EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2009-2024
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Ideals of BCK-algebras and BCI-algebras based on a
new form of fuzzy set

Eun Hwan Roh1,∗, Eunsuk Yang2, Young Bae Jun3

1 Department of Mathematics Education, Chinju National University of Education, Jinju
52673, Korea
2 Department of Philosophy, Jeonbuk National University, Jeonju 54896, Korea
3 Department of Mathematics Education, Gyeongsang National University, Jinju 52828,
Korea

Abstract. Ideals in BCK/BCI algebra based on Y ε
J -fuzzy sets are studied. The fundamental

properties of the level set of Y ε
J -fuzzy sets are investigate first. The concept of (closed) Y ε

J -fuzzy
ideals in BCK/BCI-algebras is introduces, and several properties are investigated. The relationship
between Y ε

J -fuzzy ideal and Y ε
J -fuzzy subalgebra are discussed, and also the relationship between

Y ε
J -fuzzy ideal and fuzzy ideal is identified. The characterization of (closed) Y ε

J -fuzzy ideal using
the Y-level set is established. The necessary and sufficient conditions for Y ε

J -fuzzy ideal to be
closed is explored, and conditions for Y ε

J -fuzzy subalgebra to be Y ε
J -fuzzy ideal are provided.

2020 Mathematics Subject Classifications: 03G25, 06F35, 08A72
Key Words and Phrases: subalgebra, ideal, J-operator, nonconstant factor, Y ε

J -fuzzy subalge-
bra, (closed) Y ε

J -fuzzy ideal

1. Introduction

Fuzzy sets, which are introduced by Zadeh [14], are mathematical frameworks that
are very useful in expressing and manipulating uncertainty and ambiguity of data with
applications such as pattern recognition, decision making, control systems, image pro-
cessing, data mining, expert systems, natural language processing, risk assessment and
decision analysis, etc. Various studies have been conducted since the study of fuzzy sets
in BCK-algebra began in 1991 (see [1, 5, 7–10]). Jun [6] introduce the notion of the
J-operator in the closed interval [0, 1] and investigate several properties. He used the J-
operator to create a new fuzzy set called the Y ε

J -fuzzy set and applied it to subalgebras
in BCK/BCI-algebras. He introduced the concept of the Y ε

J -fuzzy subalgebra and inves-
tigated its properties. He provided conditions for a fuzzy set to be a Y ε

J -fuzzy subalgebra,
and discussed the relationship between the fuzzy subalgebra and the Y ε

J -fuzzy subalgebra.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4933

Email addresses: ehroh9988@gmail.com (E. H. Roh),
eunsyang@jbnu.ac.kr (E. Yang), skywine@gmail.com (Y. B. Jun)

https://www.ejpam.com 2009 © 2023 EJPAM All rights reserved.



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2010

In this paper, we study the ideals of BCK/BCI-algebras based on Y ε
J -fuzzy sets. We

first investigate the underlying properties of the level sets of Y ε
J -fuzzy sets. We introduce

the concept of Y ε
J -fuzzy ideals in BCK/BCI-algebras, and investigate several properties.

We discuss the relationship between Y ε
J -fuzzy ideal and Y ε

J -fuzzy subalgebra, and also
identify the relationship between Y ε

J -fuzzy ideal and fuzzy ideal. We consider the char-
acterization of Y ε

J -fuzzy ideal using the Y-level set. We define closed Y ε
J -fuzzy ideal, and

deal with its properties. We explore the necessary and sufficient conditions for Y ε
J -fuzzy

ideal to be closed. Finally, we provide conditions for Y ε
J -fuzzy subalgebra to be Y ε

J -fuzzy
ideal.

2. Preliminaries

A BCK/BCI-algebra is an important class of logical algebras introduced by K. Iséki
(see [3] and [4]) and was extensively investigated by several researchers.

We recall the definitions and basic results required in this paper. See the books [2, 11]
for further information regarding BCK/BCI-algebras.

