EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 2, 2018, 417-430
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Subalgebras and ideals in BCK/BCI-algebras based on
Uni-hesitant fuzzy set theory

G. Muhiuddin1,∗, Shuaa Aldhafeeri2

1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
2 Department of Mathematics, College of basic Education, Public Authority for Applied
Education and Training, Kuwait

Abstract. In the present paper, the notions of uni-hesitant fuzzy algebras and uni-hesitant fuzzy
(closed) ideals in BCK-algebras and BCI-algebras are introduced, and several related properties
are investigated. Characterizations of uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed)
ideals are considered, and a new uni-hesitant fuzzy algebra (resp. uni-hesitant fuzzy (closed) ideal)
from old one is established. Relations between uni-hesitant fuzzy algebras and uni-hesitant fuzzy
(closed) ideals are discussed, and conditions for a uni-hesitant fuzzy ideal to be hesitant closed are
provided.

2010 Mathematics Subject Classifications: 06F35, 03G25, 08A72

Key Words and Phrases: Uni-hesitant fuzzy algebra, (Closed) uni-hesitant fuzzy ideal

1. Introduction

The hesitant fuzzy set which is introduced by Torra [14] is a useful generalization of
the fuzzy set that is designed for situations in which it is difficult to determine the mem-
bership of an element to a set owing to ambiguity between a few different values. The
hesitant fuzzy set permits the membership degree of an element to a set to be represented
by a set of possible values between 0 and 1 (see [14] and [15]). The hesitant fuzzy set
therefore provides a more accurate representation of people’s hesitancy in stating their
preferences over objects than the fuzzy set or its classical extensions. Hesitant fuzzy set
theory has been applied to several practical problems, primarily in the area of decision
making (see [13], [15] [16], [17], [18], [19], [20]). Furthermore, Jun et al. applied the notion
of hesitant fuzzy sets to BCK/BCI-algebras, MTL-algebras, EQ-algebras and semigroups
(see [2], [3], [4] and [5]). Recently, Muhiuddin et al. applied the notion of hesitant fuzzy
sets to residuated lattices, BCK/BCI-algebras and lattice implication algebras (see [8], [9],

∗Corresponding author.

Email addresses: chishtygm@gmail.com (G. Muhiuddin), saldhafeeri@yahoo.com (Shuaa Aldhafeeri)

http://www.ejpam.com 417 c© 2018 EJPAM All rights reserved.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 418

[10], [11] and [12]).

In this paper, we introduce the notions of uni-hesitant fuzzy algebras and uni-hesitant
fuzzy (closed) ideals in BCK/BCI-algebras, and investigate several related properties. We
consider characterizations of uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed)
ideals. Given a unihesitant fuzzy algebra (resp. uni-hesitant fuzzy (closed) ideal), we make
a new uni-hesitant fuzzy algebra (resp. uni-hesitant fuzzy (closed) ideal). We investigate
relations between uni-hesitant fuzzy algebras and uni-hesitant fuzzy (closed) ideals. We
provide conditions for a uni-hesitant fuzzy ideal to be hesitant closed.

2. Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki
and was extensively investigated by several researchers.

An algebra (X; ∗, 0) of type (2, 0) is called a BCI-algebra if it satisfies the following
conditions:

(I) (∀x, y, z ∈ X) (((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0),

(II) (∀x, y ∈ X) ((x ∗ (x ∗ y)) ∗ y = 0),

(III) (∀x ∈ X) (x ∗ x = 0),

(IV) (∀x, y ∈ X) (x ∗ y = 0, y ∗ x = 0 ⇒ x = y).

If a BCI-algebra X satisfies the following identity:

(V) (∀x ∈ X) (0 ∗ x = 0),

then X is called a BCK-algebra. Any BCK/BCI-algebra X satisfies the following axioms:

(a1) (∀x ∈ X) (x ∗ 0 = x),

(a2) (∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x),

(a3) (∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y),

(a4) (∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y)

where x ≤ y if and only if x ∗ y = 0. In a BCI-algebra X, the following hold:

(b1) (∀x, y ∈ X) (x ∗ (x ∗ (x ∗ y)) = x ∗ y) ,

(b2) (∀x, y ∈ X) (0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y)) .



