EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 2, 2020, 246-257
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On Intuitionistic Fuzzy Hyper GR-ideals in Hyper
GR-algebras

Amila P. Macodi-Ringia1,*, Gaudencio C. Petalcorin, Jr.2

1 Mathematics Department, College of Natural Sciences and Mathematics, MSU-General Santos,
General Santos City, Philippines
2 Department of Mathematics and Statistics, College of Science and Mathematics, MSU-Iligan
Institute of Technology, Iligan City, Philippines

Abstract. In this paper, fuzzy set and intuitionistic fuzzy set are applied to hyper GR-algebra.
Particularly, the fuzzy hyper GR-ideal of type 1 and the intuitionistic fuzzy hyper GR-ideal are
introduced, and a relationship between them are obtained. Moreover, some of their characteriza-
tions are established by the use of their level subsets.

2020 Mathematics Subject Classifications: 20N20, 06F35, 03G25, 03E72, 03B52, 08A72

Key Words and Phrases: Hyper GR-algebras, hyper GR-ideals, hyper GR-ideals, fuzzy hyper
GR-ideals of type 1, intuitionistic fuzzy hyper GR-ideals

1. Introduction

In 1934, hyperstructure theory was introduced in 1934 by F. Marty [13] during the
8th Congress of Scandinavian Mathematicians. Around the 40’s, several authors worked
on hypergroups, especially, in France and in the United States, but also in Italy, Russia
and Japan. Over the following decades, many important results appeared, but above
all since the 70’s onwards the most luxuriant flourishing hyperstructures has been seen.
Hyperstructures have many application to several sectors of both pure and applied sci-
ences. Davvaz et al. [5] applied this concept to elementary particles in physical theory.
While, Xin [11] applied this concept to BCI-algebras and proved that hyper BCI-algebras
are one of the generalizations of BCI-algebras. After the introduction on the concept
of hyper BCI-algebras, several researches were conducted. One of these studies is the
hyper GR-algebras. In 2016, Indangan et al. [6] introduced hyper GR-algebras and the
faithful hyper GR-algebra’s hyper operation properties were established including some
properties of hyper GR-ideals. In 2017, some hyper homomorphic properties on hyper

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i2.3660

Email addresses: amila.macodi-ringia@g.msuiit.edu.ph (A. Macodi-Ringia),
gaudencio.petalcorin@g.msuiit.edu.ph (G. Petalcorin, Jr.)

http://www.ejpam.com 246 © 2020 EJPAM All rights reserved.



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 247

GR-algebra together with the construction of the quotient hyper GR-algebra via regular
congruence relation were presented by Indangan et al. [7].

Uncertainty is an attribute of information and uncertain data are presented in various
domains. The most appropriate theory for dealing with uncertainties is the theory of
fuzzy sets developed by Zadeh [16] in 1965. It has been well developed in the context of
hyperstructure theory. Several studies were fuzzy theory is applied to hyperstructure are
fuzzy hyper BCK-ideals of hyper BCK-algebras [8], fuzzy ideals in hyper BCI-algebras
[14], fuzzy implicative hyper BCK-ideals of hyper BCK-algebras [9], and some results on
fuzzy implicative hyper GR-ideals [12]. These studies obtained some characterizations
where level subsets of fuzzy set are being used. On the contrary, fuzzy set theory has no
means to incorporate the hesitation or uncertainty in the membership degrees. Atanassov
[1, 2] introduced the concept of intuitionistic fuzzy sets in a non-empty set X which give
both a membership degree and a non-membership degree. Since then, the notion of
intuitionistic fuzzy set has been explored by researchers and a number of theoritical
and practical results have appeared. The relations between intuitionistic fuzzy sets and
algebraic hyperstructures have been already considered by many mathematicians. Some
of these studies are intuitionistic fuzzy hyper BCK-ideals of hyper BCK-algebras [3] and
intuitionistic fuzzy ideals in hyper BCI-algebras [15]. Both of these researches used level
subsets of intuitionistic fuzzy set to establish some characterizations of intuitionistic fuzzy
hyper BCK-ideals and intuitionistic fuzzy hyper BCI-ideals.

