EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 4, 2020, 939-947
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Further results on fuzzy soft BCK/BCI-algebras

Deena S. Al-Kadi1, G. Muhiuddin2,∗

1 Department of Mathematics and Statistics, Taif University, Taif 21974, Saudi Arabia
2 Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

Abstract. In this paper, we obtain further results on fuzzy soft BCK/BCI-algebras. In fact, we
introduce the notion of fuzzy soft sub-BCK/BCI-algebra and investigate related properties.

2020 Mathematics Subject Classifications: 06F35, 03G25, 06D72
Key Words and Phrases: BCK/BCI-algebra, Fuzzy BCK/BCI-algebra, soft BCK/BCI-
algebra, fuzzy soft BCK/BCI-algebra.

1. Introduction

Soft set theory was introduced initially by Molodtsov in 1999 [12] as a mathematical
tool to model uncertainty and vagueness. In [1], Ali et al. studied some operations between
two soft sets. Furthermore, they improved the definition of the complement of a soft set
then studied DeMorgan’s type results in soft set theory. Soft set theory has been applied
in different directions some are shown in [12] and other applications are shown in [5] and
[11]. Based on soft set theory many researches has been done (see for example [3, 7, 9, 14–
18]). Wong [20] used fuzzy set theory introduced by Zadeh [21] to extend general topology
to fuzzy topology. Jun [6] studied fuzzy subalgebras of BCK/BCI-algebras based on the
relations belongs to and quasi-coincidence with.

The authors in [4] introduced and studied fuzzy soft groups and fuzzy soft homomor-
phisms. Maji et al. [10] defined and studied fuzzy soft sets. Roy and Maji [19] considered
the notion of fuzzy soft set and used it to present a theoretic approach to decision making
problems and Jun et al. [8] applied the same notion to BCK/BCI-algebras. Recently, Al-
Masarwah and Ahmad [2] applied the notion of m-polar fuzzy sets to BCK/BCI-algebras.

As shown above, the BCK/BCI-algebra which is introduced by Iséki was exten-
sively investigated by several researchers and this paper gives further results on fuzzy
soft BCK/BCI-algebras. We start by recalling the definition of the algebras we are
studying and the basic definitions of soft sets and fuzzy soft sets and the definition of
some operations related. Then we investigate further results that are not studied in [8].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i4.3844

Email addresses: dak12le@hotmail.co.uk (D. Al-Kadi), chishtygm@gmail.com (G. Muhiuddin)

https://www.ejpam.com 939 c© 2020 EJPAM All rights reserved.



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 940

2. Preliminaries

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

1. (∀β, γ, δ ∈ B) (((β ∗ γ) ∗ (β ∗ δ)) ∗ (δ ∗ γ) = 0),

2. (∀β, γ ∈ B) ((β ∗ (β ∗ γ)) ∗ γ = 0),

3. (∀β ∈ B) β ∗ β = 0),

4. (∀β, γ ∈ B) (β ∗ γ = 0, γ ∗ β = 0 ⇒ β = γ).

If a BCI-algebra B satisfies (0 ∗ β = 0), (∀β ∈ B) then B is called a BCK-algebra. Any
BCK-algebra B satisfies the following properties:

• (∀β ∈ B) (β ∗ 0 = β),

• (∀β, γ, δ ∈ B) (β ≤ γ ⇒ β ∗ δ ≤ γ ∗ δ, δ ∗ γ ≤ δ ∗ β),

• (∀β, γ, δ ∈ B) ((β ∗ γ) ∗ δ = (β ∗ δ) ∗ γ),

• (∀β, γ, δ ∈ B) ((β ∗ δ) ∗ (γ ∗ δ) ≤ β ∗ γ)

where β ≤ γ if and only if β ∗ γ = 0.
Any BCI-algebra B satisfies the properties:

• (∀β, γ, δ ∈ B) (0 ∗ (0 ∗ ((β ∗ δ) ∗ (γ ∗ δ))) = (0 ∗ γ) ∗ (0 ∗ β)),

• (∀β, γ ∈ B) (0 ∗ (0 ∗ (β ∗ γ)) = (0 ∗ γ) ∗ (0 ∗ β)).

For a nonempty subset A of a BCK/BCI-algebra B, if β ∗ γ ∈ A for all β, γ ∈ A then
A is said to be a BCK/BCI-subalgebra of B.

A fuzzy set % in a BCK/BCI-algebra B which satisfies

(∀β, γ ∈ B) (%(β ∗ γ) ≥ min{%(β), %(γ)}) (1)

is said to be a fuzzy BCK/BCI-algebra.
Let % be a fuzzy set in a set B defined by:

%(γ) :=

{
k ∈ (0, 1] if γ = β,
0 if γ 6= β.