By a BCI-algebra, we mean a structure (X, ∗, 0), where 0 is a special element and ∗ is
a binary operation on X, that satisfies the following conditions:

(I) ((a ∗ b) ∗ (a ∗ c)) ∗ (c ∗ b) = 0,

(II) (a ∗ (a ∗ b)) ∗ b = 0,

(III) a ∗ a = 0,

(IV) a ∗ b = 0, b ∗ a = 0 ⇒ a = b,

for all a, b, c ∈ X. If a BCI-algebra (X, ∗, 0) satisfies the following identity:

(V) (∀a ∈ X) (0 ∗ a = 0),

then (X, ∗, 0) is called a BCK-algebra.
The order relation “≤X” in a BCK/BCI-algebra (X, ∗, 0) is defined as follows:

(∀a, b ∈ X)(a ≤X b ⇔ a ∗ b = 0). (1)

Every BCK/BCI-algebra (X, ∗, 0) satisfies the following conditions:

a ∗ 0 = a, (2)

a ≤X b ⇒ a ∗ c ≤X b ∗ c, c ∗ b ≤X c ∗ a, (3)

(a ∗ b) ∗ c = (a ∗ c) ∗ b, (4)

for all a, b, c ∈ X.
Every BCK-algebra (X, ∗, 0) satisfies:

(∀x, a ∈ X)(x ∗ a ≤X x). (5)



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2011

Every BCI-algebra (X, ∗, 0) satisfies:

(∀a, b ∈ X)(0 ∗ (a ∗ b) = (0 ∗ a) ∗ (0 ∗ b)). (6)

A BCI-algebra (X, ∗, 0) is said to be p-semisimple if 0 ∗ (0 ∗ x) = x for all x ∈ X (see
[2]).

A nonempty subset S of X is called a subalgebra of a BCK/BCI-algebra (X, ∗, 0) (see
[11]) if a ∗ y ∈ S for all a, y ∈ S. A subset A of X is called an ideal of a BCK/BCI-algebra
(X, ∗, 0) (see [11]) if it satisfies:

0 ∈ A, (7)

(∀a ∈ X) (∀y ∈ A) (a ∗ y ∈ A ⇒ a ∈ A) . (8)

An ideal A of a BCI-algebra (X, ∗, 0) is said to be closed (see [2, 11]) if it is also a
subalgebra of (X, ∗, 0).

Note that an ideal A of a BCI-algebra (X, ∗, 0) is closed if and only if 0 ∗ a ∈ A for all
a ∈ A (see [2, Proposition 1.4.4]).

Every ideal A of a BCK/BCI-algebra (X, ∗, 0) satisfies the next assertion.

(∀a, y ∈ X) (a ≤X y, y ∈ A ⇒ a ∈ A) . (9)

A fuzzy set in a set X is defined to be a function ζ : X → [0, 1]. Denote by FS(X) the
collection of all fuzzy sets in X. Define a relation “ ⊆ ” on FS(X) by

(∀ζ, ξ ∈ FS(X))(ζ ⊆ ξ ⇔ (∀a ∈ X)(ζ(a) ≤ ξ(a))).

The join (∨) and meet (∧) of ζ and ξ are defined by

(ζ ∨ ξ)(a) = max{ζ(a), ξ(a)},
(ζ ∧ ξ)(a) = min{ζ(a), ξ(a)},

respectively, for all a ∈ X. The complement of ζ, denoted by ζc, is defined by

(∀a ∈ X)(ζc(a) = 1 − ζ(a)).

A fuzzy set ζ in a set X of the form

ζ(b) :=

{
t ∈ (0, 1] if b = a,
0 if b ̸= a,

is said to be a fuzzy point with support a and value t and is denoted by ⟨at⟩.
For a fuzzy set ζ in a set X, we say that a fuzzy point ⟨at⟩ is

(i) contained in ζ, denoted by ⟨at⟩ ∈ ζ, (see [12]) if ζ(a) ≥ t.

(ii) quasi-coincident with ζ, denoted by ⟨at⟩ q ζ, (see [12]) if ζ(a) + t > 1.



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2012

If a fuzzy point ⟨at⟩ is contained in ζ or is quasi-coincident with ζ, we denote it
⟨at⟩ ∈∨q ζ. If ⟨at⟩α ζ is not established for α ∈ {∈, q,∈∨q }, it is denoted by ⟨at⟩α ζ.

Given t ∈ (0, 1] and a fuzzy set ζ in a set X, consider the following sets

(ζ, t)∈ := {a ∈ X | ⟨at⟩ ∈ ζ} and (ζ, t)q := {a ∈ X | ⟨at⟩ q ζ}

which are called the level set and the q-set of ζ related to t, respectively, in X.
Also, we consider the set

(ζ, t)∈∨q := {a ∈ X | ⟨at⟩ ∈∨q ζ}

which is called the ∈∨q -set of ζ related to t.
It is clear that (ζ, t)∈∨q = (ζ, t)∈ ∪ (ζ, t)q and (ζ, t)q ⊆ (ζ, s)q for all t, s ∈ (0, 1] with

t ≤ s.
A fuzzy set ζ in X is called a fuzzy subalgebra of a BCK/BCI-algebra (X, ∗, 0) (see

