such that µ(y) < t < µ(x) ∧µ(a) < 0.5eiπ. It is clear that xt,at ∈ µ and that yt is not in µ. Having that µ(y) + t < 1e2iπ implies that yt is not in qµ. We get that yt ∈∨qµ. • Case µ(x)∧µ(a) > 0.5eiπ. Assume, to get contradiction, that for every y ∈ H such that x ∈ a◦y we have µ(y) < µ(a) ∧µ(x) ∧ 0.5eiπ = 0.5eiπ. It is clear that x0.5eiπ,a0.5eiπ ∈ µ and that y0.5eiπ is not in µ. Having that µ(y) + 0.5eiπ < 1e2iπ implies that y0.5eiπ is not in qµ. We get that y0.5eiπ ∈∨qµ. (3 ⇒ 3′): Can be done in a similar manner to (2 ⇒ 2′). (1′ ⇒ 1): Let xt,ys ∈ µ. Then µ(x) ≤ t,µ(y) ≤ s. Let z ∈ x◦y. We consider the following cases: • Case t∧s ≤ 0.5eiπ. We get that µ(z) ≥ µ(x) ∧µ(y) ∧ 0.5eiπ ≥ t∧s∧ 0.5eiπ ≥ t∧s. Then zt∧s ∈ µ. • Case t ∧ s > 0.5eiπ. We get that µ(z) ≥ µ(x) ∧ µ(y) ∧ 0.5eiπ ≥ t ∧ s ∧ 0.5eiπ ≥ 0.5eiπ. Then µ(z) + t∧s > 1e2iπ. Thus, zt∧s ∈ q̂µ. (2′ ⇒ 2): Let xt,as ∈ µ. Then µ(x) ≤ t,µ(y) ≤ t. Let y ∈ H such that x ∈ a ◦ y. We consider the following cases: • Case t∧s ≤ 0.5eiπ. We get that µ(y) ≥ µ(x) ∧µ(a) ∧ 0.5eiπ ≥ t∧s∧ 0.5eiπ ≥ t∧s. Then yt∧s ∈ µ. • Case t ∧ s > 0.5eiπ. We get that µ(y) ≥ µ(x) ∧ µ(a) ∧ 0.5eiπ ≥ t ∧ s ∧ 0.5eiπ ≥ 0.5eiπ. Then µ(y) + t∧s ∈ µ > 1e2iπ. (3′ ⇒ 3): Can be done in a similar manner to (2′ ⇒ 2). � Theorem 4.2. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x) = rA(x)e iwA(x). Then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H if and only if rA is an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup) of H and wA is an (∈,∈∨q) π-fuzzy subhypergroup (or Hv-subgroup) of H. Int. J. Anal. Appl. 16 (5) (2018) 636 Proof. Let A be an (∈,∈ ∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H. This is equivalent to having the conditions of Theorem 4.1 satisfied and we can rewrite them as follows: (1) rA(x) ∧rA(y) ∧ 0.5 ≤ rA(z) and wA(x) ∧wA(y) ∧π ≤ wA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a ◦ y and rA(x) ∧ rA(a) ∧ 0.5 ≤ rA(y) and wA(x) ∧wA(a) ∧π ≤ wA(y); (3) For all x,a ∈ H, there exists y ∈ H such that rA(x) ∧rA(a) ∧ 0.5 ≤ rA(y) and wA(x) ∧wA(a) ∧π ≤ wA(y). The latter conditions are equivalent to having rA an (∈,∈∨q) fuzzy subhypergroup (or Hv-subgroup) of H and wA an (∈,∈ ∨q) π-fuzzy subhypergroup (or Hv-subgroup) of H as the conditions of Theorem 2.6 are satisfied for both: rA and wA. � Proposition 4.1. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x) = rA(x)e iwA(x). If A is a complex fuzzy subhypergroup (or Hv-subgroup) of H then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H. Proof. Let A be a complex fuzzy subhypergroup (or Hv-subgroup) of H. Then the following conditions are satisfied for all x,y ∈ H: (1) µA(x) ∧µA(y) ≤ µA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∧µA(a) ≤ µA(y); (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∧µA(a) ≤ µA(y). We have that: (1) µA(x) ∧µA(y) ∧ 0.5eiπ ≤ µA(x) ∧µA(y) ≤ µA(z) for all z ∈ x◦y, (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x)∧µA(a)∧0.5eiπ ≤ µA(x)∧µA(a) ≤ µA(y), (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y◦a and µA(x)∧µA(a)∧0.5eiπ ≤ µA(x)∧µA(a) ≤ µA(y). Therefore, A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H. � Remark 4.2. The converse of Proposition 4.1 is not always true. i.e., if A is an (∈,∈ ∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H then A may not be a complex fuzzy subhypergroup (or Hv-subgroup) of H. We illustrate Remark 4.2 by the following example. Int. J. Anal. Appl. 16 (5) (2018) 637 Example 4.1. Let H = {0, 1, 2} and define the Hv-group (H, +) by the following table: + 0 1 2 0 0 {1, 2} 2 1 {1, 2} 2 0 2 2 0 1 And define a complex fuzzy subset µ of H as: µ(0) = 0.8ei2π, µ(1) = 0.7ei 3π 2 and µ(2) = 0.6eiπ. Then µ is an (∈,∈∨q) complex fuzzy Hv-subgroup of H but it is not a complex fuzzy Hv-subgroup of H. Theorem 4.3. