A study of ( α,β )-complex fuzzy hyperideals in non-associative hyperrings

Abstract

In this paper, we introduce the concept of complex fuzzy left and complex fuzzy right hyperideals and discuss some basic properties of LA-hyperrings. We also introduce the concept of (α, β)-complex fuzzy hyperideals and discuss the important features of non-associative LA-hyperrings.

1 Introduction

In the introductory section we provide a brief history in the form of three subsections.

1.1 Hyperstructures

Hyperstructure theory was brought-out by Marty [1] in 1934, when he defined hypergroup, set-about analyzing their properties and exerted them to a group. Several papers and books have been compiled in this direction, see [2 –4]. Another book by Davvaz and Fotea [6] is devoted especially to the study of hyperring theory. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. Hoskova [7] discussed the abelization of quasi-hypergroups, Hv-rings and transposition Hv-groups as a categorical reflection.

Kazim and Naseeruddin [8] introduced the concept of left almost semigroups (LA-semigroups) and shifted the discussion toward non-associative structures. Yusuf gave the idea of left almost rings [9]. Hila and Dine [10] shifted the associative structures to non-associative hyperstructures and furnished the idea of LA-semihypergroup. Yaqoob et al. [11] expanded the work of Hila and Dine and characterized intra-regular left almost semihypergroup by their hyperideals, using pure left identity. Yousafzai et al. [12] showed the existence of non-associative algebraic hyperstructures. The idea of partially ordered left almost semihypergroups was developed by Yaqoob and Gulistan in [13]. Rehman et al. [14] initiated the study of LA-hyperrings and discussed its hyperideals and hypersystems in 2017. Nawaz et al. [15] introduced the concept of left almost semihyperrings.

1.2 Fuzzy sets

The fundamental concept of a fuzzy set was introduced by Zadeh in his paper [16] of 1965, provides a natural frame-work for generalizing several basic notions of algebra. Kuroki initiated the study of fuzzy semigroups [17 –19]. Dib et al. [20] discussed, fuzzy rings and fuzzy ideals in rings. Murali [21] defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. In [22], the idea of quasi-coincidence of a fuzzy point with a fuzzy set is defined. The concept of a (α, β)-fuzzy subgroup was first considered by Bhakat and Das in [23, 24] by using the “belongs to” relation (∈) and “quasi coincident with” relation (q) between a fuzzy point and a fuzzy subgroup, where α, β ∈ {∈ , q, ∈ ∨ q, ∈ ∧ q} and α ≠ ∈ ∧ q. Shabir et al. discussed (α, β)-fuzzy ideals and (∈ , ∈ ∨ q_k)-fuzzy ideals of semigroups in [25, 26]. Sen et al. [27] introduced and analyzed fuzzy semihypergroups.

1.3 Complex fuzzy sets

Buckly [28] gave the concept of fuzzy complex numbers. In 2002, Ramot et al. [29, 30] generalized the concept of fuzzy set and introduced the notion of complex fuzzy set. There are many researchers who have worked on complex fuzzy set for instance, Nguyen et al. [31] and Zhang et al. [32]. Abd Uazeez et al. [33] added the non-membership term to the idea of complex fuzzy set which is known as complex intuitionistic fuzzy sets, the range of values are extended to the unit circle in complex plan for both membership and non-membership functions instead of [0, 1], see also [34 –37].

In this paper, we introduce the concept of complex fuzzy hyperideals and discuss the basic properties of complex fuzzy LA-hyperrings. We also introduce the concept of (α, β)-complex fuzzy hyperideals and discuss the important features of a non-associative LA-hyperrings.

2 Preliminaries

In this section, we present some basic definitions and results, which are necessary to prove the main results.

Definition 2.1. [10] A hypergroupoid (H, •) is called an LA-semihypergroup, if it satisfies the following law: $(x • y) • z = (z • y) • x,$ for all x, y, z ∈ H.

Definition 2.2. [14] A hypergroupoid (H, •) is said to be an LA-hypergroup if it satisfy the following axioms:

for all x, y, z ∈ H, (x • y) • z = (z • y) • x,

for every x ∈ H, x • H = H • x = H .

Example 2.3. [14] Let H = {a, b, c} be a set with the hyperoperations ⊗₁and ⊗₂ defined as follows:

⊗₁	a	b	c
a	a	H	H
b	H	{b, c}	{b, c}
c	H	b	b

⊗₂	a	b	c
a	a	H	H
b	H	{b, c}	{b, c}
c	H	{a, b}	{a, b}

Then (H, ⊗ ₁) and (H, ⊗ ₂) are LA-hypergroups.

Definition 2.4. [14] An algebraic system (R, + , ·) is said to be an LA-hyperring, if

(R, +) is an LA-hypergroup;

(R, ·) is an LA-semihypergroup;

· is distributive with respect to +.

Example 2.5. Let R = {0, a, b} be a set with the hyperoperations " + " and " · " defined as follows:

+	0	a	b
0	0	a	b
a	b	R	{a, b}
b	a	{a, b}	R

and

.	a	b
0	0	0
a	{0, a}	{0, b}
b	{0, b}	{0, a}

Then (R, + , ·) is an LA-hyperring.

Definition 2.6. [29] A complex fuzzy set (CFS) C_f over the universe U, is an object having the form $C_{f} = {(x, μ_{C_{f}} (x)) : x \in U},$ where $μ_{C_{f}} (x) = γ_{C_{f}} (x) e^{{iw}_{C_{f}} (x)} .$ Both the amplitude and the phase terms r_{C
_f} (x), and w_{C
_f} (x) are real valued, for every x ∈ U, and the amplitude term r_{C
_f} (x) : U → [0, 1] and phase term w_{C
_f} (x) lying in the interval [0, 2π] .

3 Complex fuzzy hyperideals

Here in this section, we introduce the concept of complex fuzzy hyperideals and discuss some basic properties of complex fuzzy LA-hyperrings. R stands for an LA-hyperring in the whole paper.