Then % is called a fuzzy point with support β and value k and is denoted by βk.
For an initial universe set U , let P(U) denotes the power set of U and for a set of

parameters E, let M ⊂ E. Molodtsov [12] defined the soft set as follows.

Definition 1 ([12]). A soft set over U, is a pair (µ,M) where µ is a mapping given by

µ : M → P(U).



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 941

Clearly, a soft set is not a set. Several examples have been considered by Molodtsov
in [12].

Definition 2 ([10]). Let E be a set of parameters and M ⊆ E. A fuzzy soft set over an
initial universe set U is a pair (µ̃,M) where µ̃ is a mapping from M to the set of all fuzzy
sets in U .

In general, for every m ∈M, µ̃[m] is a fuzzy set in U and it is called fuzzy value set of
parameter m.

Definition 3 ([10]). The “union” of two fuzzy soft sets (µ̃,M) and (η̃, N) over a common

universe U , is the fuzzy soft set
(
ξ̃, Q

)
satisfying the following conditions:

(i) Q = M ∪N,

(ii) for all q ∈ Q,

ξ̃[q] =


µ̃[q] if q ∈M \N,
η̃[q] if q ∈ N \M,
µ̃[q] ∪ η̃[q] if q ∈M ∩N.

We write (µ̃,M) ∪̃ (η̃, N) =
(
ξ̃, Q

)
.

Definition 4 ([10]). For two fuzzy soft sets (µ̃,M) and (η̃, N) over a common universe
U, the (µ̃,M) “AND” (η̃, N) denoted by (µ̃,M) ∧̃ (η̃, N) is defined by

(µ̃,M) ∧̃ (η̃, N) =
(
ξ̃,M ×N

)
,

where ξ̃[m,n] = µ̃[m] ∩ η̃[n] for all (m,n) ∈M ×N.

Definition 5 ([1]). The “extended intersection” of two soft sets (µ̃,M) and (η̃, N) over a
common universe U , is the soft set (ξ̃, Q) satisfying the following conditions:

(i) Q = M ∪N,

(ii) for all q ∈ Q,

ξ̃[q] =


µ̃[q] if q ∈M \N,
η̃[q] if q ∈ N \M,
µ̃[q] ∩ η̃[q] if q ∈M ∩N.

We write (µ̃,M) ∩̃ e (η̃, N) = (ξ̃, Q).

Definition 6 ([1]). The “restricted intersection” of two soft sets (µ̃,M) and (η̃, N) over
a common universe U where M ∩N 6= ∅ is denoted by (µ̃,M) ∩̃ r (η̃, N) and is defined as
(µ̃,M) ∩̃ r (η̃, N) = (ξ̃, Q), where Q = M ∩N and for all q ∈ Q, ξ̃[q] = µ̃[q] ∩ η̃[q].



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 942

3. Fuzzy soft BCK/BCI-algebras

In what follows, B is a BCK/BCI-algebra and E is a set of parameters.

Definition 7 ([8]). For a fuzzy soft set (µ̃,M) over B where M is a subset of E, we say
that (µ̃,M) is a fuzzy soft BCK/BCI-algebra based on a parameter m over B if there
exists m ∈M such that µ̃[m] is a fuzzy BCK/BCI-algebra in B. If (µ̃,M) is a fuzzy soft
BCK/BCI-algebra based on a parameter m over B for all m ∈M, we say that (µ̃,M) is
a fuzzy soft BCK/BCI-algebra over B.

Definition 8. Let (µ̃,M) be a fuzzy soft BCK/BCI-algebra over B. Then

(1) (µ̃,M) is said to be θ-identity, where θ ∈ (0, 1], if it satisfies:

(∀m ∈M)(∀β ∈ B)

(
µ̃[m](β) =

{
θ if β = 0,
0 otherwise

)
.

(2) (µ̃,M) is said to be θ-absolute, where θ ∈ (0, 1], if µ̃[m](β) = θ for all β ∈ B and
m ∈M.