[13]) if it satisfies:

(∀x, a ∈ X)(ζ(x ∗ a) ≥ ζ(x) ∧ ζ(a)). (10)

A fuzzy set ζ in X is called a fuzzy ideal of a BCK/BCI-algebra (X, ∗, 0) (see [13]) if
it satisfies:

(∀x ∈ X)(ζ(0) ≥ ζ(x)), (11)

(∀x, a ∈ X)(ζ(x) ≥ ζ(x ∗ a) ∧ ζ(a)). (12)

In [6], Jun introduced the notion of Y ε
J -fuzzy sets based on the J-operator in the closed

interval [0, 1]. We display the basic notions about the Y ε
J -fuzzy sets.

We use the notation I instead of the closed interval [0, 1]. Let “≪” be the order relation
in I2 defined as follows:

(∀(m,n), (j, i) ∈ I2)((m,n) ≪ (j, i) ⇔ m ≤ j, n ≤ i)

For every m, ε ∈ I, we define m ∧ ε := min{m, ε} and m ∨ ε := max{m, ε}.
Consider a binary operation YJ in I given as follows:

YJ : I2 → I, (m, ε) 7→ (1 −m) ∧ (1 − ε).

We will call this binary operation YJ the J-operator in I (see [6]).
Let X be a set. Given a fuzzy set ζ in X and ε ∈ I, let ε(ζ) be a mapping defined by

ε(ζ) : X → I, x 7→ YJ(ε, ζ(x)).

It is clear that ε(ζ) is a fuzzy set in X determined by the J-operator and ε. So we can
say that ε(ζ) is a Y ε

J -fuzzy set of ζ in X (see [6]).
Given a fuzzy set ζ in X and ε ∈ (0, 1), if the Y ε

J -fuzzy set ε(ζ) of ζ is not constant on
X, then ε is said to be a nonconstant factor in (0, 1) (see [6]).

A fuzzy set ζ in X is called a Y ε
J -fuzzy subalgebra of (X, ∗, 0) (see [6]) if it satisfies:

(∀x, a ∈ X)(YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a))). (13)



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2013

3. Level sets of the Y ε
J -fuzzy set

Let ζ be a fuzzy set in X, ε ∈ I and t ∈ I \ {0, 1}. Given a Y ε
J -fuzzy set ε(ζ), we

consider the sets:

ε(ζ)t := {x ∈ X | YJ(ε, ζ(x)) ≤ t},
ε(ζ)tq := {x ∈ X | YJ(ε, ζ(x)) < 1 − t},

which is called the Y-level set and Yq-set of ε(ζ), respectively, related to t. We call t the
level degree of ε(ζ).

The Y-level set and the Yq-set of ε(ζ) related to t are calculated as follows:

ε(ζ)t = {x ∈ X | YJ(ε, ζ(x)) ≤ t}
= {x ∈ X | (1 − ε) ∧ (1 − ζ(x)) ≤ t}
= {x ∈ X | 1 − (ε ∨ ζ(x)) ≤ t}
= {x ∈ X | ε ∨ ζ(x) ≥ 1 − t}

and

ε(ζ)tq = {x ∈ X | YJ(ε, ζ(x)) < 1 − t}
= {x ∈ X | (1 − ε) ∧ (1 − ζ(x)) < 1 − t}
= {x ∈ X | ε ∨ ζ(x) > t},

respectively. The set

ε(ζ)t∈∨q := {x ∈ X | YJ(ε, ζ(x)) ≤ t or YJ(ε, ζ(x)) < 1 − t}

is called the the Y∈∨q -set of ε(ζ) related to t. It is clear that

ε(ζ)t∈∨q = ε(ζ)t ∪ ε(ζ)tq.

Proposition 1. Let ζ be a fuzzy set in X and ε ∈ I that satisfies ε ≤ ζ(x) for all x ∈ X.
Then ε(ζ)t = (ζ, t)q ∪ ζ1t where ζ1t := {x ∈ X | ζ(x) + t = 1}, and ε(ζ)tq ⊆ (ζ, t)∈.

Proof. Straightforwad.

Proposition 2. Let ζ be a fuzzy set in X and ε ∈ I. If s ≥ t in I\{0, 1}, then ε(ζ)t ⊆ ε(ζ)s

and ε(ζ)tq ⊇ ε(ζ)sq.

Proof. Straightforward.