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 419

A nonempty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x∗y ∈ S
for all x, y ∈ S. A subset A of a BCK/BCI-algebra X is called an ideal of X if it satisfies:

0 ∈ A, (1)

(∀x ∈ X) (∀y ∈ A) (x ∗ y ∈ A ⇒ x ∈ A) . (2)

An ideal A of a BCI-algebra X is said to be closed if 0 ∗ x ∈ A for all x ∈ A. We refer
the reader to the books [1, 6] for further information regarding BCK/BCI-algebras.

Torra [14] defined hesitant fuzzy sets in terms of a function that returns a set of mem-
bership values for each element in the domain. We display the basic notions on hesitant
fuzzy sets. For more details, we refer to references.

Let X be a reference set. Then we define a hesitant fuzzy set on X in terms of a
function G that when applied to X returns a subset of [0, 1], and the image of x ∈ X
under G is denoted by xG.

For a hesitant fuzzy set G on X and a subset λ of [0, 1], the set

L(G;λ) := {x ∈ X | xG ⊆ λ},

is called the uni-hesitant level set of G.

Let G and H be two hesitant fuzzy sets on X. The hesitant union G tH and hesitant
intersection G u H of G and H are defined to be hesitant fuzzy sets on X as follows:

G t H : X → P ([0, 1]), x 7→ xG ∪ xH (3)

and

G u H : X → P ([0, 1]), x 7→ xG ∩ xH, (4)

respectively.

3. Uni-hesitant fuzzy algebras in BCK/BCI-algebras

In what follows, let X denote a BCK/BCI-algebra unless otherwise specified.

Definition 1. A hesitant fuzzy set G on X is called a uni-hesitant fuzzy algebra on X if
(x ∗ y)G ⊆ xG ∪ yG for all x, y ∈ X.

Example 1. (1) Let X = {0, a, b} be a BCK-algebra with the Cayley table which is
appeared in Table 1.
Let G be a hesitant fuzzy set on X defined as follows:

G : X → P ([0, 1]), x 7→


(0, 3, 0.7) if x = 0,
(0.2, 0.8] if x = a,
(0, 1) if x = b.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 420

Table 1: Cayley table for the ∗-operation

∗ 0 a b

0 0 0 0
a a 0 a
b b b 0

It is routine to verify that G is a uni-hesitant fuzzy algebra on X.

(2) Let X = {0, a, b, c} be a BCK-algebra with the Cayley table which is appeared in
Table 2.

Table 2: Cayley table for the ∗-operation

∗ 0 a b c

0 0 0 0 0
a a 0 0 a
b b b 0 b
c c c c 0

Let H be a hesitant fuzzy set on X defined as follows:

H : X → P ([0, 1]), x 7→


(0, 3, 0.4) ∪ {0.5, 0.6} if x = 0,
(0.2, 0.4] ∪ {0.5, 0.6} if x = a,
(0.2, 0.7] if x = b,
(0.2, 0.5) ∪ [0.5, 0.6] if x = c.

It is routine to verify that H is a uni-hesitant fuzzy algebra on X.

Theorem 1. A hesitant fuzzy set G on X a uni-hesitant fuzzy algebra on X if and only if
the nonempty uni-hesitant level set L(G;λ) of G is a subalgebra of X for all λ ∈ P ([0, 1]).

Proof. Assume that G is a uni-hesitant fuzzy algebra on X. Let λ ∈ P ([0, 1]) and
x, y ∈ L(G;λ). Then xG ⊆ λ and yG ⊆ λ. It follows that (x ∗ y)G ⊆ xG ∪ yG ⊆ λ. Hence
x ∗ y ∈ L(G;λ). Therefore L(G;λ) is a subalgebra of A.

Conversely, suppose that the nonempty uni-hesitant level set of G is a subalgebra of X
for all λ ∈ P ([0, 1]). Let x, y ∈ X be such that xG = λx and yG = λy. Taking λ = λx ∪ λy
implies that x, y ∈ L(G;λ), and so x ∗ y ∈ L(G;λ). Hence

(x ∗ y)G ⊆ λ = λx ∪ λy = xG ∪ yG.