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all 0e0i < t ≤ 0.5eπi, µt = {x ∈ H : µA(x) ≥ t} 6= ∅ is a subhypergroup (or Hv-subgroup) of H. Proof. The proof is similar to that in [7]. � Proposition 4.2. Let (H,◦) be the biset hypergroup, i.e., x◦y = {x,y} for all x,y ∈ H and let µ be any homogeneous complex fuzzy subset of H. Then µ is an (∈,∈∨q) complex fuzzy subhypergroup of H. Proof. The proof follows from Proposition 4.1 and having µ a complex fuzzy subhypergroup of H [1]. � Proposition 4.3. Let (H,◦) be the total hypergroup, i.e., x ◦ y = H for all x,y ∈ H and let µ be any homogeneous complex fuzzy subset of H. Then µ is an (∈,∈ ∨q) complex fuzzy subhypergroup of H if and only if µ is a constant complex function or 0.5eiπ ≤ µ(x) ≤ 1e2iπ for all x ∈ H. Proof. It is easy to see that if if µ is a constant complex function or 0.5eiπ ≤ µ(x) ≤ 1e2iπ for all x ∈ H then µ is an (∈,∈∨q) complex fuzzy subhypergroup of H. Let µ be an (∈,∈∨q) complex fuzzy subhypergroup of H such that µ is not a constant complex function. Suppose, to get contradiction, that there exists x ∈ H such that µ(x) = t < 0.5eiπ. Then, by Theorem 4.3, µt = {x ∈ H : µA(x) ≥ t} 6= ∅ is a subhypergroup (or Hv-subgroup) of H. The latter implies that for all z ∈ H = x◦x µ(z) ≥ t. Since µ is not a constant function, it follows that we can find y ∈ H such that µ(y) = t0 6= t = µ(x). We have two cases: t0 < t and t0 > t. We consider the case t0 < t and the other case can be done in a similar manner. Having µt a subhypergroup (or Hv-subgroup) of H implies that y ∈ H = x◦x ⊆ µt. � Definition 4.4. Let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or Hv-group) H. Then µ is called a complex fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H if for all x,y ∈ H, the following conditions are satisfied: Int. J. Anal. Appl. 16 (5) (2018) 638 (1) µA(x) ∧µA(y) ∧β ≤ µA(z) ∨α for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∧µA(a) ∧β ≤ µA(y) ∨α; (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∧µA(a) ∧β ≤ µA(y) ∨α. Remark 4.3. If α = 0e0i,β = 1ei2π, then we obtain the complex fuzzy subhypergroup (or Hv-subgroup). And if α = 0e0i,β = 0.5eiπ, we have an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H. Theorem 4.4. Let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or Hv-group) H. Then µ is a complex fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H if and only if µt 6= ∅ is a subhypergroup (or Hv-subgroup) of H for all t ∈]α,β]. Proof. The proof is similar to that in [7]. � 4.2. Generalized complex anti-fuzzy Hv-subgroups. Notation 4.3. Let (H,◦) be a hypergroup (or Hv-group) and A be a complex fuzzy subset of H with mem- bership function µA(x) = rA(x)e iwA(x). We say, for all 0e0i ≤ t = seiθ < 1e2πi, that: (1) xt ∈ µA if rA(x) ≤ s and wA(x) ≤ θ; (2) xt ∈ q̂µA if rA(x) + s < 1 and wA(x) + θ < 2π; (3) xt ∈∨q̂µA if xt ∈ µA or xt ∈ qµA. Otherwise, we say that xt ∈∨qµA. Definition 4.5. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x). Then A is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup (or Hv- subgroup) of H if for all t,s ∈ [0e0i, 1e2πi[ and x,y ∈ H, the following conditions hold: (1) xt,ys ∈ µ implies zt∨s ∈∨q̂µ for all z ∈ x◦y; (2) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ H such that x ∈ a◦y; (3) xt,as ∈ µ implies yt∨s ∈∨q̂µ for some y ∈ H such that x ∈ y ◦a. Theorem 4.5. Let (H,◦) be a hypergroup (or Hv-group) and A be a fuzzy subset of H with membership function µA(x). Then A is an (∈,∈ ∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all x,y ∈ H, the following conditions hold: (1) µA(x) ∨µA(y) ∨ 0.5eiπ ≥ µA(z) for all z ∈ x◦y}; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∨µA(a) ∨ 0.5eiπ ≥ µA(y); (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∨µA(a) ∨ 0.5eiπ ≥ µA(y). Proof. The proof is similar to that of Theorem 3.2. � Theorem 4.6. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x) = rA(x)e iwA(x). Then A is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup Int. J. Anal. Appl. 16 (5) (2018) 639 (or Hv-subgroup) of H if and only if rA is an (∈,∈∨q̂) fuzzy subhypergroup (or Hv-subgroup) of H and wA is an (∈,∈∨q̂) π-anti-fuzzy subhypergroup (or Hv-subgroup) of H. Proof. Let A be an (∈,∈ ∨q̂) complex fuzzy subhypergroup (or Hv-subgroup) of H. This is equivalent to having the conditions of Theorem 4.5 satisfied and we can rewrite them as follows: (1) rA(x) ∨rA(y) ∨ 0.5 ≥ rA(z) and wA(x) ∨wA(y) ∨π ≥ wA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a ◦ y and rA(x) ∨ rA(a) ∨ 0.5 ≥ rA(y) and wA(x) ∨wA(a) ∨π ≥ wA(y); (3) For all x,a ∈ H, there exists y ∈ H such that rA(x) ∨rA(a) ∨ 0.5 ≥ rA(y) and wA(x) ∨wA(a) ∨π ≥ wA(y). The latter conditions are equivalent to having rA is an (∈,∈∨q̂) anti-fuzzy subhypergroup (or Hv-subgroup) of H and wA is an (∈,∈∨q̂) π-anti-fuzzy subhypergroup (or Hv-subgroup) of H as the conditions of Theorem 3.2 are satisfied for both: rA and wA. � Theorem 4.7. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H if and only if its complement, Ac, is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup (or Hv-subgroup) of H. Proof. The proof results from Theorems 3.3, 4.2 and 4.6. � Proposition 4.4. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x) = rA(x)e iwA(x). If A is a complex anti-fuzzy subhypergroup (or Hv-subgroup) of H then A is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or Hv-subgroup) of H. Proof. Let A be a complex anti-fuzzy subhypergroup (or Hv-subgroup) of H. Then the following conditions are satisfied for all x,y ∈ H: (1) µA(x) ∨µA(y) ≥ µA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∨µA(a) ≥ µA(y); (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∨µA(a) ≥ µA(y). We have that: (1) µA(x) ∨µA(y) ∨ 0.5eiπ ≥ µA(x) ∨µA(y) ≥ µA(z) for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x)∨µA(a)∨0.5eiπ ≥ µA(x)∨µA(a) ≥ µA(y); (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y◦a and µA(x)∨µA(a)∨0.5eiπ ≥ µA(x)∨µA(a) ≥ µA(y). Int. J. Anal. Appl. 16 (5) (2018) 640 Therefore, A is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or Hv-subgroup) of H. � Remark 4.4. The converse of Proposition 4.1 is not always true, i.e., if A is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup (or Hv-subgroup) of H then A may not be a complex anti-fuzzy subhypergroup (or Hv-subgroup) of H. We illustrate Remark 4.4 by the following example. Example 4.2. Let H = {0, 1, 2} and define the Hv-group (H, +) by the following table: + 0 1 2 0 0 {1, 2} 2 1 {1, 2} 2 0 2 2 0 1 And define a complex fuzzy subset µ of H as: µ(0) = 0.2ei0, µ(1) = 0.3ei π 2 and µ(2) = 0.4eiπ. Then µ is an (∈,∈∨q̂) complex anti-fuzzy Hv-subgroup of H but it is not a