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2014

4. Y ε
J -fuzzy ideals

We begin this section by looking at a characterization of Y ε
J -fuzzy subalgebra by Y-

level set. In what follows, let (X, ∗, 0) be a BCK-algebra or a BCI-algebra, and ε ∈ (0, 1)
unless otherwise specified.

Theorem 1. A fuzzy set ζ in X is a Y ε
J -fuzzy subalgebra of (X, ∗, 0) if and only if the

nonempty Y-level set ε(ζ)t of ε(ζ) is a subalgebra of (X, ∗, 0) for all t ∈ I \ {0, 1}.

Proof. Assume that ζ is a Y ε
J -fuzzy subalgebra of (X, ∗, 0) and let t ∈ I \ {0, 1} be

such that ε(ζ)t ̸= ∅. Let x, y ∈ ε(ζ)t. Then YJ(ε, ζ(x)) ≤ t and YJ(ε, ζ(y)) ≤ t, which
imply from (13) that

YJ(ε, ζ(x ∗ y)) ≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(y)) ≤ t.

Hence x ∗ y ∈ ε(ζ)t, and therefore ε(ζ)t is a subalgebra of (X, ∗, 0).
Conversely, suppose that the nonempty Y-level set ε(ζ)t is a subalgebra of (X, ∗, 0)

for all t ∈ I \ {0, 1}. If (13) is not valid, then

YJ(ε, ζ(b ∗ c)) > t ≥ YJ(ε, ζ(b)) ∨ YJ(ε, ζ(c))

for some b, c ∈ X and t ∈ I \ {0, 1}. Hence b, c ∈ ε(ζ)t and b ∗ c /∈ ε(ζ)t, which is a
contradiction. Therefore YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a)) for all x, a ∈ X, which
shows that ζ is a Y ε

J -fuzzy subalgebra of (X, ∗, 0).

Definition 1. A fuzzy set ζ in X is called a Y ε
J -fuzzy ideal if it satisfies:

(∀x ∈ X)(YJ(ε, ζ(0)) ≤ YJ(ε, ζ(x))), (14)

(∀x, a ∈ X)(YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))). (15)

Example 1. Let X = {0, 1, 2, a, b} be a set with the binary operation “∗” given by Table
1.

Table 1: Cayley table for the binary operation “∗”

∗ 0 1 2 a b

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



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2015

Then (X, ∗, 0) is a BCI-algebra (see [2]). Define a fuzzy set ζ in X as follows:

ζ : X → [0, 1], y 7→


0.68 if y = 0,
0.61 if y = 1,
0.46 if y = 2,
0.54 if y = a,
0.46 if y = b.

It is routine to verify that ζ is a Y ε
J -fuzzy ideal of (X, ∗, 0) for all ε ∈ (0, 1).

Proposition 3. Every Y ε
J -fuzzy ideal ζ of (X, ∗, 0) satisfies:

(∀x, a ∈ X)(x ≤X a ⇒ YJ(ε, ζ(x)) ≤ YJ(ε, ζ(a))). (16)

(∀x, a, y ∈ X)(x ∗ a ≤X y ⇒ YJ(ε, ζ(x)) ≤ YJ(ε, ζ(a)) ∨ YJ(ε, ζ(y))). (17)

Proof. Let ζ be a Y ε
J -fuzzy ideal of (X, ∗, 0) and let x, a ∈ X be such that x ≤X a.

Then x ∗ a = 0, and so

YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

= YJ(ε, ζ(0)) ∨ YJ(ε, ζ(a))

= YJ(ε, ζ(a))

by (14) and (15). Thus (16) is valid. Let x, a, y ∈ X be such that x ∗ a ≤X y. Then

YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ((x ∗ a) ∗ y)) ∨ YJ(ε, ζ(y))

= YJ(ε, ζ(0)) ∨ YJ(ε, ζ(y))

= YJ(ε, ζ(y)),

and thus YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a)) ≤ YJ(ε, ζ(y)) ∨ YJ(ε, ζ(a)). This
completes the proof.

Corollary 1. If ζ is a fuzzy ideal of (X, ∗, 0), then its Y ε
J -fuzzy set ε(ζ) satisfies:

(∀x, a ∈ X)(x ≤X a ⇒ ε(ζ)(x) ≤ ε(ζ)(a)).

(∀x, a, y ∈ X)(x ∗ a ≤X y ⇒ ε(ζ)(x) ≤ ε(ζ)(a) ∨ ε(ζ)(y)).