Therefore G is a uni-hesitant fuzzy algebra on X.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 421

Proposition 1. Every uni-hesitant fuzzy algebra G on X satisfies 0G ⊆ xG for all x ∈ X.

Proof. Since x ∗ x = 0 for all x ∈ X, it is straightforward.

Proposition 2. Let X be a BCI-algebra. If G is a uni-hesitant fuzzy algebra on X, then
(x ∗ (0 ∗ y))G ⊆ xG ∪ yG for all x, y ∈ X.

Proof. Using Proposition 1, we have

(x ∗ (0 ∗ y))G ⊆ xG ∪ (0 ∗ y)G ⊆ xG ∪ 0G ∪ yG = xG ∪ yG

for all x, y ∈ X.

Proposition 3. For any uni-hesitant fuzzy algebra G on X, we have

(∀x, y ∈ X) ((x ∗ y)G ⊆ yG ⇔ xG = 0G) .

Proof. Assume that (x ∗ y)G ⊆ yG for all x, y ∈ X. Taking y = 0 induces xG =
(x ∗ 0)G ⊆ 0G. It follows from Proposition 1 that xG = 0G for all x ∈ X.

Conversely, suppose that xG = 0G for all x ∈ X. Then

(x ∗ y)G ⊆ xG ∪ yG = 0G ∪ yG = yG

for all x, y ∈ X.

Theorem 2. Given a uni-hesitant fuzzy algebra G on X, the hesitant fuzzy set G∗ on X
defined by

G∗ : X → P ([0, 1]), x 7→
{
xG if x ∈ L(G;λ),
(0, 1) otherwise

is a uni-hesitant fuzzy algebra on X.

Proof. If G is a uni-hesitant fuzzy algebra on X, then L(G;λ) is a subalgebra of A for
all λ ∈ P ([0, 1]) with L(G;λ) 6= ∅ by Theorem 1. Let x, y ∈ X. If x, y ∈ L(G;λ), then
x ∗ y ∈ L(G;λ) and so

(x ∗ y)G∗ = (x ∗ y)G ⊆ xG ∪ yG = xG∗ ∪ yG∗.

If x /∈ L(G;λ) or y /∈ L(G;λ), then xG∗ = (0, 1) or yG∗ = (0, 1). Thus

(x ∗ y)G∗ ⊆ (0, 1) = xG∗ ∪ yG∗.

Therefore G∗ is a uni-hesitant fuzzy algebra on X.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 422

4. Uni-hesitant fuzzy ideals

Definition 2. A hesitant fuzzy set G on X is called a uni-hesitant fuzzy ideal on X if it
satisfies 0G ⊆ xG for all x ∈ X and

(∀x, y ∈ X) (xG ⊆ (x ∗ y)G ∪ yG) . (5)

Example 2. Let X = {0, 1, 2, a, b} be a BCI-algebra with the Cayley table which is
appeared in Table 3.

Table 3: Cayley table for the ∗-operation

∗ 0 1 2 a b

0 0 0 0 a a
1 1 0 1 b a
2 2 2 0 a a
a a a a 0 0
b b a b 1 0

Let H be a hesitant fuzzy set on X defined as follows:

H : X → P ([0, 1]), x 7→


{0.2, 0.4, 0.6, 0.8} if x = 0,
[0.2, 0.3) ∪ {0.4, 0.6, 0.8} if x = 1,
(0.1, 0.2] ∪ {0.4} ∪ [0.6, 0.8] if x = 2,
(0.1, 0.4] ∪ [0.6, 0.8] if x = a,
[0.1, 0.4] ∪ [0.6, 0.9) if x = b.

It is routine to verify that H is a uni-hesitant fuzzy ideal on X.

Example 3. Consider a BCI-algebra X = {2n | n ∈ Z} with a binary operation “÷”
(usual division). Let G be a hesitant fuzzy set on X given as follows:

G : X → P ([0, 1]), x 7→
{
λ1 if n ≥ 0,
λ2 if n < 0,

where λ1 and λ2 are subsets of [0, 1] with λ1 ( λ2. Then G is a uni-hesitant fuzzy ideal on
X,

Lemma 1. Every uni-hesitant fuzzy ideal G on X satisfies the following condition.