In this paper, we intoduce fuzzy hyper GR-ideals of type 1 and intuitionistic fuzzy
hyper GR-ideals including some of thier properties and characterizations by following
the works of Borzooei et al. [3], Jun et al. [8, 9], Nisar et al. [14], and Palaniappan et al.
[15].

2. Preliminaries

Let H be a nonempty set with a hyperoperation “~”. For any two subsets A and B of
H and x ∈ H, we define

A ~ B =
⋃

a∈A,b∈B

a ~ b,A ~ x = A ~ {x}, and x ~ B = {x} ~ B.

Moreover, x � y is defined by 0 ∈ x ~ y and A � B is defined by for all a ∈ A, there exist
b ∈ B such that a� b. The symbol “�”is called a hyperorder on H.

Definition 2.1. [10] Let H be a nonempty set endowed with a hyperoperation ~ and a
constant 0. Then (H,~, 0) is called a hyper BCK-algebra if it satisfies the following axioms,
for all x, y, z ∈ H:

(i) (x ~ z) ~ (y ~ z)� x ~ y;

(ii) (x ~ y) ~ z = (x ~ z) ~ y;

(iii) x ~H � {x}; and



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 248

(iv) x� y and y� x imply x = y.

Definition 2.2. [11] Let H be a nonempty set and ~ be a hyperoperation on H. Then
(H,~, 0) is called a hyper BCI-algebra if it contains a constant 0 ∈ H and satisfies the
following axioms, for all x, y, z ∈ H:

(i) (x ~ z) ~ (y ~ z)� x ~ y;

(ii) (x ~ y) ~ z = (x ~ z) ~ y;

(iii) x� x;

(iv) x� y and y� x imply x = y; and

(v) 0 ~ (0 ~ x)� x, x , 0;

Definition 2.3. [6] Let H be a nonempty set and ~ be a hyperoperation on H. If H contains
a constant 0 and the following axioms

(HGR1) (x ~ z) ~ (y ~ z)� x ~ y,

(HGR2) (x ~ y) ~ z = (x ~ z) ~ y,

(HGR3) x� x,

(HGR4) 0 ~ (0 ~ x)� x, x , 0, and

(HGR5) (x ~ y) ~ z� y ~ z

are satisfied for all x, y, z ∈ H, then (H,~, 0) is said to be a hyper GR-algebra.
For the sake of simplicity, we say H is a hyper GR-algebra.

Definition 2.4. [6] A subset I of a hyper GR-algebra H is called a hyper GR-ideal of H if it
contains 0 and for all x, y ∈ H, x ~ y ⊆ I and y ∈ I imply that x ∈ I.

Definition 2.5. [16] A fuzzy set µ of a nonempty set M is a function µ : M→ [0, 1].

Definition 2.6. [4] Let µ be a fuzzy set of M. For a fixed t ∈ [0, 1], the set µt = {x ∈M|µ(x) ≥
t} is called a level subset of µ.

Definition 2.7. [1] An intuitionistic fuzzy set A in a nonempty set H is an object having the
form A = {(x, µA(x), γA(x))|x ∈ H} where the function µA : H → [0, 1] and γA : H → [0, 1]
denote the degree of membership and the degree of nonmembership, respectively, and
for all x ∈ H,

0 ≤ µA(x) + γA(x) ≤ 1.

Furthermore, we have πA(x) = 1−µA(x)−γA(x) called the intuitionistic fuzzy set index or
hesitation margin of x in A. πA(x) is the degree of indeterminancy of x ∈ H to intuitionistic
fuzzy set A and πA(x) ∈ [0, 1]. πA(x) expresses the lack of knowledge of whether x belongs
to intuitionistic fuzzy set A or not.



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 249

We shall use the symbol A = (µA, γA) for the intuitionistic fuzzy set A = {(x, µA(x), γA(x))|x ∈
H}.

Definition 2.8. [15] For an intuitionistic fuzzy set A = (µA, γA) in H and s, t ∈ [0, 1], the
set A〈t,s〉 = {x ∈ H|µA(x) ≥ t, γA(x) ≤ s} is called a level subset of A.