Theorem 2. In a BCK-algebra (X, ∗, 0), every Y ε
J -fuzzy ideal is a Y ε

J -fuzzy subalgebra
for all ε ∈ (0, 1).

Proof. Let ζ be a Y ε
J -fuzzy ideal of a BCK-algebra (X, ∗, 0) for all ε ∈ (0, 1). The

combination of (5) and (16) induces YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ(x)), and so

YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a)).

Therefore ζ is a Y ε
J -fuzzy subalgebra of (X, ∗, 0).

In a BCI-algebra, Theorem 2 may not be true as seen in the following example.



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2016

Example 2. Let (X, ∗, 0) be a BCI-algebra and (Z,−, 0) the adjoint BCI-algebra of the
additive group (Z,+, 0) of integers. Then (Y,⊛, (0, 0)) is a BCI-algebra (see [2]) where
Y = X × Z and ⊛ is a binary operation in Y given as follows:

(∀(x, a), (y, b) ∈ Y )((x, a) ⊛ (y, b) = (x ∗ y, a− b)).

Define a fuzzy set ζ in Y as follows:

ζ : Y → [0, 1], c 7→


0.87 if c = (0, 0),
0.73 if c ∈ X × N0,
0.42 otherwise

wher N0 is the set of all nonnegative integes. It is routine to verify that ζ is a Y ε
J -fuzzy

ideal of (Y,⊛, (0, 0)) for ε = 0.61. We can observe that

YJ(ε, ζ((0, 3) ⊛ (0, 7))) = YJ(0.61, ζ(0,−4))

= (1 − 0.61) ∧ (1 − 0.42) = 0.39

and

YJ(ε, ζ(0, 3)) ∨ YJ(ε, ζ(0, 7))

= ((1 − 0.61) ∧ (1 − 0.73)) ∨ ((1 − 0.61) ∧ (1 − 0.73)) = 0.27.

Hence YJ(ε, ζ((0, 3) ⊛ (0, 7))) ≰ YJ(ε, ζ(0, 3)) ∨ YJ(ε, ζ(0, 7)) for ε = 0.61, which shows
that ζ is not a Y ε

J -fuzzy subalgebra of (Y,⊛, (0, 0)).

The following example shows that there exists ε ∈ (0, 1) such that a Y ε
J -fuzzy subal-

gebra may not be a Y ε
J -fuzzy ideal.

Example 3. (i) Let X = {0, b1, b2, b3} be a set with a binary operation “∗” given by Table
2.

Table 2: Cayley table for the binary operation “ ∗ ”

∗ 0 b1 b2 b3
0 0 0 0 0
b1 b1 0 0 b1
b2 b2 b1 0 b2
b3 b3 b3 b3 0

Then X is a BCK-algebra (see [11]). A fuzzy set ζ in X defined by

ζ : X → [0, 1], x 7→


0.63 if x = 0,
0.54 if x = b1,
0.42 if x = b2,
0.49 if x = b3



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2017

is a Y ε
J -fuzzy subalgebra of (X, ∗, 0) for ε = 0.52. But it is not a Y ε

J -fuzzy ideal of (X, ∗,
0) for ε = 0.52 since

YJ(ε, ζ(b2)) = YJ(0.52, 0.42) = (1 − 0.52) ∧ (1 − 0.42)

= 0.48 ≰ 0.46 = (1 − 0.52) ∧ (1 − 0.54)

= ((1 − 0.52) ∧ (1 − 0.54)) ∨ ((1 − 0.52) ∧ (1 − 0.54))

= YJ(0.52, ζ(b1)) ∨ YJ(0.52, ζ(b1))

= YJ(ε, ζ(b2 ∗ b1)) ∨ YJ(ε, ζ(b1)).

(ii) Consider the BCI-algebra (X, ∗, 0) in Example 1 and let ζ be a fuzzy set in X
given as follows:

ζ : X → [0, 1], y 7→


0.78 if y = 0,
0.54 if y = 1,
0.37 if y = 2,
0.65 if y = a,
0.37 if y = b.

Then ζ is a Y ε
J -fuzzy subalgebra of (X, ∗, 0) for ε = 0.49. We can observe that

YJ(ε, ζ(1)) = YJ(0.49, 0.54) = (1 − 0.49) ∧ (1 − 0.54) = 0.46

and

YJ(ε, ζ(1 ∗ a)) ∨ YJ(ε, ζ(a)) = YJ(ε, ζ(a)) ∨ YJ(ε, ζ(a))

= YJ(ε, ζ(a)) = YJ(0.49, 0.65) = (1 − 0.49) ∧ (1 − 0.65) = 0.35.