(∀x, y ∈ X) (x ∗ y = 0 ⇒ xG ⊆ yG) . (6)

Proof. It is straightforward by (5) and Proposition 1.

Proposition 4. Every uni-hesitant fuzzy ideal G on X satisfies:



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 423

(1) (∀x, y, z ∈ X) ((x ∗ y)G ⊆ (x ∗ z)G ∪ (z ∗ y)G) .

(2) (∀x, y ∈ X) ((x ∗ y)G = 0G ⇒ xG ⊆ yG) .

Proof. Since ((x ∗ y) ∗ (x ∗ z)) ∗ (z ∗ y) = 0, it follows from Lemma 1 that

((x ∗ y) ∗ (x ∗ z))G ⊆ (z ∗ y)G.

Hence

(x ∗ y)G ⊆ ((x ∗ y) ∗ (x ∗ z))G ∪ (x ∗ z)G ⊆ (x ∗ z)G ∪ (z ∗ y)G

for all x, y, z ∈ X.
(2) If (x ∗ y)G = 0G, then

xG ⊆ (x ∗ y)G ∪ yG = 0G ∪ yG = yG

for all x, y ∈ X.

Proposition 5. For any uni-hesitant fuzzy ideal G on X, the following conditions are
equivalent:

(1) (∀x, y ∈ X) ((x ∗ y)G ⊆ ((x ∗ y) ∗ y)G) .

(2) (∀x, y, z ∈ X) (((x ∗ z) ∗ (y ∗ z))G ⊆ ((x ∗ y) ∗ z)G) .

Proof. Assume that (1) is valid and let x, y, z ∈ X. Since

((x ∗ (y ∗ z)) ∗ z) ∗ z = ((x ∗ z) ∗ (y ∗ z)) ∗ z ≤ (x ∗ y) ∗ z,

that is, (((x ∗ (y ∗ z)) ∗ z) ∗ z) ∗ ((x ∗ y) ∗ z) = 0, it follows from (a3), (1) and (6) that

((x ∗ z) ∗ (y ∗ z))G = ((x ∗ (y ∗ z)) ∗ z)G
⊆ (((x ∗ (y ∗ z)) ∗ z) ∗ z)G ⊆ ((x ∗ y) ∗ z)G.

Conversely, suppose that (2) holds. If we take y = z in (2), then

((x ∗ z) ∗ z)G ⊇ ((x ∗ z) ∗ (z ∗ z))G = ((x ∗ z) ∗ 0)G = (x ∗ z)G

by (III) and (a1). This proves (1).

Theorem 3. Every uni-hesitant fuzzy ideal on a BCK-algebra is a uni-hesitant fuzzy
algebra.

Proof. Let G be a uni-hesitant fuzzy ideal on a BCK-algebraX. Note that (x∗y)∗x = 0
for all x, y ∈ X. Using Lemma 1 and (5), we have

(x ∗ y)G ⊆ xG ⊆ (x ∗ y)G ∪ yG ⊆ xG ∪ yG

for all x, y ∈ X. Hence G is a uni-hesitant fuzzy algebra on X.

If X is a BCI-algebra, then Theorem 3 is not true as seen in the following example.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 424

Example 4. Let (Y, ∗, 0) be a BCI-algebra and let (Z,−, 0) be the adjoint BCI-algebra of
the additive group (Z,+, 0) of integers. Then X := Y ×Z is a BCI-algebra with a binary
operation ⊗ defined as follows:

(∀(x,m), (y, n) ∈ X) ((x,m)⊗ (y, n) = (x ∗ y,m− n)) .

For a subset A = Y × (N ∪ {0}) of X, let G be a hesitant fuzzy set on X given by

G : X → P ([0, 1]), (x,m) 7→
{
λ if x ∈ A,
[0, 1] otherwise

where λ ∈ P ([0, 1]) with λ 6= [0, 1]. Then G is a uni-hesitant fuzzy ideal on X. Note that
(0, 2) ∈ A and (0, 3) ∈ A, but (0, 2)⊗ (0, 3) = (0,−1) /∈ A. Thus

((0, 2)⊗ (0, 3))G = [0, 1] * λ = (0, 2)G ∪ (0, 3)G.