3. Fuzzy Hyper GR-ideals of Type 1

Definition 3.1. A fuzzy set µ in a hyper GR-algebra H is a fuzzy hyper GR-ideal of type 1 if
for all x, y ∈ H,

(F1) µ(0) ≥ µ(x) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}

.

Example 3.2. Let H = [0, 1] such that for any a, b ∈ [0, 1],

a ~ b =
{

[0, 0.3], if b , 0 or a = 0 = b;
{a}, if a , 0 and b = 0.

It can be seen that H is a hyper GR-algebra. Define a fuzzy set µ in H by

µ(a) =
{

l, if a = 0
k, if a , 0.

where k, l ∈ [0, 1] and k < l. By routine calculations, we see thatµ is a fuzzy hyper GR-ideal
of type 1 in H.

Example 3.3. Consider the hyper GR-algebra H in Example 3.2. Define a fuzzy set µ in H
by

µ(x) =


1, if x = 0,
0.3 + x, if x ∈ (0, 0.3],
0.7, if x ∈ (0.3, 1].

It can be shown that µ is a fuzzy hyper GR-ideal of type 1 in H.

Proposition 3.4. Let H be a hyper GR-algebra. If µ is a fuzzy hyper GR-ideal of type 1 in H such
that inf

u∈x~y
µ(u) = µ(0) = inf

u∈y~x
µ(u) for x , y, then µ(x) = µ(y).

Proof. Let x, y ∈ H such that x , y and inf
u∈x~y

µ(u) = µ(0) = inf
u∈y~x

µ(u). By F1,

µ(x) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}
= min{µ(0), µ(y)} = µ(y) and

µ(y) ≥ min
{

inf
u∈y~x

µ(u), µ(x)
}
= min{µ(0), µ(x)} = µ(x).



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 250

Thus, µ(x) = µ(y). �

Theorem 3.5. A fuzzy set µ in a hyper GR-algebra H is a fuzzy hyper GR-ideal of type 1 if and
only if µt is a hyper GR-ideal of H whenever µt , ∅ and t ∈ [0, 1].

Proof. Suppose µ is a fuzzy hyper GR-ideal of type 1. Let t ∈ [0, 1] such that µt , ∅.
Then there exists a ∈ µt. By F1,µ(0) ≥ µ(a) ≥ t. Then, 0 ∈ µt. Let x, y ∈ H such that x~y ⊆ µt
and y ∈ µt. Then µ(y) ≥ t and µ(u) ≥ t for any u ∈ x~y. This implies that t is a lowerbound

of {µ(u) : u ∈ x~y}. Thus, inf
u∈x~y

µ(u) ≥ t. By F1,µ(x) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}
≥ min{t, t} = t.

Hence, x ∈ µt and so µt is a hyper GR-ideal of H.
Conversely, let µt be a hyper GR-ideal of H for any t ∈ [0, 1]. Let x ∈ H and let

k ∈ [0, 1] such that k = µ(x). Since 0 ∈ µk, µ(0) ≥ k = µ(x). Moreover, let x, y, z ∈ H and

let l ∈ [0, 1] such that l = min
{

inf
u∈x~y

µ(u), µ(y)
}

. Since µ(y) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}
= l,

y ∈ µl. Let w ∈ x ~ y. Then µ(w) ≥ inf
u∈x~y

µ(u) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}
= l. It follows

that w ∈ µl and so x ~ y ⊆ µl. Since µl is a hyper GR-ideal of H, x ∈ µl. It implies that

µ(x) ≥ l = min
{

inf
u∈x~y

µ(u), µ(y)
}

. Thus, µ is a fuzzy hyper GR-ideal of type 1. �

Corollary 3.6. For any nonempty subset A of H, let µA be a fuzzy set in hyper GR-algebra H
defined by

µA(x) =
{

n, if x ∈ A,
m, otherwise,

for all x ∈ H where n,m ∈ [0, 1] with n > m. Then A is a hyper GR-ideal of H if and only if µA is
a fuzzy hyper GR-ideal of type 1 in H.