Remark 3.1. C_fF (R) denote the collection of all complex fuzzy subsets of R.

Definition 3.2. Let C_f and C_g be any two complex fuzzy subsets of R. Then

Containment: C_f ⊆ C_g ⇔ γ_{C
_f} (x) ≤ γ_{C
_g} (x) , w_{C
_f} (x) ≤ w_{C
_g} (x) ,

Equal: C_f = C_g ⇔ γ_{C
_f} (x) = γ_{C
_g} (x) , w_{C
_f} (x) = w_{C
_g} (x) ,

Union: C_f ∪ C_g ⇔ max {γ_{C
_f} (x) , γ_{C
_g} (x)} , max {w_{C
_f} (x) , w_{C
_g} (x)} ,

Intersection: C_f ∩ C_g ⇔ min {γ_{C
_f} (x) , γ_{C
_g} (x)} , min {w_{C
_f} (x) , w_{C
_g} (x)} ,

Complement: $C_{f}^{c} = {(x, μ_{C_{f}}^{c} (x)), x \in R}$ of R is a function from R into the closed unit interval [0, 1] , i.e., $μ_{C_{f}}^{c} R \to [0, 1]$ with $μ_{C_{f}}^{c} (x) = γ_{C_{f}}^{c} (x) e^{{iw}_{C_{f}}^{c} (x)},$ where $i = \sqrt{- 1},$ $γ_{C_{f}}^{c} (x) = 1 - γ_{C_{f}} (x)$ and $w_{C_{f}}^{c} (x) = 2 π - w_{C_{f}} (x) .$

Definition 3.3. Let A≠ ∅ be a subset of R. The complex characteristic function of A is denoted by Cχ_A and defined by $C χ_{A} (a) = {\begin{matrix} 1 e^{i (2 π)} if a \in A \\ 0 e^{i (0)} otherwise . \end{matrix}$

Note that R can be considered as a complex fuzzy subset of itself and we write R = Cχ_R, i.e., Cχ_R (x) =1e^i(2π) for all x ∈ R.

Definition 3.4. Let A be a subset of R. The complex fuzzy level set of A is denoted by C_f(t,s) and defined by $C_{f (t, s)} = {x \in R | γ_{C_{f}} (x) \geq t and w_{C_{f}} (x) \geq s}$ where γ_{C
_f} (x) , t ∈ [0, 1] and w_{C
_f} (x) , s ∈ [0, 2π] .

Definition 3.5. Let C_f = {(x, μ_{C
_f} (x)) , x ∈ R} and C_g = {(x, μ_Cg (x)) , x ∈ R} be two complex fuzzy subsets of R. Then

the sum C_f ⊕ C_g is defined by $\begin{matrix} (C_{f} \oplus C_{g}) (x) \\ = & \underset{x \in a + b}{⋁} {{γ_{C_{f}} (a) \land γ_{C_{g}} (b)} e^{i {w_{C_{f}} (a) \land w_{C_{g}} (b)}}} \end{matrix}$ and (C_f ⊕ C_g) (x) = 0eⁱ⁽⁰⁾ if x cannot be ex-pressible as an element in a + b for all x ∈ H .

the product C_f ⊗ C_g is defined by $\begin{matrix} (C_{f} \otimes C_{g}) (x) \\ = & \underset{x \in a \cdot b}{⋁} {{γ_{C_{f}} (a) \land γ_{C_{g}} (b)} e^{i {w_{C_{f}} (a) \land w_{C_{g}} (b)}}} \end{matrix}$ and (C_f ⊗ C_g) (x) =0eⁱ⁽⁰⁾ if x cannot be expressible as an element in a · b for all x ∈ H .

the intrinsic product C_f ⊙ C_g is defined by $\begin{matrix} (C_{f} ⊙ C_{g}) (x) \\ = & \underset{x \in Σ_{i = 1}^{n} a_{i} \cdot b_{i}}{⋁} {{\begin{matrix} γ_{C_{f}} (a_{i}) \\ \land γ_{C_{g}} (b_{i}) \end{matrix}} e^{i {\begin{matrix} w_{C_{f}} (a_{i}) \\ \land w_{C_{g}} (b_{i}) \end{matrix}}}} \end{matrix}$ and (C_f ⊙ C_g) (x) =0eⁱ⁽⁰⁾ if x cannot be expressible as an element in $Σ_{i = 1}^{n} a_{i} \circ b_{i}$ for all x ∈ H .

Lemma 3.6. Let A and B be any non-empty subset of R. Then the following properties hold.

If A ⊆ B then Cχ_A ⊆ Cχ_B,

Cχ_A ⊙ Cχ_B = Cχ_A·B,

Cχ_A ∪ Cχ_B = Cχ_A∪B,

Cχ_A ∩ Cχ_B = Cχ_A∩B.

Proof. Proof is straightforward.□

Proposition 3.7. If C_f, C_g, C_h are complex fuzzy subsets of R, then (C_f ⊙ C_g) ⊙ C_h = (C_h ⊙ C_g) ⊙ C_f.

