EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 9-18
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Inf-hesitant fuzzy subalgebras and ideals in
BCK/BCI-algebras

G. Muhiuddin1,∗, Abdulaziz M. Alanazi1, Mohamed E. A. Elnair1, K. P. Shum2

1 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
2 Institute of Mathematics, Yunnan University, Kunming 650091, People’s Republic of
China

Abstract. In the present paper, we introduce the notions of Inf-hesitant fuzzy subalgebras and
Inf-hesitant fuzzy ideals in BCK/BCI-algebras and investigate their relations and properties.
In addition, we discuss the characterizations of Inf-hesitant fuzzy subalgebras and Inf-hesitant
fuzzy ideals in BCK/BCI-algebras.

2020 Mathematics Subject Classifications: 06F35, 03G25, 08A72

Key Words and Phrases: p-semisimple BCI-algebra; Inf-hesitant fuzzy subalgebras; Inf-
hesitant fuzzy ideals.

1. Introduction

The motivation for introducing hesitant fuzzy sets is that it is sometimes difficult to
determine the membership of an element into a set and in some circumstances this dif-
ficulty is caused by a doubt between a few different values. For example, two experts
discuss the membership of x into A, and one wants to assign 0.3 and the other 0.4. So, the
uncertainty on the possible values is somehow limited. Torra [25] proposed the concept of
hesitant fuzzy sets as a new generalization of fuzzy sets [33], which allows the membership
of an element of a set to be represented by several possible values. They also discussed
relationships among hesitant fuzzy sets and other generalizations of fuzzy sets such as
intuitionistic fuzzy sets, type-2 fuzzy sets, and fuzzy multisets. Some set theoretic oper-
ations such as union, intersection and complement on hesitant fuzzy sets have also been
proposed by Torra [25]. Hesitant fuzzy sets can be used as an efficient mathematical tool
for modeling peoples hesitancy in daily life than the other classical extensions of fuzzy sets.
Hesitant fuzzy sets are a very useful to express peoples hesitancy in daily life and a very
useful tool to deal with uncertainty, which can be accurately and perfectly described in

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3575

Email addresses: chishtygm@gmail.com (G. Muhiuddin), am.alenezi@ut.edu.sa (A. M. Alanazi),
abomunzir124@gmail.com (M. E. A. Elnair), kpshum@ynu.edu.cn (K. P. Shum),

http://www.ejpam.com 9 c© 2020 EJPAM All rights reserved.



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 10

terms of the opinions of decision makers. After the pioneering work of Torra, the hesitant
fuzzy has received much attention from many authors in many fields for eg. Xu and Xia
[30] proposed a variety of distance measures for hesitant fuzzy sets, based on which the
corresponding similarity measures can be obtained. They investigated the connections of
the aforementioned distance measures and further develop a number of hesitant ordered
weighted distance measures and hesitant ordered weighted similarity measures. A number
of research papers have been appeared on hesitant fuzzy set theory in decision making
problem etc. (see [23, 27–29, 31]). Fuzzy set theory plays an important role in the devel-
opment of hesitant fuzzy sets theory. Muhiuddin et al. have applied the fuzzy set theory
and related notions to different algebraic structures (see for e.g., [15–18, 18, 19, 19–22]).
In recent years, a number of research papers have been devoted to the study of fuzzy sets
theory and related concepts on different algebraic structures (see e.g., [5–8, 24]). Recently,
hesitant fuzzy sets theory have been applied to different algebraic structures on various
aspects viz., Jun et al. have applied the hesitant fuzzy sets theory to BCK/BCI-algebras
and semigroups (see [2–4]). Also, Muhiuddin et al. have applied the hesitant fuzzy sets
theory to residuated lattices, lattice implication algebras and BCK/BCI-algebras (see [10–
14]).

In this paper, we introduce some new types of hesitant fuzzy subalgebras and ideals
in BCK/BCI-algebras, and investigate their relations and properties. Finally, we discuss
the characterizations of these new types of hesitant fuzzy subalgebras and hesitant fuzzy
ideals in BCK/BCI-algebras.