Proof. Note that

(µA)t =


∅, if n < t ≤ 1,
A, if m < t ≤ n,
H, if 0 ≤ t ≤ m

(1)

are all possible level subsets of µA where t ∈ [0, 1]. Then, (µA)t = A is a nonempty level
subsets of µA. Thus, by Theorem 3.5, A is a hyper GR-ideal of H if and only if µA is a
fuzzy hyper GR-ideal of type 1 in H. �

Theorem 3.7. If µ is a fuzzy hyper GR-ideal of type 1 of a hyper GR-algebra H, then the set
A = {x ∈ H|µ(x) = µ(0)} is a hyper GR-ideal of H.

Proof. Suppose µ is a fuzzy hyper GR-ideal of type 1 of a hyper GR-algebra H. Let
x, y ∈ H such that x ~ y ⊆ A and y ∈ A. Then µ(y) = µ(0) and µ(u) = µ(0) for all u ∈ x ~ y.

By the hypothesis, µ(0) ≥ µ(x) ≥ min
{

inf
u∈x~y

µ(u), µ(y)
}
= µ(0). Hence, µ(x) = µ(0) and so

x ∈ A. Thus, A is a hyper GR-ideal of H. �



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 251

4. Intuitionistic Fuzzy Hyper GR-ideals

Definition 4.1. An intuitionistic fuzzy set A = (µA, γA) in a hyper GR-algebra H is an
intuitionistic fuzzy hyper GR-ideal if for all x, y ∈ H the following hold:

(IFGR1) µA(0) ≥ µA(x) and γA(0) ≤ γA(x);

(IFGR2) µA(x) ≥ min
{

inf
u∈x~y

µA(u), µA(y)
}

; and

(IFGR3) γA(x) ≤ max

 sup
v∈x~y

γA(v), γA(y)

.

For the sake of simplicity, we shall use the symbol A = (µA, γA) for the intuitionistic
fuzzy set A = {(x, µA(x), γA(x))|x ∈ H}.

Example 4.2. Consider the hyper GR-algebra H in Example 3.3 and its fuzzy set µ. Let
A = (µA, γA) in H be an intuitionistic fuzzy set where µA = µ and

γA(x) =


0, if x = 0,
0.7 − x, if x ∈ (0, 0.3],
0.1, if x ∈ (0.3, 1].

By routine calculations, A is an intuitionistic fuzzy hyper GR-ideal in H.

Theorem 4.3. Let A = (µA, γA) be an intuitionistic fuzzy set in a hyper GR-algebra H. A〈t,s〉 is
a hyper GR-ideal of H if and only if A is an intuitionistic fuzzy hyper GR-ideal of H whenever
A〈t,s〉 , ∅ and t, s ∈ [0, 1].

Proof. Suppose A〈t,s〉 is a hyper GR-ideal of H for any t, s ∈ [0, 1]. Let x ∈ H and let
k, l ∈ [0, 1] such that k = µA(x) and l = γA(x). Since A〈k,l〉 is a hyper GR-ideal of H, 0 ∈ A〈k,l〉.
Then µA(0) ≥ k = µA(x) and γA(0) ≤ l = γA(x). Moreover, let x, y ∈ H and let t̃, s̃ ∈ [0, 1]

such that t̃ = min
{

inf
u∈x~y

µA(u), µA(y)
}

and s̃ = max

 sup
v∈x~y

γA(v), γA(y)

. Suppose w ∈ x~y.

Then,

µA(w) ≥ inf
u∈x~y

µA(u) ≥ min
{

inf
u∈x~y

µA(u), µA(y)
}
= t̃ and

γA(w) ≤ sup
v∈x~y

γA(v) ≤ max

 sup
v∈x~y

γA(v), γA(y)

 = s̃.

These imply that w ∈ A
〈t̃,s̃〉 and so x ~ y ⊆ A

〈t̃,s̃〉. Note that

µA(y) ≥ min
{

inf
u∈x~y

µA(u), µA(y)
}
= t̃ and

γA(y) ≤ max

 sup
v∈x~y

γA(v), γA(y)

 = s̃.