Proof. Let C_f, C_g, C_h are complex fuzzy subsets of R and x ∈ R. Now $\begin{matrix} ((C_{f} ⊙ C_{g}) ⊙ C_{h}) (x) \\ = & \underset{x \in Σ_{i = 1}^{n} a_{i} \cdot b_{i}}{⋁} {(C_{f} ⊙ C_{g}) (a_{i}) \land C_{h} (b_{i})} \\ = & \underset{x \in Σ_{i = 1}^{n} a_{i} \cdot b_{i}}{⋁} {\begin{matrix} (\underset{a_{i} \in Σ_{i = 1}^{n} c_{i} \cdot d_{i}}{⋁} {C_{f} (c_{i}) \land C_{g} (d_{i})}) \\ \land C_{h} (b_{i}) \end{matrix}} \\ = & \underset{x \in Σ_{i = 1}^{n} (c_{i} \cdot d_{i}) \cdot b_{i}}{⋁} {{C_{f} (c_{i}) \land C_{g} (d_{i})} \land C_{h} (b_{i})} \\ = & \underset{x \in Σ_{i = 1}^{n} (b_{i} \cdot d_{i}) \cdot c_{i}}{⋁} {{C_{h} (b_{i}) \land C_{g} (d_{i})} \land C_{f} (c_{i})} \\ = & \underset{x \in Σ_{i = 1}^{n} a_{i} \cdot c_{i}}{⋁} {\begin{matrix} (\underset{a_{i} \in Σ_{i = 1}^{n} b_{i} \cdot d_{i}}{⋁} {C_{h} (b_{i}) \land C_{g} (d_{i})}) \\ \land C_{f} (c_{i}) \end{matrix}} \\ = & \underset{x \in Σ_{i = 1}^{n} a_{i} \cdot b_{i}}{⋁} {(C_{h} ⊙ C_{g}) (a_{i}) \land C_{f} (b_{i})} \\ = & ((C_{h} ⊙ C_{g}) ⊙ C_{f}) (x) . \end{matrix}$ Thus (C_f ⊙ C_g) ⊙ C_h = (C_h ⊙ C_g) ⊙ C_f.□

Proposition 3.8. If C_f, C_g, C_h are complex fuzzy subsets of R, then

(C_f ⊕ C_g) ⊕ C_h = (C_h ⊕ C_g) ⊕ C_f.

(C_f ⊗ C_g) ⊗ C_h = (C_h ⊗ C_g) ⊗ C_f.

Proof. Proof is similar to the proof of Proposition 3.□

Proposition 3.9. Let R be an LA-hyperring with pure left identity e . Then

C_f ⊙ (C_g ⊙ C_h) = C_g ⊙ (C_f ⊙ C_h) ,

(C_f ⊙ C_g) ⊙ (C_h ⊙ C_k) = (C_f ⊙ C_h) ⊙ (C_g ⊙ C_k) ,

(C_f ⊙ C_g) ⊙ (C_h ⊙ C_k) = (C_k ⊙ C_h) ⊙ (C_g ⊙ C_f) ,

for all complex fuzzy subset C_f, C_g, C_h and C_k of R.

Definition 3.10. A complex fuzzy subset C_f of R is called complex fuzzy LA-subhyperring of R if

$inf_{z \in a + b} C_{f} (z) \geq C_{f} (a) \land C_{f} (b)$ ,

$inf_{z \in a \cdot b} C_{f} (z) \geq C_{f} (a) \land C_{f} (b),$

for all a, b ∈ R.

Example 3.11. Let R ={ 0, a, b } be a set with the hyperoperation + and · defined as follows:

+	0	a	b
0	0	R	R
a	R	{a, b}	{a, b}
b	R	R	R

and

.	a	b
0	0	0
a	R	b
b	R	R

Then (R, + , ·) is an LA-hyperring. One can see that (R, +) and (R, ·) both satisfy left invertive law and also both are non associative, as {a, b } = a + (a + b) ≠ (a + a) + b = R and {b } = a · (a · b) ≠ (a · a) · b = R [14]. Now let C_f be a complex fuzzy subset of R such that

C_{f} (x) = {\begin{matrix} 0.8 e^{i (1.6 π)} & if x = 0 \\ 0.3 e^{i (0.9 π)} & if x \in {a, b} \end{matrix},

then clearly C_f is a complex fuzzy LA-subhyperring of R .

Definition 3.12. A complex fuzzy subset C_f of R is called a complex fuzzy left hyperideal of R if

$inf_{z \in a + b} C_{f} (z) \geq C_{f} (a) \land C_{f} (b),$

$inf_{z \in a \cdot b} C_{f} (z) \geq C_{f} (b),$ for all a, b ∈ R.

Definition 3.13. A complex fuzzy subset C_f of R is called a complex fuzzy right hyperideal of R if

$inf_{z \in a + b} C_{f} (z) \geq C_{f} (a) \land C_{f} (b),$

$inf_{z \in a \cdot b} C_{f} (z) \geq C_{f} (a),$ for all a, b ∈ R.

A complex fuzzy subset C_f of R is called a complex fuzzy hyperideal of R if C_f is a complex fuzzy right hyperideal and a complex fuzzy left hyperideal of R.

Example 3.14. Let R ={ 0, a, b, c, d, e } be a set with the hyperoperation + and · defined by Table † and Table ‡, respectively.

Table 1
Cayley table for R ={ 0, a, b, c, d, e } in Example 3.

+ 0 a b c d e

0 0 {0, a} b {b, c} d {d, e}

a {0, a} a {b, c} c {d, e} e

b d {d, e} {0, b, d} R {b, d} {b, c, d, e}

c {d, e} e R {a, c, e} {b, c, d, e} {c, e}

d b {b, c} {b, d} {b, c, d, e} {0, b, d} R

e {b, c} c {b, c, d, e} {c, e} R {a, c, e}

+	0	a	b	c	d	e
0	0	{0, a}	b	{b, c}	d	{d, e}
a	{0, a}	a	{b, c}	c	{d, e}	e
b	d	{d, e}	{0, b, d}	R	{b, d}	{b, c, d, e}
c	{d, e}	e	R	{a, c, e}	{b, c, d, e}	{c, e}
d	b	{b, c}	{b, d}	{b, c, d, e}	{0, b, d}	R
e	{b, c}	c	{b, c, d, e}	{c, e}	R	{a, c, e}

Table 2

Cayley table for R ={ 0, a, b, c, d, e } in Example 3.

·	a	b	c	d	e
0	0	0	0	0	0
a	{0, a}	0	{0, a}	0	{0, a}
b	0	b	b	d	d
c	{0, a}	b	{b, c}	d	{d, e}
d	0	d	d	b	b
e	{0, a}	d	{d, e}	b	{b, c}

Then (R, + , ·) is an LA-hyperring. Now let C_f be a complex fuzzy subset of R such that $C_{f} (x) = {\begin{matrix} 0.8 e^{i (1.6 π)} & if x = 0 \\ 0.6 e^{i (0.9 π)} & if x = a \\ 0.3 e^{i (0.4 π)} & if x \in {b, c, d, e} \end{matrix},$ then clearly C_f is a complex fuzzy hyperideal of R .