Hence YJ(ε, ζ(1)) ≰ YJ(ε, ζ(1 ∗ a))∨ YJ(ε, ζ(a)), and therefore ζ is not a Y ε
J -fuzzy ideal of

(X, ∗, 0) for ε = 0.49.

Theorem 3. Let ζ be a fuzzy set in X. If ζ(x) ≤ ε for all x ∈ X, then ζ is a Y ε
J -fuzzy

ideal of (X, ∗, 0).

Proof. Let ζ be a fuzzy set in X that satisfies ζ(x) ≤ ε for all x ∈ X. Then 1 − ε ≤
1 − ζ(x) for all x ∈ X. Hence

YJ(ε, ζ(0)) = (1 − ε) ∧ (1 − ζ(0)) = 1 − ε = (1 − ε) ∧ (1 − ζ(x)) = YJ(ε, ζ(x))

for all x ∈ X. Also, we have

YJ(ε, ζ(x)) = 1 − ε = YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

for all x, a ∈ X. Therefore ζ is a Y ε
J -fuzzy ideal of (X, ∗, 0).

Let ζ be a fuzzy set in X. If there exists z ∈ X that satisfies ζ(z) > ε, then ζ may not
be a Y ε

J -fuzzy ideal of (X, ∗, 0) as shown in the example below.



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2018

Table 3: Cayley table for the binary operation “ ∗ ”

∗ 0 b1 b2 b3 b4
0 0 0 0 0 0
b1 b1 0 0 0 b1
b2 b2 b1 0 0 b2
b3 b3 b1 b1 0 b3
b4 b4 b4 b4 b4 0

Example 4. Let X = {0, b1, b2, b3, b4} be a set with a binary operation “∗” given by Table
3.
Then (X, ∗, 0) is a BCK-algebra and so a BCI-algebra (see [11]). Consider a fuzzy set ζ
in X given as follows:

ζ : X → [0, 1], y 7→


0.93 if y = 0,
0.46 if y = b1,
0.77 if y = b2,
0.58 if y = b3,
0.35 if y = b4.

If ε := 0.53, then YJ(ε, ζ(0)) ≤ YJ(ε, ζ(x)) for all x ∈ X. But

YJ(ε, ζ(b1 ∗ b3)) ∨ YJ(ε, ζ(b3)) = YJ(0.53, 0.93) ∨ YJ(0.53, 0.58)

= 0.07 ∨ 0.42 = 0.42 < 0.47 = YJ(ε, ζ(b1)).

Hence ζ is not a Y ε
J -fuzzy ideal of (X, ∗, 0) for ε = 0.53.

Theorem 4. Every fuzzy ideal of (X, ∗, 0) is a Y ε
J -fuzzy ideal of (X, ∗, 0) for all ε ∈ (0, 1).

Proof. Let ζ be a fuzzy ideal of (X, ∗, 0) and let ε ∈ (0, 1). Then ζc(0) ≤ ζc(x) and
ζc(x) ≤ ζc(x ∗ a) ∨ ζc(a) for all x, a ∈ X. Hence

YJ(ε, ζ(0)) = (1 − ε) ∧ ζc(0) ≤ (1 − ε) ∧ ζc(x) = YJ(ε, ζ(x))

and

YJ(ε, ζ(x)) = (1 − ε) ∧ ζc(x)

≤ (1 − ε) ∧ (ζc(x ∗ a) ∨ ζc(a))

= ((1 − ε) ∧ ζc(x ∗ a)) ∨ ((1 − ε) ∧ ζc(a))

= YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

for all x, a ∈ X. Therefore ζ is a Y ε
J -fuzzy ideal of (X, ∗, 0) for all ε ∈ (0, 1).



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2019

Theorem 5. If ζ is a Y ε
J -fuzzy ideal of (X, ∗, 0) for some nonconstant factor ε ∈ (0, 1),

then it is a fuzzy ideal of (X, ∗, 0).

Proof. Assume that ζ is a Y ε
J -fuzzy ideal of (X, ∗, 0) for some nonconstant factor

ε ∈ (0, 1). Then

(1 − ε) ∧ (1 − ζ(0)) = YJ(ε, ζ(0)) ≤ YJ(ε, ζ(x)) = (1 − ε) ∧ (1 − ζ(x))

for all x ∈ X. Hence 1 − ζ(0) ≤ 1 − ζ(x), and so ζ(0) ≥ ζ(x) for all x ∈ X. For every
x, a ∈ X, we have

(1 − ε) ∧ (1 − ζ(x)) = YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

= ((1 − ε) ∧ (1 − ζ(x ∗ a))) ∨ ((1 − ε) ∧ (1 − ζ(a)))

= (1 − ε) ∧ ((1 − ζ(x ∗ a)) ∨ (1 − ζ(a))).