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. A BCK-algebra X is said to be positive implicative if
it satisfies:

(∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z)) . (1)



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 11

A BCK-algebra X is said to be implicative if it satisfies:

(∀x, y ∈ X) (x = x ∗ (y ∗ x)) . (2)

Any BCK/BCI-algebra X satisfies the following conditions:

(∀x ∈ X) (x ∗ 0 = x) , (3)

(∀x, y, z ∈ X) (x ≤ y ⇒ x ∗ z ≤ y ∗ z, z ∗ y ≤ z ∗ x) , (4)

(∀x, y, z ∈ X) ((x ∗ y) ∗ z = (x ∗ z) ∗ y) , (5)

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ≤ x ∗ y) (6)

where x ≤ y if and only if x ∗ y = 0.
Any BCI-algebra X satisfies the following conditions:

(∀x, y, z ∈ X) (0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) = (0 ∗ y) ∗ (0 ∗ x)) , (7)

(∀x, y ∈ X) (0 ∗ (0 ∗ (x ∗ y)) = (0 ∗ y) ∗ (0 ∗ x)) , (8)

(∀x ∈ X) (0 ∗ (0 ∗ (0 ∗ x)) = 0 ∗ x) . (9)

A BCI-algebra X is said to be p-semisimple (see [1]) if 0 ∗ (0 ∗ x) = x for all x ∈ X.
Every p-semisimple BCI-algebra X satisfies:

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) = x ∗ y) . (10)

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, (11)

(∀x ∈ X) (x ∗ y ∈ A, y ∈ A ⇒ x ∈ A) . (12)

A subset A of a BCI-algebra X is called a p-ideal of X (see [32]) if it satisfies (11) and

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ∈ A, y ∈ A ⇒ x ∈ A) . (13)

Note that every p-ideal is an ideal, but the converse is not true in general (see [32]).
Note that an ideal A of a BCI-algebra X is a p-ideal of X if and only if the following
assertion is valid:

(∀x, y, z ∈ X) ((x ∗ z) ∗ (y ∗ z) ∈ A ⇒ x ∗ y ∈ A) . (14)

We refer the reader to the books [1, 9] for further information regarding BCK/BCI-
algebras.



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 12

3. Inf-hesitant fuzzy subalgebras and ideals

Torra [25] introduced a new extension for fuzzy sets to manage those situations in
which several values are possible for the definition of a membership function of a fuzzy
set.

Definition 1 ([25, 26]). Let X be a reference set. A hesitant fuzzy set on X is defined in
terms of a function that when applied to X returns a subset of [0, 1], which can be viewed
as the following mathematical representation:

H := {(x, h(x)) | x ∈ X}

where h : X → P ([0, 1]).

In what follows, the power set of [0, 1] is denoted by P ([0, 1]) and

P ∗([0, 1]) = P ([0, 1]) \ {∅}.

For any element D ∈ P ∗([0, 1]), the infimum of D is denoted by inf D. For any hesitant
fuzzy set H := {(x, h(x)) | x ∈ X} and D ∈ P ∗([0, 1]), consider the set

Inf[H;D] := {x ∈ X | inf h(x) ≥ inf D} .

Definition 2. Let X be a BCK/BCI-algebra. Given an element D ∈ P ∗([0, 1]), a hesitant
fuzzy set H := {(x, h(x)) | x ∈ X} is called an Inf-hesitant fuzzy subalgebra of X related to
D (briefly, D-Inf-hesitant fuzzy subalgebra of X if the set Inf[H;D] is a subalgebra of X
whenever it is non-empty. If H := {(x, h(x)) | x ∈ X} is a D-Inf-hesitant fuzzy subalgebra
of X for all D ∈ P ∗([0, 1]) with Inf[H;D] 6= ∅, then we say that H := {(x, h(x)) | x ∈ X}
is an Inf-hesitant fuzzy subalgebra of X.

Example 1. (1) Let X = {0, a, b, c} be a BCK-algebra with the following Cayley table:

∗ 0 a b c

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

Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(0, (0.8, 1]), (a, (0.3, 0.5) ∪ {0.9}), (b, [0.5, 0.7]), (c, (0.3, 0.5) ∪ {0.7})} .

Since inf h(0) = 0.8, inf h(a) = 0.3 = inf h(c) and inf h(b) = 0.5, it is routine to verify
that H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy subalgebra of X.



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 13

(2) Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table:

∗ 0 a b c d

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

Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(0, {0.8, 0.9}), (a, [0.2, 0.9)), (b, (0.7, 0.8]), (c, {0.5} ∪ (0.7, 0.9)), (d, [0.1, 0.5])} .

Note that inf h(0) = 0.8, inf h(a) = 0.2, inf h(b) = 0.7, inf h(c) = 0.5 and inf h(d) = 0.1.
It is easy to check that H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy subalgebra of X.

(3) Consider a BCI-algebra X = {0, 1, a, b, c} with the following Cayley table.