Example 1. Consider a BCI-algebra B = {0, 1, 2, β, γ} with the following Cayley table:

∗ 0 1 2 β γ

0 0 0 0 β β
1 1 0 1 γ β
2 2 2 0 β β
β β β β 0 0
γ γ β γ 1 0

(1) Let M = {m1,m2,m3,m4} be a set of parameters and define a fuzzy soft set (µ̃,M)
as follows:

µ̃ 0 1 2 β γ

m1 0.03 0 0 0 0
m2 0.03 0 0 0 0
m3 0.03 0 0 0 0
m4 0.03 0 0 0 0

Then (µ̃,M) is a 0.03-identity fuzzy soft BCI-algebra over B.
(2) Let N = {n1, n2, n3} be a set of parameters and define a fuzzy soft set (η̃, N) as

follows:

η̃ 0 1 2 β γ

n1 0.4 0.4 0.4 0.4 0.4
n2 0.4 0.4 0.4 0.4 0.4
n3 0.4 0.4 0.4 0.4 0.4

Then (η̃, N) is a 0.4-absolute fuzzy soft BCI-algebra over B.



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 943

Theorem 1. Let π : B → C be a homomorphism of BCK/BCI-algebras. If a fuzzy soft
BCK/BCI-algebra (µ̃,M) over B satisfies:

(∀m ∈M)(∀β ∈ B)

(
µ̃[m](β) =

{
θ if β ∈ Kerπ,
0 otherwise,

)
(2)

then (π(µ̃),M) is a θ-identity fuzzy soft BCK/BCI-algebra over C.

Proof. Let m ∈ M and γ ∈ C. If γ = 0C (the zero element of C), then 0B ∈ Kerπ
where 0B is the zero element of B and so

π(µ̃)[m](γ) = π(µ̃[m])(0C) = sup
β∈π−1(0C)

µ̃[m](β) = sup
β∈Kerπ

µ̃[m](β) = θ.

If γ 6= 0C , then π(µ̃)[m](γ) = 0. Therefore, (π(µ̃),M) is a θ-identity fuzzy soft BCK/BCI-
algebra over C.

Theorem 2. Let π : B → C be a homomorphism of BCK/BCI-algebras. If (µ̃,M) is a
θ-absolute fuzzy soft BCK/BCI-algebra over B, then (π(µ̃),M) is a θ-absolute fuzzy soft
BCK/BCI-algebra over C.

Proof. Direct.

Definition 9. Let (µ̃,M) and (η̃, N) be two fuzzy soft BCK/BCI-algebras over B. We
say that (µ̃,M) is a fuzzy soft sub-BCK/BCI-algebra of (η̃, N) if

(1) M ⊆ N,

(2) µ̃[m] is a fuzzy sub-BCK/BCI-algebra of η̃[m] for all m ∈ M, that is, µ̃[m] is a
fuzzy BCK/BCI-algebra satisfying the condition:

(∀β ∈ B) (µ̃[m](β) ≤ η̃[m](β)) .

Example 2. Consider a BCK-algebra B = {0, 1, 2, 3, 4} with the following Cayley table:

∗ 0 1 2 3 4

0 0 0 0 0 0
1 1 0 1 1 0
2 2 2 0 2 0
3 3 3 3 0 0
4 4 4 4 4 0

Let N = {n1, n2, n3, n4, n5} be a set of parameters and let (η̃, N) be a fuzzy soft set over
B given as follows:

η̃ 0 1 2 3 4

n1 0.9 0.7 0.5 0.4 0.2
n2 0.8 0.7 0.6 0.4 0.3
n3 0.9 0.8 0.6 0.5 0.3
n4 0.7 0.6 0.4 0.3 0.1
n5 0.9 0.6 0.5 0.6 0.5



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 944

Then, (η̃, N) is a fuzzy soft BCK-algebra over B. Now let M = {n2, n5} be a subset of
N. Define a soft set (µ̃,M) over B as follows:

µ̃ 0 1 2 3 4

n2 0.78 0.67 0.56 0.34 0.23
n5 0.89 0.56 0.45 0.56 0.45

Then, (µ̃,M) is a fuzzy soft sub-BCK-algebra of (η̃, N).

The following theorem is obvious.

Theorem 3. Let (µ̃,M) and (η̃,M) be fuzzy soft BCK/BCI-algebras over B. If µ̃[m] ⊆
η̃[m] for all m ∈M, then (µ̃,M) is a fuzzy soft sub-BCK/BCI-algebra of (η̃,M).

Lemma 1 ([8]). If (µ̃,M) and (η̃, N) are fuzzy soft BCK/BCI-algebras over B, then the
extended intersection of (µ̃,M) and (η̃, N) is a fuzzy soft BCK/BCI-algebra over B.

Theorem 4. Let
(
ξ̃, Q

)
be a fuzzy soft BCK/BCI-algebra over B. If (µ̃,M) and (η̃, N)

are fuzzy soft sub-BCK/BCI-algebras of
(
ξ̃, Q

)
, then so is the extended intersection of

(µ̃,M) and (η̃, N).