complex anti-fuzzy Hv-subgroup of H. Theorem 4.8. Let (H,◦) be a hypergroup (or Hv-group) and A be a (homogeneous) complex fuzzy subset of H with membership function µA(x). Then A is an (∈,∈∨q̂) complex fuzzy subhypergroup (or Hv-subgroup) of H if and only if for all 0.5eπi ≤ t < 1e2πi, µt = {x ∈ H : µA(x) ≤ t} 6= ∅ is a subhypergroup (or Hv-subgroup) of H. Proof. Let µA be an (∈,∈ ∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H and let 0.5eπi ≤ t < 1e2πi. We need to show that a◦µt = µt◦a = µt for all a ∈ µt. We prove that a◦µt = µt and µt◦a = µt can be done in a similar manner. Let x ∈ µt. Then µA(x) ≤ t and µA(a) ≤ t. Since µA is an (∈,∈∨q) complex fuzzy subhypergroup (or Hv-subgroup) of H, it follows by Theorem 4.5 that µA(z) ≤ µA(x) ∨µA(a) ∨ 0.5eπi = leqt∨0.5eπi ≤ t for all z ∈ x◦a. The latter implies that a◦µt ⊆ µt. To prove that µt ⊆ a◦µt, let x ∈ µt. Then, by Theorem 4.5, there exists y ∈ H such that x ∈ a◦y and µA(y) ≤ µA(x)∨µA(a)∨0.5eπi =≤ t∨0.5eπi ≤ t. Thus, y ∈ µt and a◦µt ⊆ µt. Conversely, let 0.5eπi ≤ t < 1e2πi and µt = {x ∈ H : µA(x) ≥ t} 6= ∅ be a subhypergroup (or Hv-subgroup) of H. Let t0 = µA(x) ∨ µA(y) ∨ 0.5eπi. Then x,y ∈ µt0 . Since µt0 is a subhypergroup (or Hv-subgroup) of H, it follows that for all z ∈ x◦y, we have z ∈ µt0 . i.e., µA(x) ∨µA(y) ∨ 0.5eπi = t0 ≥ µA(z). Now let x,a ∈ H and t1 = µA(x) ∨µA(x) ∨ 0.5eπi. Then x,a ∈ µt1 . Since µt1 is a subhypergroup (or Hv-subgroup) of H, it follows that there exists y ∈ µt1 such that x ∈ a◦y. i.e., µA(x) ∨µA(a) ∨ 0.5eπi = t1 ≥ µA(y). � Proposition 4.5. Let (H,◦) be the biset hypergroup, i.e., x◦y = {x,y} for all x,y ∈ H and let µ be any homogeneous complex fuzzy subset of H. Then µ is an (∈,∈∨q̂) complex anti-fuzzy subhypergroup of H. Int. J. Anal. Appl. 16 (5) (2018) 641 Proof. The proof follows from Proposition 4.4 and having µ a complex anti-fuzzy subhypergroup of H ( [1]). � Proposition 4.6. Let (H,◦) be the total hypergroup, i.e., x ◦ y = H for all x,y ∈ H and let µ be any homogeneous complex fuzzy subset of H. Then µ is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup of H if and only if µ is a constant complex function or µ(x) ≤ 0.5eiπ for all x ∈ H. Proof. Let µ is an (∈,∈ ∨q̂) complex anti-fuzzy subhypergroup of H. The statement is equivalent, by Theorem 4.7, we get that µc is an (∈,∈ ∨q) complex fuzzy subhypergroup of H. The latter is equivalent, using Proposition 4.3, to that µc is a constant complex function or µc(x) ≥ 0.5eiπ for all x ∈ H. � Definition 4.6. Let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or Hv-group) H. Then µ is called a complex anti-fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H if for all x,y ∈ H, the following conditions are satisfied: (1) µA(x) ∨µA(y) ∨α ≥ µA(z) ∧β for all z ∈ x◦y; (2) For all x,a ∈ H, there exists y ∈ H such that x ∈ a◦y and µA(x) ∨µA(a) ∨α ≥ µA(y) ∧β; (3) For all x,a ∈ H, there exists y ∈ H such that x ∈ y ◦a and µA(x) ∨µA(a) ∨α ≥ µA(y) ∧β. Remark 4.5. If α = 0e0i,β = 1e2πi, we get the complex anti-fuzzy subhypergroup (or Hv-subgroup). And if α = 0.5eiπ,β = 1ei2π, we get an (∈,∈∨q̂) complex fuzzy subhypergroup of H. Theorem 4.9. Let 0e0i ≤ α < β ≤ 1e2πi and µ be a (homogeneous) complex fuzzy subset of a hypergroup (or Hv-group) H. Then µ is a complex anti-fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H if and only if µt 6= ∅ is a subhypergroup (or Hv-subgroup) of H for all t ∈ [α,β[. Proof. Let µ is a complex fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H. We