Theorem 3.15. Let A be a non-empty subset of R. Then the following properties hold.

A is an LA-subhyperring of R if and only if $C_{χ_{A}}$ is a complex fuzzy LA-subhyperring of R.

A is a left (right, two-sided) hyperideal of R if and only if $C_{χ_{A}}$ is a complex fuzzy left (right, two-sided) hyperideal of R.

Proof. Proof is straightforward.□

Theorem 3.16. Let C_f be a complex fuzzy subset of R, then the following assertions are true.

C_f is a complex fuzzy LA-subhyperring of R if and only if C_f ⊕ C_f ⊆ C_f and C_f ⊗ C_f ⊆ C_f,

C_f is a complex fuzzy left hyperideal of R if and only if C_f ⊕ C_f ⊆ C_f and R ⊗ C_f ⊆ C_f,

C_f is a complex fuzzy right hyperideal of R if and only if C_f ⊕ C_f ⊆ C_f and C_f ⊗ R ⊆ C_f,

C_f is a complex fuzzy hyperideal of R if and only if C_f ⊕ C_f ⊆ C_f, R ⊗ C_f ⊆ C_f and C_f ⊗ R ⊆ C_f.

Proof. (1) Suppose that, C_f is a complex fuzzy LA-subhyperring of R. Let x ∈ R. If $\begin{matrix} (C_{f} \oplus C_{f}) (x) & = & 0 e^{i 0} \leq C_{f} (x), \\ and (C_{f} \otimes C_{f}) (x) & = & 0 e^{i 0} \leq C_{f} (x) . \end{matrix}$ This implies C_f ⊕ C_f ⊆ C_f and C_f ⊗ C_f ⊆ C_f. Otherwise, for x ∈ a · b, we have $\begin{matrix} (C_{f} \oplus C_{f}) (x) & = & \underset{x \in a + b}{⋁} {C_{f} (a) \land C_{f} (b)} \\ \leq & \underset{x \in a + b}{⋁} {inf_{x \in a + b} C_{f} (x)} \\ \leq & \underset{x \in a + b}{⋁} {C_{f} (x)} \\ since C_{f} is a complex \\ fuzzy LA - subhyperring . \\ = & C_{f} (x) . \end{matrix}$ and $\begin{matrix} (C_{f} \otimes C_{f}) (x) & = & \underset{x \in a \cdot b}{⋁} {C_{f} (a) \land C_{f} (b)} \\ \leq & \underset{x \in a \cdot b}{⋁} {inf_{x \in a \cdot b} C_{f} (x)} \\ \leq & \underset{x \in a \cdot b}{⋁} {C_{f} (x)} \\ since C_{f} is a complex \\ fuzzy LA - subhyperring . \\ = & C_{f} (x) . \end{matrix}$ Conversely, assume that C_f ⊕ C_f ⊆ C_f and C_f ⊗ C_f ⊆ C_f. Thus, for all x, y ∈ R, we have $\begin{matrix} inf_{z \in x + y} C_{f} (z) & \geq & inf_{z \in x + y} (C_{f} \oplus C_{f}) (z) \\ = & inf_{z \in x + y} {\underset{z \in x + y}{⋁} {C_{f} (x) \land C_{f} (y)}} \\ \geq & C_{f} (x) \land C_{f} (x) . \end{matrix}$ and $\begin{matrix} inf_{z \in x \cdot y} C_{f} (z) & \geq & inf_{z \in x \cdot y} (C_{f} \otimes C_{f}) (z) \\ = & inf_{z \in x \cdot y} {\underset{z \in x \cdot y}{⋁} {C_{f} (x) \land C_{f} (y)}} \\ = & C_{f} (x) \land C_{f} (x) . \end{matrix}$ This implies C_f is a complex fuzzy LA-subhyperring of R. The other cases can be seen in similar way.□

Lemma 3.17. If C_f, C_g are the complex fuzzy LA-subhyperrings of R with pure left identity e, then C_f ⊗ C_g is a complex fuzzy LA-subhyperring of R.

Proof. Proof is straightforward.□

Lemma 3.18. If R contains pure left identity e, then every complex fuzzy right hyperideal of R is a complex fuzzy hyperideal of R.

Proof. Let a, b ∈ R and let z ∈ a · b ⊆ ((e · a) · b) ⊆ ((b · a) · e) . $\begin{matrix} inf_{z \in a \cdot b} C_{f} (z) & \geq & inf_{t \in ((b \cdot a) \cdot e)} C_{f} (t) \\ \geq & inf_{r \in b \cdot a} C_{f} (r) \\ \geq & C_{f} (b) . \end{matrix}$ Thus C_f is a complex fuzzy hyperideal of R.□

Theorem 3.19. If C_f is a complex fuzzy right hyperideal of R and C_g is a complex fuzzy left hyperideal of R with pure left identity e, then C_f ⊗ C_g ⊆ C_f ∩ C_g.

Proof. Let C_f, C_g be complex fuzzy hyperideals of R and x ∈ R. If $(C_{f} \otimes C_{g}) (x) = 0 e^{i (0)} \leq (C_{f} \cap C_{g}) (x),$ then (C_f ⊗ C_g) (x) ≤ (C_f ∩ C_g) (x) , otherwise for x ∈ a · b, we have $\begin{matrix} (C_{f} \otimes C_{g}) (x) \\ = & \underset{x \in a \cdot b}{⋁} {C_{f} (a) \land C_{g} (b)} \\ \leq & \underset{x \in a \cdot b}{⋁} {inf_{x \in a \cdot b} C_{f} (x) \land inf_{x \in a \cdot b} C_{g} (x)} \\ = & \underset{x \in a \cdot b}{⋁} inf_{x \in a \cdot b} {(C_{f} \cap C_{g}) (x)} \\ = & (C_{f} \cap C_{g}) (x) . \end{matrix}$ Hence this shows that C_f ⊗ C_g ⊆ C_f ∩ C_g.□

Theorem 3.20. Let R be an LA-hyperring and C_f a complex fuzzy right hyperideal of R . Then C_f ⊗ C_f is a complex fuzzy hyperideal of R.