Therefore G is not a uni-hesitant fuzzy algebra on X.

Proposition 6. Every uni-hesitant fuzzy ideal G on X satisfies the following condition:

(∀x, y, z ∈ X) ((x ∗ y) ∗ z = 0 ⇒ xG ⊆ yG ∪ zG) . (7)

Proof. Let x, y, z ∈ X be such that (x ∗ y) ∗ z = 0. Then

(x ∗ y)G ⊆ ((x ∗ y) ∗ z)G ∪ zG = 0G ∪ zG = zG

by (5) and Proposition 1. It follows that

xG ⊆ (x ∗ y)G ∪ yG ⊆ yG ∪ zG

for all x, y, z ∈ X. This completes the proof.

We provide conditions for a hesitant fuzzy set to be a hesitant fuzzy ideal.

Proposition 7. If a hesitant fuzzy set G on X satisfies Proposition 1 and (7), then G is
a uni-hesitant fuzzy ideal on X.

Proof. It is straightforward by (II) and (7).

The following could be easily proved by induction.

Corollary 1. Let G be a hesitant fuzzy set on X satisfying Proposition 1. Then G is a
uni-hesitant fuzzy ideal on X if and only if

(∀x, a1, a2, · · · , an ∈ X)

x ∗ n∏
i=1

ai = 0 ⇒ xG ⊆
⋃

i=1,2,··· ,n
aiG

 ,

where x ∗
n∏

i=1
ai = (· · · (x ∗ a1) ∗ · · · ) ∗ an.



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 425

Proposition 8. Let X be a BCK-algebra such that

(x ∗ a) ∗ b = 0, (8)

a ∗
n∏

i=1

ai = 0, (9)

b ∗
m∏
j=1

bj = 0, (10)

for all x, a, b, a1, a2, · · · , an, b1, b2, · · · , bm ∈ X. If G is a uni-hesitant fuzzy ideal on X,
then

xG ⊆
⋃

i=1,2,··· ,n
j=1,2,··· ,m

(aiG ∪ bjG)

for all x, a1, a2, · · · , an, b1, b2, · · · , bm ∈ X.

Proof. Using (8) and (a3), we have (x ∗ b) ∗ a = 0. It follows from (a2), (a3) and (9)
that (

x ∗
n∏

i=1

ai

)
∗ b ≤ a ∗

n∏
i=1

ai = 0

so that

(
x ∗

n∏
i=1

ai

)
∗ b = 0, i.e., x ∗

n∏
i=1

ai ≤ b. Using (a2), (a3) and (10), we have

(
x ∗

n∏
i=1

ai

)
∗

m∏
j=1

bj ≤ b ∗
m∏
j=1

bj = 0,

and so

(
x ∗

n∏
i=1

ai

)
∗

m∏
j=1

bj = 0. Thus, by Corollary 1, we have

xG ⊆
⋃

i=1,2,··· ,n
j=1,2,··· ,m

(aiG ∪ bjG)

for all x, a1, a2, · · · , an, b1, b2, · · · , bm ∈ X.

Theorem 4. For a hesitant fuzzy set G on X, the following are equivalent.

(i) G is a uni-hesitant fuzzy ideal on X.

(ii) The nonempty uni-hesitant level set L(G;λ) of G is an ideal of X for all λ ∈ P ([0, 1]).

Proof. Assume that G is a uni-hesitant fuzzy ideal on X. Let λ ∈ P ([0, 1]) be such
that L(G;λ) 6= ∅. Then xG ⊆ λ for some x ∈ X. It follows from Proposition 1 that



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 426

0G ⊆ xG ⊆ λ. Hence 0 ∈ L(G;λ). Let x, y ∈ X be such that x ∗ y ∈ L(G;λ) and
y ∈ L(G;λ). Then (x ∗ y)G ⊆ λ and yG ⊆ λ. It follows from (5) that

xG ⊆ (x ∗ y)G ∪ yG ⊆ λ.