Then, y ∈ A
〈t̃,s̃〉. Since A

〈t̃,s̃〉 is a hyper GR-ideal of H, x ∈ A
〈t̃,s̃〉. It follows that



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 252

µA(x) ≥ t̃ = min
{

inf
u∈x~y

µA(u), µA(y)
}

and

γA(x) ≤ s̃ = max

 sup
v∈x~y

γA(v), γA(y)

.

By Definition 4.1, A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal in H.
Conversely, suppose A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal in H. Then,

µA(0) ≥ µA(x) for all x ∈ H. Let t, s ∈ [0, 1]. Since A〈t,s〉 , ∅, there exists z ∈ A〈t,s〉 such
that µA(0) ≥ µA(z) ≥ t and γA(0) ≤ γA(z) ≤ s. Thus, 0 ∈ A〈t,s〉. Let x, y ∈ H such that
x ~ y ⊆ A〈t,s〉 and y ∈ A〈t,s〉. Then, µA(y) ≥ t, γA(y) ≤ s, µA(u) ≥ t and γA(v) ≤ s for any
u, v ∈ x~ y. It follows that t is a lowerbound for {µA(u) : u ∈ x~ y} and s is an upperbound
for {γA(v) : v ∈ x ~ y}. Then, inf

u∈x~y
µA(u) ≥ t and sup

v∈x~y
γA(v) ≤ s. By IFGR2 and IFGR3,

µA(x) ≥ min
{

inf
u∈x~y

µA(u), µA(y)
}
≥ min{t, t} = t and

γA(x) ≤ max

 sup
v∈x~y

γA(v), γA(y)

 ≤ max{s, s} = s.

Hence, x ∈ A〈t,s〉 and so A〈t,s〉 is a hyper GR-ideal of H. �

Lemma 4.4. Let µ : H→ [0, 1] be a fuzzy set and S ⊆ H. Then

(a) 1 − sup
x∈S

µ(x) = inf
x∈S

(
1 − µ(x)

)
, and

(b) 1 − inf
x∈S
µ(x) = sup

x∈S
(1 − µ(x)).

Proof. Let x ∈ S.

(a) Since µ(x) ≤ sup
x∈S

µ(x), 1−µ(x) ≥ 1− sup
x∈S

µ(x). Then, 1− sup
x∈S

µ(x) is a lowerbound for

{1−µ(x)|x ∈ S}. It implies that 1−sup
x∈S

µ(x) ≤ inf
x∈S

(
1 − µ(x)

)
. Since inf

x∈S
(1−µ(x)) ≤ 1−µ(x),

µ(x) ≤ 1 − inf
x∈S

(1 − µ(x)). Thus, 1 − inf
x∈S

(1 − µ(x)) is an upperbound for {µ(x) : x ∈ S}.

Then sup
x∈S

µ(x) ≤ 1 − inf
x∈S

(1 − µ(x)) and so inf
x∈S

(1 − µ(x)) ≤ 1 − sup
x∈S

µ(x). Therefore,

1 − sup
x∈S

µ(x) = inf
x∈S

(1 − µ(x)).

(b) Note that µ(x) ≥ inf
x∈S
µ(x). Then, 1 − µ(x) ≤ 1 − inf

x∈S
µ(x). This implies that 1 − inf

x∈S
µ(x)

is an upperbound for {1 − µ(x)|x ∈ S}. It follows that sup
x∈S

(1 − µ(x)) ≤ 1 − inf
x∈S
µ(x).

Since 1 − µ(x) ≤ sup
x∈S

(1 − µ(x)), 1 − sup
x∈S

(1 − µ(x)) ≤ µ(x). Then, 1 − sup
x∈S

(1 − µ(x)) is

a lowerbound for {µ(x)|x ∈ S}. This implies that 1 − sup
x∈S

(1 − µ(x)) ≤ inf
x∈S
µ(x) and so

1 − inf
x∈S
µ(x) ≤ sup

x∈S
(1 − µ(x)). Hence, 1 − inf

x∈S
µ(x) = sup

x∈S
(1 − µ(x)). �

The following corollary follows from Lemma 4.4.