Proof. Proof is straightforward.□

Remark 3.21. Let C_f be a complex fuzzy left hyperideal of R, then C_f ⊗ C_f is a complex fuzzy hyperideal of R.

For an LA-hyperring R, the complex fuzzy subset R of R is defined as R (x) =1e^i2π, for all x ∈ R .

Lemma 3.22. Let R be an LA-hyperring with pure left identity e . Then R ⊗ R = R.

Proof. Every x in R can be written as x ∈ e · x, where e is the pure left identity in R. So $\begin{matrix} (R \otimes R) (x) & = & ⋁_{x \in e \cdot x} {R (e) \land R (x)} \\ = & 1 e^{i 2 π} \\ = & R (x) . \end{matrix}$

This implies that R ⊗ R = R . □

4 (α, β)-Complex fuzzy hyperideals

Here in this section we generalized the concept of (α, β)-fuzzy hyperideals by introducing the idea of (α, β)-complex fuzzy hyperideals in LA-hyperrings based on our newly defined concept of complex fuzzy point. We then discussed the important features of a non-associative LA-hyperrings by using (α, β)-complex fuzzy hyperideals. Here, α, β denotes any one of ∈, q, ∈ ∨ q or ∈ ∧ q unless otherwise specified.

Definition 4.1. A complex fuzzy subset C_f over R of the form $C_{f} (y) = {\begin{matrix} {se}^{i θ} & if y = x \\ 0 e^{i 0} & if y \neq x \end{matrix}$ is said to be a complex fuzzy point with support x and value se^iθ and is denoted by x_{se
^iθ}, where s ∈ (0, 1] and θ ∈ (0, 2π]. For a complex fuzzy point x_{se
^iθ} and complex fuzzy set C_f over R, we define;

x_se
^iθ ∈ C_f if and only if C_f (x) ≥ se^iθ which implies that γ_{C
_f} (x) ≥ s and w_{C
_f} (x) ≥ θ,

x_se
^iθqC_f if and only if C_f (x) + se^iθ ≥ 1e^i(2π) which implies that γ_{C
_f} (x) + s > 1 and w_{C
_f} (x) + θ > 2π,

x_se
^iθ ∈ ∨ qC_f if and only if x_se
^iθ ∈ C_f or x_se
^iθqC_f,

$x_{{se}^{i θ}} \bar{\in} C_{f}$ means that x_se
^iθ ∈ C_f does not hold.

Definition 4.2. A complex fuzzy subset C_f of R is called an (α, β)-complex fuzzy LA-subhyperring of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}}αC_f, and y_{s
₂
e
^{iθ
₂}}αC_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}}βC_f, for each z ∈ x + y .

x_{s
₁
e
^{iθ
₁}}αC_f, and y_{s
₂
e
^{iθ
₂}}αC_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}}βC_f, for each z ∈ x · y .

Let C_f be a complex fuzzy subset of R such that C_f (x) ≤ (0.5, π) which implies that γ_{C
_f} (x) ≤0.5 and w_{C
_f} (x) ≤ π for all x ∈ R. Let x ∈ R be such that x_{se
^iθ} ∈ ∧ qC_f . Then C_f (x) ≥ (s, θ) implies that $γ_{C_{f}} (x) \geq s and w_{C_{f}} (x) \geq θ .$ Also C_f (x) + (s, θ) ≥1e^i(2π) implies that $γ_{C_{f}} (x) + s > 1 and w_{C_{f}} (x) + θ > 2 π .$ It follows that $\begin{matrix} 1 & < & γ_{C_{f}} (x) + s \\ \leq & γ_{C_{f}} (x) + γ_{C_{f}} (x) \\ = & 2 γ_{C_{f}} (x) \end{matrix}$ and $\begin{matrix} 2 π & < & w_{C_{f}} (x) + θ \\ \leq & w_{C_{f}} (x) + w_{C_{f}} (x) \\ = & 2 w_{C_{f}} (x) \end{matrix}$ so that γ_{C
_f} (x) >0.5 and w_{C
_f} (x) > π . This means that {x_{se
^iθ} | x_{se
^iθ} ∈ ∧ qC_f } = ∅ . Therefore the case α = ∈ ∧ q is omitted.

Definition 4.3. A complex fuzzy subset C_f of R is called an (α, β)-complex fuzzy left hyperideal of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}}αC_f and y_{s
₂
e
^{iθ
₂}}αC_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}}βC_f, for each z ∈ x + y .

x ∈ R and y_se
^iθαC_f

⇒z_{se
^iθ}βC_f, for each z ∈ x · y .

Definition 4.4. A complex fuzzy subset C_f of R is called an (α, β)-complex fuzzy right hyperideal of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}}αC_f, and y_{s
₂
e
^{iθ
₂}}αC_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}}βC_f, for each z ∈ x + y .

x_se
^iθαC_f and y ∈ R

⇒z_{se
^iθ}βC_f, for each z ∈ x · y .

A complex fuzzy subset C_f of R is called an (α, β)-complex fuzzy hyperideal of R if it is both an (α, β)-complex fuzzy left hyperideal and (α, β)-complex fuzzy right hyperideal of R .

Lemma 4.5. A complex fuzzy subset C_f of R is a complex fuzzy LA-subhyperring of R if and only if, for all s₁, s₂ ∈ (0, 1] and θ₁, θ₂ ∈ (0, 2π] , it satisfies,

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ C_f for each z ∈ x + y,

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ C_f for each z ∈ x · y.