∗ 0 1 a b c

0 0 0 c c a
1 1 0 c c a
a a a 0 0 c
b b a 1 0 c
c c c a a 0

Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(0, [0.8, 0.9]), (1, (0.6, 0.7]), (a, [0.5, 0.6]), (b, [0.5, 0.6]), (c, [0.3, 0.7])}.

Then H := {(x, h(x)) | x ∈ X} is a D1-Inf-hesitant fuzzy subalgebra of X with D1 :=
[0.55, 0.65]. But it is not a D2-Inf-hesitant fuzzy subalgebra of X with D2 := [0.4, 0.6]
since Inf[H;D2] = {0, 1, a, b} is not a subalgebra of X.

(4) Consider a BCK-algebra X = {0, a, b, c, d} with the following Cayley table.

∗ 0 a b c d

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

Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(0, [0.7, 0.8]), (a, (0.6, 0.7]), (b, [0.3, 0.6]), (c, [0.5, 0.7]), (d, [0.2, 0.4])}.

Then H := {(x, h(x)) | x ∈ X} is a D1-Inf-hesitant fuzzy subalgebra of X with D1 :=
[0.2, 0.4]. If we take D2 := (0.4, 0.6], then Inf[H;D2] = {0, a, c} which is not a subalgebra
of X. Hence H := {(x, h(x)) | x ∈ X} is not a D2-Inf-hesitant fuzzy subalgebra of X.



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 14

Theorem 1. A hesitant fuzzy set H := {(x, h(x)) | x ∈ X} on a BCK/BCI-algebra X
is an Inf-hesitant fuzzy subalgebra of X if and only if the following assertion is valid:

(∀x, y ∈ X) (inf h(x ∗ y) ≥ min{inf h(x), inf h(y)}) . (15)

Proof. Assume that H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy subalgebra of X.
Assume that there exists Q ∈ P ∗([0, 1]) such that

inf h(x ∗ y) < inf Q ≤ min{inf h(x), inf h(y)}.

Then x, y ∈ Inf[H;D] and x ∗ y /∈ Inf[H;D]. This is a contradiction, and so

inf h(x ∗ y) ≥ min{inf h(x), inf h(y)}

for all x, y ∈ X.
Conversely, suppose that (15) is valid. Let D ∈ P ∗([0, 1]) and x, y ∈ Inf[H;D]. Then

inf h(x) ≥ inf D and inf h(y) ≥ inf D. It follows from (15) that

inf h(x ∗ y) ≥ min{inf h(x), inf h(y)} ≥ inf D

and that x ∗ y ∈ Inf[H;D]. Hence the set Inf[H;D] is a subalgebra of X, and so H :=
{(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy subalgebra of X.

Lemma 1. If H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy subalgebra of a BCK/BCI-
algebra X, then

(∀x ∈ X) (inf h(0) ≥ inf h(x)) . (16)

Proof. Using (III) and (15), we have

inf h(0) = inf h(x ∗ x) ≥ min {inf h(x), inf h(x)} = inf h(x)

for all x ∈ X.

Proposition 1. Let H := {(x, h(x)) | x ∈ X} be an Inf-hesitant fuzzy subalgebra of a
BCK-algebra X. For any elements a1, a2, · · · , an ∈ X, if there exists ak ∈ {a1, a2, · · · , an}
such that a1 = ak, then

(∀x ∈ X) (inf h((· · · ((a1 ∗ a2) ∗ a3) ∗ · · · ) ∗ an) ≥ inf h(x)) .

Proof. Using (5), (III) and (IV), we have (· · · ((a1 ∗ a2) ∗ a3) ∗ · · · ) ∗ an = 0. Thus the
desired result follows from Lemma 1.

Definition 3. Let X be a BCK/BCI-algebra. Given an element D ∈ P ∗([0, 1]), a hesitant
fuzzy set H := {(x, h(x)) | x ∈ X} is called an Inf-hesitant fuzzy ideal of X related to D
(briefly, D-Inf-hesitant fuzzy ideal of X) if the set Inf[H;D] is an ideal of X whenever
it is non-empty. If H := {(x, h(x)) | x ∈ X} is a D-Inf-hesitant fuzzy ideal of X for
all D ∈ P ∗([0, 1]) with Inf[H;D] 6= ∅, then we say that H := {(x, h(x)) | x ∈ X} is an
Inf-hesitant fuzzy ideal of X.



G. Muhiuddin et al. / Eur. J. Pure Appl. Math, 13 (1) (2020), 9-18 15

Example 2. (1) The hesitant fuzzy set H := {(x, h(x)) | x ∈ X} in Example 1(1) is an
Inf-hesitant fuzzy ideal of X.