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 253

Corollary 4.5. Let µ : H→ [0, 1] be a fuzzy set and S ⊆ H. Then

(a) 1 −max
x∈S

µ(x) = min
x∈S

(1 − µ(x)),

(b) 1 −min
x∈S

µ(x) = max
x∈S

(1 − µ(x)).

Lemma 4.6. An intuitionistic fuzzy set A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal
in a hyper GR-algebra H if and only if the fuzzy sets µA and γ̄A are fuzzy hyper GR-ideals of type
1 in H.

Proof. Suppose A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal in H. Clearly,
µA is a fuzzy hyper GR-ideal of type 1 in H and γA(x) ≥ γA(0) for all x ∈ H. Then,
γ̄A(x) = 1 − γA(x) ≤ 1 − γA(0) = γ̄A(0). Let x, y ∈ H. Then,

γ̄A(x) = 1 − γA(x) ≥ 1 −max

 sup
v∈x~y

γA(v), γA(y)

 . (2)

Case 1. Suppose max

 sup
v∈x~y

γA(v), γA(y)

 = sup
v∈x~y

γA(v).

Then, 1 −max

 sup
v∈x~y

γA(v), γA(y)

 = 1 − sup
v∈x~y

γA(v). By Corollary 4.5,

1 − sup
v∈x~y

γA(v) = inf
v∈x~y

(1 − γA(v)) = inf
v∈x~y

γ̄A(v) ≥ min
{

inf
v∈x~y

γ̄A(v), γ̄A(y)
}
.

By (2), we have γ̄A(x) ≥ min
{

inf
v∈x~y

γ̄A(v), γ̄A(y)
}

.

Case 2. Suppose max

 sup
v∈x~y

γA(v), γA(y)

 = γA(y).

Then,

1 −max

 sup
v∈x~y

γA(v), γA(y)

 = 1 − γA(y) = γ̄A(y) ≥ min
{

inf
v∈x~y

γ̄A(v), γ̄A(y)
}
.

It follows from (2) that γ̄A(x) ≥ min
{

inf
v∈x~y

γ̄A(v), γ̄A(y)
}

. Therefore, γ̄A is a fuzzy hyper

GR-ideal of type 1 in H.
Conversely, suppose µA and γ̄A are fuzzy hyper GR-ideals of type 1. Let x ∈ H.

Clearly, γ̄A(x) ≤ γ̄A(0). Then, 1 − γA(x) ≤ 1 − γA(0) and so γA(x) ≥ γA(0). Let x, y ∈ H. By
IFGR3 and Lemma 4.4,

1 − γA(x) =γ̄A(x) ≥ min
{

inf
v∈x~y

γ̄A(v), γ̄A(y)
}



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 254

=min
{

inf
v∈x~y

(1 − γA(v)), 1 − γA(y)
}

=min

1 − sup
v∈x~y

γA(v), 1 − γA(y)


=1 −max

 sup
v∈x~y

γA(v), γA(y)

 .
It follows that −γA(x) ≥ −max

 sup
v∈x~y

γA(v), γA(y)

 and so

γA(x) ≤ max

 sup
v∈x~y

γA(v), γA(y)

.

Therefore, A is an intuitionistic fuzzy hyper GR-ideal in H. �

Theorem 4.7. Let A = (µA, γA) be an intuitionistic fuzzy set in a hyper GR-algebra H. Then, A
is an intuitionistic fuzzy hyper GR-ideal in H if and only if  = (µA, µ̄A) and à = (γ̄A, γA) are
intuitionistic fuzzy hyper GR-ideals of H.

Proof. Suppose A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal in H. By
Lemma 4.6, µA and γ̄A are fuzzy hyper GR-ideals of type 1 in H. Let x, y ∈ H. Then,
µ̄A(x) = 1 − µA(x) ≥ 1 − µA(0) = µ̄A(0). By Lemma 4.4,

µ̄A(x) = 1 − µA(x)

≤ 1 −min
{

inf
u∈x~y

µA(u), µA(y)
}

= max
{

1 − inf
u∈x~y

µA(u), 1 − µA(y)
}

= max

 sup
u∈x~y

(1 − µA(u)), µ̄A(y)


= max

 sup
u∈x~y

µ̄A(u), µ̄A(y)

 .
Hence by Definition 4.1,  = (µA, µ̄A) and à = (γ̄A, γA) are intuitionistic fuzzy hyper
GR-ideals in H.