Proof. Suppose C_f is a complex fuzzy LA-subhyperring of R. Let x, y ∈ R and s₁, s₂ ∈ (0, 1] and θ₁, θ₂ ∈ (0, 2π] be such that x_{s
₁
e
^{iθ
₁}} ∈ C_f and y_{s
₂
e
^{iθ
₂}} ∈ C_f . Then C_f (x) ≥ s₁e^{iθ
₁} implies that $γ_{C_{f}} (x) \geq s_{1} and w_{C_{f}} (x) \geq θ_{1},$ and C_f (y) ≥ s₂e^{iθ
₂} implies that $γ_{C_{f}} (x) \geq s_{2} and w_{C_{f}} (x) \geq θ_{2} .$ Since C_f is a complex fuzzy LA-subhyperring of R, $γ_{C_{f}} (z) \geq min {γ_{C_{f}} (x), γ_{C_{f}} (y)} \geq min {t_{1}, t_{2}}$ and $w_{C_{f}} (z) \geq min {w_{C_{f}} (x), w_{C_{f}} (y)} \geq min {s_{1}, s_{2}}$ for each z ∈ x · y . Also $γ_{C_{f}} (z) \geq min {γ_{C_{f}} (x), γ_{C_{f}} (y)} \geq min {t_{1}, t_{2}}$ and $w_{C_{f}} (z) \geq min {w_{C_{f}} (x), w_{C_{f}} (y)} \geq min {s_{1}, s_{2}}$ for each z ∈ x + y . Hence z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ C_f for each z ∈ x + y, z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ C_f for each z ∈ x · y and s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π].

Conversely, assume that C_f satisfy the given conditions. We show that γ_{C
_f} (z)≥ min { γ_{C
_f} (x) , γ_{C
_f} (y) }, w_{C
_f} (z)≥ min { w_{C
_f} (x) , w_{C
_f} (y) } for each z ∈ x · y and $\begin{matrix} γ_{C_{f}} (z) & \geq & min {γ_{C_{f}} (x), γ_{C_{f}} (y)}, \\ w_{C_{f}} (z) & \geq & min {w_{C_{f}} (x), w_{C_{f}} (y)}, \end{matrix}$ for each z ∈ x + y . On contrary assume that there exist x, y ∈ R such that γ_{C
_f} (z)< min { γ_{C
_f} (x) , γ_{C
_f} (y) }, w_{C
_f} (z)< min { w_{C
_f} (x) , w_{C
_f} (y) } for each z ∈ x · y and $\begin{matrix} γ_{C_{f}} (z) & < & min {γ_{C_{f}} (x), γ_{C_{f}} (y)}, \\ w_{C_{f}} (z) & < & min {w_{C_{f}} (x), w_{C_{f}} (y)}, \end{matrix}$ for each z ∈ x + y . Let t ∈ (0, 1] and s ∈ (0, 2π] be such that γ_{C
_f} (z)< t ≤ min { γ_{C
_f} (x) , γ_{C
_f} (y) }, w_{C
_f} (z)< s ≤ min { w_{C
_f} (x) , w_{C
_f} (y) } for each z ∈ x · y. Then x_t ∈ C_f and y_t ∈ C_f but $z_{t} \bar{\in} C_{f}$ for each z ∈ x · y and $z_{t} \bar{\in} C_{f}$ for each z ∈ x + y . This contradicts our hypothesis. Similarly we can show it for each z ∈ x + y . Hence C_f is a complex fuzzy LA-subhyperring of R .□

Lemma 4.6. A complex fuzzy LA-subhyperring C_f of R is a complex fuzzy left hyperideal of R if and only if y_{se
^iθ} ∈ C_f and x ∈ R ⇒ z_{se
^iθ} ∈ C_f, for all z ∈ x · y and for all s ∈ (0, 1] , θ ∈ (0, 2π] .

Proof. Suppose C_f is a fuzzy left hyperideal of R . Let y_{se
^iθ} ∈ C_f, then C_f (y) ≥ se^iθ implies that γ_{C
_f} (y) ≥ s and w_{C
_f} (y) ≥ θ . Since C_f is a complex fuzzy left hyperideal of R, so $γ_{C_{f}} (z) \geq γ_{C_{f}} (y) \geq s and w_{C_{f}} (z) \geq w_{C_{f}} (y) \geq θ$ for each z ∈ x · y . Hence z_{se
^iθ}βC_f for each z ∈ x · y . Conversely, suppose that C_f satisfy the given condition. We show that γ_{C
_f} (z) ≥ γ_{C
_f} (y) and w_{C
_f} (z) ≥ w_{C
_f} (y) for each z ∈ x · y . On the contrary assume that there exist x, y ∈ R such that γ_{C
_f} (z) < γ_{C
_f} (y) and w_{C
_f} (z) < w_{C
_f} (y) for each z ∈ x · y . Let s ∈ (0, 1] and θ ∈ (0, 2π] be such that γ_{C
_f} (z) < s ≤ γ_{C
_f} (y) and w_{C
_f} (z) < θ ≤ w_{C
_f} (y) for each z ∈ x · y . This implies that y_{se
^iθ} ∈ C_f. But ${(z)}_{{se}^{i θ}} \bar{\in} C_{f}$ which contradicts our hypothesis. So γ_{C
_f} (z) ≥ γ_{C
_f} (y) and w_{C
_f} (z) ≥ w_{C
_f} (y) for each z ∈ x · y . Hence C_f is a complex fuzzy left hyperideal of R .□

Remark 4.7. Every complex fuzzy left (right) hyperideal of R is an (∈ , ∈)-complex fuzzy left (right) hyperideal of R but converse is not true.

Theorem 4.8. Let C_f be a nonzero (α, β)-complex fuzzy LA-subhyperring of R . Then the set C_{f
₀} = {x ∈ R|γ_{C
_f} (x) >0 and w_{C
_f} (x) >0} is an LA-subhyperring of R.