It follows that 1 − ζ(x) ≤ ((1 − ζ(x ∗ a)) ∨ (1 − ζ(a))) = 1 − (ζ(x ∗ a) ∧ ζ(a)). Thus
ζ(x) ≥ ζ(x ∗ a) ∧ ζ(a). Therefore ζ is a fuzzy ideal of (X, ∗, 0).

Theorem 6. A fuzzy set ζ in X is a Y ε
J -fuzzy ideal of (X, ∗, 0) if and only if the nonempty

Y-level set ε(ζ)t of ε(ζ) is an ideal of (X, ∗, 0) for all t ∈ I \ {0, 1}
Proof. Assume that ζ is a Y ε

J -fuzzy ideal of (X, ∗, 0) and let t ∈ I \ {0, 1} be such
that ε(ζ)t ̸= ∅. If 0 /∈ ε(ζ)t, then YJ(ε, ζ(0)) > t ≥ YJ(ε, ζ(b)) for some b ∈ X, which
contradicts (14). Hence 0 /∈ ε(ζ)t. Let x, y ∈ X be such that x ∗ y ∈ ε(ζ)t and y ∈ ε(ζ)t.
Then YJ(ε, ζ(x ∗ y)) ≤ t and YJ(ε, ζ(y)) ≤ t. It follows from (15) that

YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ y)) ∨ YJ(ε, ζ(y)) ≤ t.

Hence x ∈ ε(ζ)t, which shows that ε(ζ)t is an ideal of (X, ∗, 0).
Conversely, suppose that the nonempty Y-level set ε(ζ)t of ε(ζ) is an ideal of (X, ∗,

0) for all t ∈ I \ {0, 1}. If there exists c ∈ X such that YJ(ε, ζ(0)) > YJ(ε, ζ(c)), then
YJ(ε, ζ(0)) > t ≥ YJ(ε, ζ(c)) for some t ∈ I \ {0, 1}. It follows that c ∈ ε(ζ)t, that
is, ε(ζ)t ̸= ∅. Hence 0 ∈ ε(ζ)t, and so YJ(ε, ζ(0)) ≤ t, which is a contradiction. Thus
YJ(ε, ζ(0)) ≤ YJ(ε, ζ(x)) for all x ∈ X. Suppose that (15) is not valid. Then

YJ(ε, ζ(x)) > t ≥ YJ(ε, ζ(x ∗ a)) ∨ YJ(ε, ζ(a))

for some x, a ∈ X and t ∈ I \ {0, 1}. It follows that x ∗ a ∈ ε(ζ)t and a ∈ ε(ζ)t, but
x /∈ ε(ζ)t. This is a contradiction, and thus (15) is valid. Therefore ζ is a Y ε

J -fuzzy ideal
of (X, ∗, 0).

We provide conditions for Y ε
J -fuzzy subalgebra to be Y ε

J -fuzzy ideal.

Theorem 7. If a Y ε
J -fuzzy subalgebra ζ of (X, ∗, 0) satisfies the condition (17), then it

is a Y ε
J -fuzzy ideal of (X, ∗, 0).

Proof. Let ζ be a Y ε
J -fuzzy subalgebra ζ of (X, ∗, 0) that satisfies the condition (17).

The combination of (III) and (13) induces YJ(ε, ζ(0)) ≤ YJ(ε, ζ(x)) for all x ∈ X. For
every x, y ∈ X, we have x ∗ (x ∗ y) ≤X y by (III), (1) and (4). It follows from (17) that
YJ(ε, ζ(x)) ≤ YJ(ε, ζ(x ∗ y)) ∨ YJ(ε, ζ(y)) for all x, y ∈ X. Therefore ζ is a Y ε

J -fuzzy ideal
of (X, ∗, 0).



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2020

5. Closed Y ε
J -fuzzy ideals in BCI-algebras

In this section, let (X, ∗, 0) denote a BCI-algebra. We recall that any Y ε
J -fuzzy ideal

may not be a Y ε
J -fuzzy subalgebra in BCI-algebras (cf. Example 2). This is a motivation

for the definition below.