Proof. It is proved by Definition 9 and Lemma 1.

Lemma 2 ([8]). Let (µ̃,M) and (η̃, N) be fuzzy soft BCK/BCI-algebras over B. If M
and N are disjoint, then the union (µ̃,M) ∪̃ (η̃, N) is a fuzzy soft BCK/BCI-algebra
over B.

Theorem 5. Let
(
ξ̃, Q

)
be a fuzzy soft BCK/BCI-algebra over B. If (µ̃,M) and (η̃, N)

are fuzzy soft sub-BCK/BCI-algebras of
(
ξ̃, Q

)
, then so is the union of (µ̃,M) and (η̃, N)

whenever M and N are disjoint.

Proof. It is proved by Definition 9 and Lemma 2.

Lemma 3 ([8]). If (µ̃,M) and (η̃, N) are two fuzzy soft BCK/BCI-algebras over B, then
(µ̃,M) ∧̃ (η̃, N) is a fuzzy soft BCK/BCI-algebra over B.

Theorem 6. Let
(
ξ̃, Q

)
be a fuzzy soft BCK/BCI-algebra over B. If (µ̃,M) and (η̃, N)

are fuzzy soft sub-BCK/BCI-algebras of
(
ξ̃, Q

)
, then (µ̃,M) ∧̃ (η̃, N) is a fuzzy soft sub-

BCK/BCI-algebra of
(
ξ̃, Q

)
.

Proof. It is proved by Definition 9 and Lemma 3.

Let (µ̃,M) be a fuzzy soft set over B and let k ∈ [0, 1]. For a parameter m in M,
consider the following sets:



Al-Kadi, Muhiuddin / Eur. J. Pure Appl. Math, 13 (4) (2020), 939-947 945

(µ̃,M)≥km := {β ∈ B | µ̃[m](β) ≥ k} ,

and

(µ̃,M)≥k := {β ∈ B | µ̃[m](β) ≥ k for all m ∈M} .

Obviously, (µ̃,M)≥k =
⋂

m∈M
(µ̃,M)≥km .

Theorem 7. For a fuzzy soft set (µ̃,M) over B, the following two statements are equiv-
alent:

(1) (µ̃,M) is a fuzzy soft BCK/BCI-algebra over B based on a parameter m ∈M.

(2) (µ̃,M)≥km is a subalgebra of B for all k ∈ [0, 1] with (µ̃,M)≥km 6= ∅.

Proof. (1)⇒ (2). Assume that (µ̃,M) is a fuzzy soft BCK/BCI-algebra over B based
on a parameter m ∈M. Let k ∈ [0, 1] such that (µ̃,M)≥km 6= ∅. Let β, γ ∈ (µ̃,M)≥km . Then,
µ̃[m](β) ≥ k and µ̃[m](γ) ≥ k. Thus, µ̃[m](β ∗ γ) ≥ min {µ̃[m](β), µ̃[m](γ)} ≥ k which
implies that β ∗ γ ∈ (µ̃,M)≥km . Therefore, (µ̃,M)≥km is a subalgebra of B.

(2) ⇒ (1). Suppose that the second assertion is valid and that (µ̃,M) is not a fuzzy
soft BCK/BCI-algebra over B based on a parameter m ∈M. Then,

µ̃[m](β ∗ γ) < k0 ≤ min {µ̃[m](β), µ̃[m](γ)}

for some β, γ ∈ B and k0 ∈ (0, 1]. It follows that β, γ ∈ (µ̃,M)≥k0m but β ∗ γ /∈ (µ̃,M)≥k0m ,
which is a contradiction. Hence, (µ̃,M) is a fuzzy soft BCK/BCI-algebra over B based
on a parameter m ∈M.

Corollary 1. A fuzzy soft set (µ̃,M) over B is a fuzzy soft BCK/BCI-algebra over B if
and only if (µ̃,M)≥k is a subalgebra of B for all k ∈ [0, 1] with (µ̃,M)≥k 6= ∅.

4. Conclusion

The main goal of the present paper is to obtain further results on fuzzy softBCK/BCI-
algebras. In fact, the notion of fuzzy soft sub-BCK/BCI-algebra is introduced and related
properties are investigated.

In our future study, we intend to apply the notions of the present paper to different
algebras such as BL-algebras, MTL-algebras, R0 -algebras, MV -algebras, EQ-algebras and
lattice implication algebras etc.

Acknowledgements

The authors would like to express their sincere thanks to the learned referee(s) for
valuable comments and several useful suggestions.



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