Proof. Let x, y ∈ C_{f
₀} . Then γ_{C
_f} (x) >0 and w_{C
_f} (x) >0 . Let γ_{C
_f} (z) = 0 and w_{C
_f} (z) = 0 for each z ∈ x · y . If α ∈ { ∈ , ∈ ∨ q } , then x_{γ_{C
_f}(x)}αγ_{C
_f} and y_{w_{C
_f}(y)}αw_{C
_f}, but $\begin{matrix} γ_{C_{f}} (z) & = & 0 < {\begin{matrix} min {γ_{C_{f}} (x), γ_{C_{f}} (y)}, γ_{C_{f}} (z) \\ + min {γ_{C_{f}} (x), γ_{C_{f}} (y)} \end{matrix}} \\ \leq & 0 + 1 \\ = & 1 \end{matrix}$ and $\begin{matrix} w_{C_{f}} (z) & = & 0 < {\begin{matrix} min {w_{C_{f}} (x), w_{C_{f}} (y)}, w_{C_{f}} (z) \\ + min {w_{C_{f}} (x), w_{C_{f}} (y)} \end{matrix}} \\ \leq & 0 + 2 π \\ = & 2 π . \end{matrix}$ So ${(z)}_{{min {γ_{C_{f}} (x), γ_{C_{f}} (x)}, min {w_{C_{f}} (y), w_{C_{f}} (y)}}} \bar{β} C_{f}$ for every β∈ { ∈ , q, ∈ ∨ q, ∈ ∧ q } and z ∈ x · y . This is a contradiction. Hence γ_{C
_f} (z) > 0 and w_{C
_f} (z) > 0 that is z ∈ x · y ∈ C_{f
₀} . Also x_1e^i(2π)qC_f and y_1e^i(2π)qC_f but ${(z)}_{1 e^{i (2 π)}} \bar{β} C_{f}$ for every β∈ { ∈ , q, ∈ ∨ q, ∈ ∧ q } and z ∈ x · y . Hence γ_{C
_f} (z) > 0 and w_{C
_f} (z) > 0 that is z ∈ x · y ∈ C_{f
₀} . Similarly we can show it for z ∈ x + y . Thus C_{f
₀} is an LA-subhyperring of R .□

Theorem 4.9. Let C_f be a nonzero (α, β)-complex fuzzy left (resp . , right) hyperideal of R . Then the set C_{f
₀} = {x ∈ R|γ_{C
_f} (x) >0 and w_{C
_f} (x) >0} is a left (resp . right) hyperideal of R .

Proof. Similar to the proof of Theorem 4.□

Theorem 4.10. Let L be a left (resp . right) hyperideal of R and let C_f be a complex fuzzy subset in R such that, $γ_{C_{f}} (x) = {\begin{matrix} 0 & if x \in R - L \\ \geq 0.5 & if x \in L \end{matrix}$ and $w_{C_{f}} (x) = {\begin{matrix} 0 & if x \in R - L \\ \geq π & if x \in L \end{matrix} .$ Then C_f is a (q, ∈ ∨ q)-fuzzy left (resp . right) hyperideal of R and also C_f is a (∈ , ∈ ∨ q)-fuzzy left (resp . right) hyperideal of R .

Proof. Proof is straightforward.□

Theorem 4.11. Let A be a LA-subhyperring of R and let C_f be a complex fuzzy subset in R such that, $γ_{C_{f}} (x) = {\begin{matrix} 0 & if x \in R - L \\ \geq 0.5 & if x \in L \end{matrix}$ and $w_{C_{f}} (x) = {\begin{matrix} 0 & if x \in R - L \\ \geq π & if x \in L \end{matrix} .$ Then C_f is an (q, ∈ ∨ q)-complex fuzzy LA-subhyperring of R and also f is an (∈ , ∈ ∨ q)-fuzzy LA-subhyperring of R .

Proof. Proof is straightforward.□

5 (∈ , ∈ ∨ q)-Complex Fuzzy hyperideals

In this section we investigate some useful results of LA-hyperrings by using the properties of (∈ , ∈ ∨ q)-complex fuzzy hyperideals.

Definition 5.1. A complex fuzzy subset C_f of R is called an (∈ , ∈ ∨ q)-complex fuzzy LA-subhyperring of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x + y .

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x · y .

Lemma 5.2. Let C_f be a complex fuzzy subset of R. Then C_f is an (∈ , ∈ ∨ q)-complex fuzzy LA-subhyperring of R if and only if

$inf_{z \in x + y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}},$

$inf_{z \in x \cdot y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} .$

Proof. Let C_f be an (∈ , ∈ ∨ q)-complex fuzzy LA-subhyperring of R . On the contrary,

(i) let for any s ∈ (0, 1] and θ ∈ (0, 2π] such that $\begin{matrix} inf_{z \in x + y} C_{f} (z) & < & {se}^{i θ} \\ < & min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} . \end{matrix}$ Then C_f (x) > se^iθ and C_f (y) > se^iθ implies that x_{se
^iθ} ∈ C_f and y_{se
^iθ} ∈ C_f . But $z_{{se}^{i θ}} \bar{\in \lor q} C_{f}$ for each z ∈ x + y, which is a contradiction. Thus $inf_{z \in x + y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} .$ (ii) let for any s ∈ (0, 1] and θ ∈ (0, 2π] such that $\begin{matrix} inf_{z \in x \cdot y} C_{f} (z) & < & {se}^{i θ} \\ < & min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} . \end{matrix}$ Then C_f (x) > se^iθ and C_f (y) > se^iθ implies that x_{se
^iθ} ∈ C_f and y_{se
^iθ} ∈ C_f . But $z_{{se}^{i θ}} \bar{\in \lor q} C_{f}$ for each z ∈ x · y, which is a contradiction. Thus $inf_{z \in x \cdot y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} .$ Conversely. (i) Assume that $inf_{z \in x + y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}}$ for all x, y, z ∈ R . Let x_{s
₁
e
^{iθ
₁}} ∈ C_f and y_{s
₂
e
^{iθ
₂}} ∈ C_f for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π]. Then C_f (x) ≥ s₁e^{iθ
₁} and C_f (y) ≥ s₂e^{iθ
₂} . Thus, we have $\begin{matrix} inf_{z \in x + y} C_{f} (z) & \geq & min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} \\ \geq & min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}, 0.5 e^{i π}} . \end{matrix}$ If min { s₁e^{iθ
₁}, s₂e^{iθ
₂} } ≥ 0.5e^iπ, then $inf_{z \in x + y} C_{f} (z) \geq 0.5 e^{i π},$ which implies that $inf_{z \in x + y} C_{f} (z) + min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}} > 0.5 e^{i π} .$ If min { s₁e^{iθ
₁}, s₂e^{iθ
₂} } < 0.5e^iπ, then $inf_{z \in x + y} C_{f} (z) \geq min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}} .$ Therefore, z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x + y .