Conversely, let  = (µA, µ̄A) and à = (γ̄A, γA) be intuitionistic fuzzy hyper GR-ideals
in H. Then by Lemma 4.6, µA and γ̄A are fuzzy hyper GR-ideals of type 1 in H. Thus by
Lemma 4.6, A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal in H. �



A. Macodi-Ringia, G. Petalcorin, Jr. / Eur. J. Pure Appl. Math, 13 (2) (2020), 246-257 255

Theorem 4.8. For any subset I of a hyper GR-algebra H, let A(I) = (µA(I), γA(I)) be an intuition-
istic fuzzy set in H defined by the following, respectively:

(µA(I))(x) =
{

k1, if x ∈ I
k2, otherwise

(γA(I))(x) =
{

m1, if x ∈ I
m2, otherwise

for all x ∈ H, where k1, k2,m1,m2 ∈ [0, 1] with k1 > k2, m1 < m2, ki + mi ≤ 1 for i = 1, 2.
Then, I is a hyper GR-ideal of H if and only if A(I) = (µA(I), γA(I)) is an intuitionistic fuzzy hyper
GR-ideal in H.

Proof. Note that the level subsets of µA(I) are

(
µA(I)

)
t1
=


∅, if k1 < t1 ≤ 1
I, if k2 < t1 ≤ k1
H, if 0 ≤ t1 ≤ k2.

(3)

Since

(γ̄A(I))(x) =
{

1 −m1, if x ∈ I
1 −m2, otherwise

and 1 −m1 > 1 −m2,

(
γ̄A(I)

)
t2
=


∅, if 1 −m1 < t2 ≤ 1
I, if 1 −m2 < t2 ≤ 1 −m1
H, if 0 ≤ t2 ≤ 1 −m2.

Suppose I is a hyper GR-ideal of H. Then the nonempty level subsets
(
µA(I)

)
t1

and
(
γ̄A(I)

)
t2

are hyper GR-ideals of H. By Theorem 3.5, µA(I) and γ̄A(I) are fuzzy hyper GR-ideals of
type 1. By Lemma 4.6, A(I) =

(
µA(I), γA(I)

)
is an intuitionistic fuzzy hyper GR-ideal in H.

Conversely, suppose A(I) =
(
µA(I), γA(I)

)
is an intuitionistic fuzzy hyper GR-ideal in H.

By Lemma 4.6, µA(I) and γ̄A(I) are fuzzy hyper GR-ideals in H. It follows from Theorem
3.5 that I =

(
µA(I)

)
t1

is a hyper GR-ideal of H. �

Theorem 4.9. If A = (µA, γA) is an intuitionistic fuzzy hyper GR-ideal of a hyper GR-algebra
H, then the set I = {x ∈ H|µA(x) = µA(0) and γA(x) = γA(0)} is a hyper GR-ideal of H.

Proof. Let A = (µA, γA) be an intuitionistic fuzzy hyper GR-ideal in H. Clearly, 0 ∈ I.
Let x, y ∈ H such that x~y ⊆ I and y ∈ I. Then, µA(y) = µA(0), γA(y) = γA(0), µA(u) = µA(0)
and γA(v) = γA(0) for any u, v ∈ x ~ y. By IFGR1 and IFGR2,

µA(0) ≥ µA(x) ≥ min
{

inf
u∈x~y

µA(u), µA(y)
}
= µA(0) and



REFERENCES 256

γA(0) ≤ γA(x) ≤ max

 sup
v∈x~y

γA(v), γA(y)

 = γA(0).

This implies that µA(x) = µA(0) and γA(x) = γA(0) and so x ∈ I. Hence, I is a hyper
GR-ideal of H. �

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