(2) Let (Y, ∗, 0) be a BCI-algebra and (Z,+, 0) an additive group of integers. Let
(Z,−, 0) be the adjoint BCI-algebra of (Z,+, 0) and let X := Y × Z. Then (X,⊗, (0, 0))
is a BCI-algebra where the operation ⊗ is given by

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

For a subset A := Y × N0 of X where N0 is the set of nonnegative integers, let H :=
{(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(x, (0.5, 1]), (y, [0.4, 0.9]) | x ∈ A, y ∈ X \A} .

Then H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy ideal of X.
(3) Let X = {0, a, b, c, d} be a BCK-algebra with the following Cayley table:

∗ 0 a b c d

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

Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on X defined by

H = {(0, [0.8, 1)), (a, [0.4, 0.7]), (b, {0.3} ∪ (0.4, 0.6]), (c, [0.6, 0.9]), (d, [0.1, 0.5])} .

If D1 := [0.5, 0.8), then Inf[H;D1] = {0, c} which is not an ideal of X since b ∗ c = 0 ∈
Inf[H;D1] but b /∈ Inf[H;D1]. Thus H := {(x, h(x)) | x ∈ X} is not a D1-Inf-hesitant
fuzzy ideal of X. We can easily verify that H := {(x, h(x)) | x ∈ X} is a D2-Inf-hesitant
fuzzy ideal of X with D2 = [0.25, 0.5].

Theorem 2. A hesitant fuzzy set H := {(x, h(x)) | x ∈ X} on a BCK/BCI-algebra X
is an Inf-hesitant fuzzy ideal of X if and only if it satisfies (16) and

(∀x, y ∈ X) (inf h(x) ≥ min{inf h(x ∗ y), inf h(y)}) . (17)

Proof. Let H := {(x, h(x)) | x ∈ X} be an Inf-hesitant fuzzy ideal of X. If (16) is not
valid, then there exists D ∈ P ∗([0, 1]) and a ∈ X such that inf h(0) < inf D ≤ inf h(a). It
follows that a ∈ Inf[H;D] and 0 /∈ Inf[H;D]. This is a contradiction, and so (16) is valid.
Now assume that there exist a, b ∈ X such that inf h(a) < min{inf h(a∗b), inf h(b)}. Then
there exists K ∈ P ∗([0, 1]) such that

inf h(a) < inf K ≤ min{inf h(a ∗ b), inf h(b)},

which implies that a ∗ b ∈ Inf[H;K], b ∈ Inf[H;K] but a /∈ Inf[H;K]. This is a contradic-
tion, and thus (17) holds.



REFERENCES 16

Conversely, suppose that H := {(x, h(x)) | x ∈ X} satisfies two conditions (16) and
(17). Let K ∈ P ∗([0, 1]) be such that Inf[H;K] 6= ∅. Obviously, 0 ∈ Inf[H;K]. Let
x, y ∈ X be such that x ∗ y ∈ Inf[H;K] and y ∈ Inf[H;K]. Then inf h(x ∗ y) ≥ inf K and
inf h(y) ≥ inf K. It follows from (17) that

inf h(x) ≥ min{inf h(x ∗ y), inf h(y)} ≥ inf K

and that x ∈ Inf[H;K]. Hence Inf[H;K] is an ideal of X for all K ∈ P ∗([0, 1]), and
therefore H := {(x, h(x)) | x ∈ X} is an Inf-hesitant fuzzy ideal of X.

Theorem 3. Let H := {(x, h(x)) | x ∈ X} be a hesitant fuzzy set on a BCI-algebra X
defined by

H = {(x,D), (y,E) | x ∈ B, y ∈ X \B, inf D ≥ inf E}

where D,E ∈ P ∗([0, 1]) and B is the BCK-part of X. Then H := {(x, h(x)) | x ∈ X} is
an Inf-hesitant fuzzy ideal of X.

Proof. Since 0 ∈ B, we have inf h(0) = inf D ≥ inf h(x) for all x ∈ X. Let x, y ∈ X. If
x ∈ B, then it is clear that

inf h(x) ≥ min{inf h(x ∗ y), inf h(y)}.

Assume that x ∈ X \B. Since B is an ideal of X, it follows that x∗y ∈ X \B or y ∈ X \B
and that

inf h(x) = min{inf h(x ∗ y), inf h(y)}.

Therefore H := {(x, h(x)) | x ∈ X} is an Int-hesitant fuzzy ideal of X by Theorem 2.

Acknowledgements

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

This research was partially supported by the research grant S-0198-1440, Deanship of
Scientific Research, University of Tabuk, Tabuk-71491, Saudi Arabia.

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