(ii) Assume that $inf_{z \in x \cdot y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}}$ for all x, y, z ∈ R . Let x_{s
₁
e
^{iθ
₁}} ∈ C_f and y_{s
₂
e
^{iθ
₂}} ∈ C_f for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π]. Then C_f (x) ≥ s₁e^{iθ
₁} and C_f (y) ≥ s₂e^{iθ
₂} . Thus, we have $\begin{matrix} inf_{z \in x \cdot y} C_{f} (z) & \geq & min {C_{f} (x), C_{f} (y), 0.5 e^{i π}} \\ \geq & min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}, 0.5 e^{i π}} . \end{matrix}$ If min { s₁e^{iθ
₁}, s₂e^{iθ
₂} } ≥ 0.5e^iπ, then $inf_{z \in x \cdot y} C_{f} (z) \geq 0.5 e^{i π},$ which implies that $inf_{z \in x \cdot y} C_{f} (z) + min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}} > 0.5 e^{i π} .$ If min { s₁e^{iθ
₁}, s₂e^{iθ
₂} } < 0.5e^iπ, then $inf_{z \in x \cdot y} C_{f} (z) \geq min {s_{1} e^{i θ_{1}}, s_{2} e^{i θ_{2}}} .$ Therefore, z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x · y . Thus, from the both cases we see that C_f is an (∈ , ∈ ∨ q)-complex fuzzy LA-subhyperring of R .□

Definition 5.3. A complex fuzzy subset C_f of R is called an (∈ , ∈ ∨ q)-complex fuzzy right hyperideal of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x + y .

x_{s
₁
e
^{iθ
₁}} ∈ C_f ⇒ z_{s
₁
e
^{iθ
₁}} ∈ ∨ qC_f, for each z ∈ x · y .

Definition 5.4. A complex fuzzy subset C_f of R is called an (∈ , ∈ ∨ q)-complex fuzzy left hyperideal of R if for all s₁, s₂ ∈ (0, 1], θ₁, θ₂ ∈ (0, 2π] and x, y ∈ R, the following conditions hold:

x_{s
₁
e
^{iθ
₁}} ∈ C_f, and y_{s
₂
e
^{iθ
₂}} ∈ C_f

⇒z_{(s₁∧s₂)e^{i(θ₁∧θ₂)}} ∈ ∨ qC_f, for each z ∈ x + y .

y_{s
₁
e
^{iθ
₁}} ∈ C_f ⇒ z_{s
₁
e
^{iθ
₁}} ∈ ∨ qC_f, for each z ∈ x · y .

Lemma 5.5. Let C_f be a complex fuzzy subset of R. Then C_f is an (∈ , ∈ ∨ q)-complex fuzzy left hyperideal of R if and only if

$inf_{z \in x + y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}},$

$inf_{z \in x \cdot y} C_{f} (z) \geq min {C_{f} (y), 0.5 e^{i π}}$ .

Proof. Proof is similar to the proof of Theorem 5.□

Lemma 5.6. Let C_f be a complex fuzzy subset of R. Then C_f is an (∈ , ∈ ∨ q)-complex fuzzy right hyperideal of R if and only if

$inf_{z \in x + y} C_{f} (z) \geq min {C_{f} (x), C_{f} (y), 0.5 e^{i π}},$

$inf_{z \in x \cdot y} C_{f} (z) \geq min {C_{f} (x), 0.5 e^{i π}}$ .

Proof. Proof is similar to the proof of Theorem 5.□

Lemma 5.7. Intersection of (∈ , ∈ ∨ q)-complex fuzzy left (resp., right) hyperideals of R is an (∈ , ∈ ∨ q)-complex fuzzy left (resp., right) hyperidealof R .

Proof. Proof is straightforward.□

Lemma 5.8. The characteristic function $χ_{T}$ is an (∈ , ∈ ∨ q)-complex fuzzy left (resp., right) hyperideal of R if and only if T is a left (resp., right) hyperideal of R .

Proof. Proof is straightforward.□

6 Conclusion

This paper generalize the idea of fuzzy hyperideals and (α, β)-fuzzy hyperideals by introducing the concept of (α, β)-complex fuzzy hyperideals. All types of (α, β)-fuzzy hyperideals are the special case of (α, β)-complex fuzzy hyperideals as shown in the following Figure 1 and Figure 2.

Fig.1

Generalization of sets

Fig.2

Generalization of fuzzy hyperideals

We discussed different types of complex fuzzy hyperideals and (α, β)-complex fuzzy hyperideals with some interesting results. In future we are aiming to discuss the following types of generalized complex fuzzy hyperideals in detail,

(i) (∈ , ∈ ∨ q_k)-complex fuzzy hyperideals,

(ii) $(\bar{\in}, \bar{\in} \lor \bar{q_{k}})$ -complex fuzzy hyperideals,

(iii) (∈ _γ, ∈ _γ ∨ q_δ)-complex fuzzy hyperideals,

(iv) $({\bar{\in}}_{γ}, {\bar{\in}}_{γ} \lor \bar{q_{δ}})$ -complex fuzzy hyperideals.

Footnotes

Acknowledgments

The 2nd author would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project Number 1440-62.

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