Thus x ∈ L(G;λ). Therefore L(G;λ) is an ideal of X.
Conversely suppose that the nonempty uni-hesitant level set of G is an ideal of X for

all λ ∈ P ([0, 1]). Then 0 ∈ L(G;λ). If there exists a ∈ X such that 0G * aG, then 0G * λ
for λ = aG \ 0G. Hence 0 /∈ L(G;λ), a contradiction. Therefore 0G ⊆ xG for all x ∈ X.
Let x, y ∈ X be such that (x ∗ y)G = λ1 and yG = λ2. Let us take λ = λ1 ∪ λ2. Then
x ∗ y ∈ L(G;λ) and y ∈ L(G;λ). Since L(G;λ) is an ideal of X, it follows from (2) that
x ∈ L(G;λ). Hence xG ⊆ λ = λ1 ∪ λ2 = (x ∗ y)G ∪ yG. Consequently, G is a uni-hesitant
fuzzy ideal on X.

5. Uni-hesitant fuzzy closed ideals in BCI-algebras

Definition 3. A uni-hesitant fuzzy ideal G on a BCI-algebra X is said to be hesitant
closed if the following inclusion is valid.

(∀x ∈ X) ((0 ∗ x)G ⊆ xG) . (11)

Obviously, every uni-hesitant fuzzy closed ideal is a uni-hesitant fuzzy algebra.

Example 5. The uni-hesitant fuzzy ideal H in Example 2 is hesitant closed.

Note that every uni-hesitant fuzzy ideal on a BCK-algebra X is hesitant closed. The
following example shows that there exists a uni-hesitant fuzzy ideal on a BCI-algebra X
which is not hesitant closed.

Example 6. In Example 3, the uni-hesitant fuzzy ideal G on X is not hesitant closed
since (

1÷ 22
)
G = 2−2G = λ2 * λ1 = 1G ∪ 22G.

We provide conditions for a uni-hesitant fuzzy ideal to be hesitant closed.

Theorem 5. Let G be a uni-hesitant fuzzy ideal on a BCI-algebra X. Then G is hesitant
closed if and only if G is a uni-hesitant fuzzy algebra on X.

Proof. Assume that G is hesitant closed. Then (0 ∗ x)G ⊆ xG for all x ∈ X. It follows
from (5) that

(x ∗ y)G ⊆ ((x ∗ y) ∗ x)G ∪ xG = (0 ∗ y)G ∪ xG ⊆ xG ∪ yG

for all x, y ∈ X. Hence G is a uni-hesitant fuzzy algebra on X.
Conversely, suppose that G is a uni-hesitant fuzzy algebra on a BCI-algebra X. Then

(0 ∗ x)G ⊆ 0G ∪ xG = xG



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 427

for all x ∈ X. Therefore G is hesitant closed.

By the similar way to Theorem 4, we have a characterization of a uni-hesitant fuzzy
closed ideal.

Theorem 6. For a hesitant fuzzy set G on a BCI-algebra X, the following are equivalent.

(i) G is a uni-hesitant fuzzy closed ideal on X.

(ii) The nonempty uni-hesitant level set L(G;λ) of G is a closed ideal of X for all λ ∈
P ([0, 1]).

Theorem 7. The hesitant union of two uni-hesitant fuzzy closed ideals on a BCI-algebra
X is a uni-hesitant fuzzy closed ideal on X.

Proof. Let G and H be uni-hesitant fuzzy closed ideals on a BCI-algebra X. Then
L(G;λ) and L(H;λ) are closed ideals of X for all λ ∈ P ([0, 1]) whenever they are nonempty.
Thus 0 ∈ L(G;λ) ∩ L(H;λ), and so 0(G t H) = 0G ∪ 0H ⊆ λ. Thus 0 ∈ L(G t H;λ). Let
x, y ∈ X be such that x ∗ y ∈ L(G t H;λ) and y ∈ L(G t H;λ). Then

(x ∗ y)G ∪ (x ∗ y)H = (x ∗ y)(G t H) ⊆ λ and yG ∪ yH = y(G t H) ⊆ λ.