need to show that a◦µt = µt ◦a = µt for all a ∈ µt. We prove that a◦µt = µt and µt ◦a = µt can be done in a similar manner. Let x ∈ µt. Then µ(x) ≤ t and µ(a) ≤ t. Since µ is a complex anti-fuzzy subhypergroup (or Hv-subgroup) with thresholds (α,β) of H, it follows that µ(z) ∧β ≤ µ(x) ∨µ(a) ∨α = t∨α ≤ t for all z ∈ x◦a. Since β ≥ t, it follows that µ(z) ≤ t. The latter implies that a◦µt ⊆ µt. To prove that µt ⊆ a◦µt, let x ∈ µt. Then, there exists y ∈ H such that x ∈ a◦y and µ(y) ∧β ≤ µ(x) ∨µ(a) ∨α = t∨α ≤ t. Thus, y ∈ µt and a◦µt ⊆ µt. Conversely, let α < t ≤ β and µt = {x ∈ H : µ(x) ≤ t} 6= ∅ be a subhypergroup (or Hv-subgroup) of H. Suppose that there exists z ∈ x◦y such that µ(z) ∧β > µ(x) ∨µ(y) ∨α = t. It is clear that x,y ∈ µt and z is not in µt which contradicts our hypothesis that µt is a subhypergroup (or Hv-subgroup) of H. Thus, Condition 1. of Definition 4.6 is satisfied. Now assume that there exist a,x ∈ H such that for all y ∈ H, x ∈ a ◦ y, we have µ(y) ∧ β > µ(x) ∨ µ(a) ∨ α = t0. It is clear that x,a ∈ µt0 and y is not in µt0 Int. J. Anal. Appl. 16 (5) (2018) 642 which contradicts our hypothesis that µt0 is a subhypergroup (or Hv-subgroup) of H. Thus, Condition 2. of Definition 4.6 is satisfied. We can prove Condition 3. in a similar manner. � 5. Conclusion This paper contributed to the study of fuzzy subhyperstructures by introducing the concepts of gener- alized complex fuzzy (anti-fuzzy) Hv-subgroups and investigating their properties. References [1] M. Al-Tahan, B. Davvaz, Complex fuzzy Hv-subgroups of an Hv-group, submitted. [2] P. Corsini, Join spaces, power sets, fuzzy sets, Proceedings of the Fifth Int. Congress of Algebraic Hyperstructures and Appl., 1993, Iasi, Romania, Hadronic Press, 1994. [3] P. Corsini, A new connection between hypergroups and fuzzy sets, Southeast Asian Bull. Math. 27(2003), 221-229. [4] P. Corsini and V. Leoreanu, Applications of Hyperstructures Theory, Advances in Mathematics, Kluwer Academic Pub- lisher, 2003. [5] B. Davvaz, Fuzzy Hv-groups, Fuzzy sets Syst. 101 (1999), 191-195. [6] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. viii+200 pp. [7] B. Davvaz and I. Cristea, Fuzzy Algebraic Hyperstructures- An introduction, Studies in Fuzziness and Soft Computing 321. Cham: Springer, 2015. [8] F. Marty, Sur une generalization de la notion de group, In 8th Congress Math. Scandenaves, (1934), 45-49. [9] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512?17. [10] D. Ramot, R. Milo, M. Friedman and A. Kandel, Complex fuzzy logic, IEEE trans. fuzzy syst. 10(2) (2002), 171-186. [11] D. Ramot, M. Friedman, G. Langholz and A. Kandel, Complex fuzzy sets, IEEE trans. fuzzy syst. 11(4) (2003), 450-461. [12] T. Vougiouklis, Hyperstructures and Their Representations, Aviani editor. Hadronic Press, Palm Harbor, USA, 1994. [13] T. Vougiouklis, A new class of hyperstructures, J. Combin. Inform. Syst. Sci. 20 (1995), 229?35. [14] T. Vougiouklis, The fundamental relation in hyperrings. The general hyperfield, In: Proc of the 4th int. congress on algebraic hyperstructures and appl. (A.H.A 1990). World Sientific, Xanthi, Greece, pp 203-211. [15] L. A. Zadeh, Fuzzy sets, Inf. Control 8 (1965), 338-353. 1. Introduction 2. Preliminaries 3. Generalized traditional fuzzy subhyperstructures 4. Generalized complex fuzzy and anti-fuzzy subhyperstructures 4.1. Generalized complex fuzzy Hv-subgroups 4.2. Generalized complex anti-fuzzy Hv-subgroups 5. Conclusion References