Hence (x ∗ y)G ⊆ λ, (x ∗ y)H ⊆ λ, yG ⊆ λ and yH ⊆ λ. It follows from (5) that
xG ⊆ (x ∗ y)G ∪ yG and xH ⊆ (x ∗ y)H ∪ yH. Hence

x(G t H) = xG ∪ xH
⊆ ((x ∗ y)G ∪ yG) ∪ ((x ∗ y)H ∪ yH)

= ((x ∗ y)G ∪ (x ∗ y)H) ∪ (yG ∪ yH)

= (x ∗ y)(G t H) ∪ y(G t H)

and (0 ∗ y)(G tH) = (0 ∗ y)G ∪ (0 ∗ y)H ⊆ yG ∪ yH = y(G tH). Thus x ∈ L(G tH;λ) and
0 ∗ y ∈ L(G tH;λ). Therefore L(G tH;λ) is a closed ideal of X. It follows from Theorem
6 that G t H is a uni-hesitant fuzzy closed ideal on X.

Theorem 8. If G is a uni-hesitant fuzzy closed ideal on a BCI-algebra X, then the set

A := {x ∈ X | xG = 0G}

is a closed ideal of X.

Proof. Clearly 0 ∈ A. Let x, y ∈ X be such that x ∗ y ∈ A and y ∈ A. Then
(x ∗ y)G = 0G = yG. It follows from (5) and (11) that xG ⊆ (x ∗ y)G ∪ yG = 0G and
(0 ∗ y)G ⊆ yG = 0G. Since 0G ⊆ xG for all x ∈ X, we have xG = 0G and (0 ∗ y)G = 0G,
that is, x ∈ A and 0 ∗ y ∈ A. Therefore A is a closed ideal of X.

Let X be a BCI-algebra and B(X) := {x ∈ X | 0 ≤ x}. For any x ∈ X and n ∈ N, we
define xn by

x1 = x, xn+1 = x ∗ (0 ∗ xn).



G. Muhiuddin, S. Aldhafeeri / Eur. J. Pure Appl. Math, 11 (2) (2018), 417-430 428

If there is an n ∈ N such that xn ∈ B(X), then we say that x is of finite periodic (see [7]),
and we denote its period |x| by

|x| = min{n ∈ N | xn ∈ B(X)}.

Otherwise, x is of infinite period and denoted by |x| =∞.

Theorem 9. If X is a BCI-algebra in which every element is of finite period, then every
uni-hesitant fuzzy ideal on X is hesitant closed.

Proof. Let G be a uni-hesitant fuzzy ideal on X. For any x ∈ X, assume that |x| = n.
Then xn ∈ B(X). Note that(

0 ∗ xn−1
)
∗ x =

(
0 ∗ (0 ∗ (0 ∗ xn−1))

)
∗ x

= (0 ∗ x) ∗
(
0 ∗ (0 ∗ xn−1)

)
= 0 ∗

(
x ∗ (0 ∗ xn−1)

)
= 0 ∗ xn = 0,

and so
((

0 ∗ xn−1
)
∗ x
)
G = 0G ⊆ xG by Proposition 1. It follows from (5) that(

0 ∗ xn−1
)
G ⊆

((
0 ∗ xn−1

)
∗ x
)
G ∪ xG ⊆ xG. (12)

Also, note that (
0 ∗ xn−2

)
∗ x =

(
0 ∗ (0 ∗ (0 ∗ xn−2))

)
∗ x

= (0 ∗ x) ∗
(
0 ∗ (0 ∗ xn−2)

)
= 0 ∗

(
x ∗ (0 ∗ xn−2)

)
= 0 ∗ xn−1,

which implies from (12) that(
(0 ∗ xn−2) ∗ x

)
G =

(
0 ∗ xn−1

)
G ⊆ xG.

Using (5), we have (
0 ∗ xn−2

)
G ⊆

((
0 ∗ xn−2

)
∗ x
)
G ∪ xG ⊆ xG.

Continuing this process, we have (0 ∗ x)G ⊆ xG for all x ∈ X. Therefore G is hesitant
closed.

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for
their valuable comments and several useful suggestions.



REFERENCES 429

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