Definition 2. A Y ε
J -fuzzy ideal ζ of (X, ∗, 0) is said to be closed if it is also a Y ε

J -fuzzy
subalgebra of (X, ∗, 0).

Example 5. Let X = {0, b1, b2, b3, b4} be a set with a binary operation “∗” given by Table
4.

Table 4: Cayley table for the binary operation “ ∗ ”

∗ 0 b1 b2 b3 b4
0 0 0 0 b3 b3
b1 b1 0 0 b3 b3
b2 b2 b2 0 b4 b3
b3 b3 b3 b3 0 0
b4 b4 b4 b3 b2 0

Then (X, ∗, 0) is a BCI-algebra (see [2]). Let ζ be a fuzzy set in X given by

ζ : X → [0, 1], y 7→


0.78 if y = 0,
0.63 if y ∈ {b1, b2},
0.47 otherwise,

It is routine to check that ζ is a closed Y ε
J -fuzzy ideal of (X, ∗, 0) for ε := 0.46.

Theorem 8. A fuzzy set ζ in X given by

ζ : X → [0, 1], y 7→
{

s1 if y ∈ {x ∈ X | 0 ≤X x},
s2 otherwise,

where s1 > s2 in (0, 1), is a closed Y ε
J -fuzzy ideal of (X, ∗, 0).

Proof. The Y-level set ε(ζ)t is calculated as follows:

ε(ζ)t =


∅ if 0 < t < 1 − s1,
{x ∈ X | 0 ≤X x} if 1 − s1 ≤ t < 1 − s2,
X if 1 − s2 ≤ t < 1.

Let A := {x ∈ X | 0 ≤X x}, and let y, z ∈ A. Then 0 ≤X y and 0 ≤X z, i.e., 0 ∗ y = 0
and 0 ∗ z = 0. Hence 0 ∗ (y ∗ z) = (0 ∗ y) ∗ (0 ∗ z) = 0 by (III) and (6), and so 0 ≤X y ∗ z,



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2021

i.e., y ∗ z ∈ A. Thus A is a subalgebra of (X, ∗, 0). It is clear that 0 ∈ A. Let y, z ∈ X be
such that y ∗ z ∈ A and z ∈ A. Then

0 = 0 ∗ (y ∗ z) = (0 ∗ y) ∗ (0 ∗ z) = (0 ∗ y) ∗ 0 = 0 ∗ y

by (2) and (6). Hence y ∈ A, which shows that A is an ideal of (X, ∗, 0). Therefore A is
a closed ideal of (X, ∗, 0). By the combination of Theorems 1 and 6, we conclude that ζ
is a closed Y ε

J -fuzzy ideal of (X, ∗, 0).

Lemma 1. A fuzzy set ζ in X is a closed Y ε
J -fuzzy ideal of (X, ∗, 0) if and only if the

nonempty Y-level set ε(ζ)t of ε(ζ) is a closed ideal of (X, ∗, 0) for all t ∈ I \ {0, 1}.

Proof. Assume that ζ is a closed Y ε
J -fuzzy ideal of (X, ∗, 0) and let t ∈ I \ {0, 1} be

such that ε(ζ)t ̸= ∅. Then ε(ζ)t is an ideal of (X, ∗, 0) by Theorem 6. Let x ∈ ε(ζ)t. Then

YJ(ε, ζ(0 ∗ x))
(13)

≤ YJ(ε, ζ(0)) ∨ YJ(ε, ζ(x))
(14)

≤ YJ(ε, ζ(x)) ≤ t,

and so 0 ∗ x ∈ ε(ζ)t. Hence ε(ζ)t is a closed ideal of (X, ∗, 0).
Conversely, suppose that the nonempty Y-level set ε(ζ)t of ε(ζ) is a closed ideal of (X,

∗, 0) for all t ∈ I \ {0, 1}. Then ε(ζ)t is an ideal of (X, ∗, 0), and thus ζ is a Y ε
J -fuzzy

ideal of (X, ∗, 0) by Theorem 6. If ζ is not a Y ε
J -fuzzy subalgebra of (X, ∗, 0), then

YJ(ε, ζ(x ∗ a)) > YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a))

for some x, a ∈ X. Selecting t := YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a)) induces x, a ∈ ε(ζ)t and
x ∗ a /∈ ε(ζ)t, which is a contradiction. Hence

YJ(ε, ζ(x ∗ a)) ≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(a))

for all x, a ∈ X, which shows that ζ is a Y ε
J -fuzzy subalgebra of (X, ∗, 0). Consequently,

ζ is a closed Y ε
J -fuzzy ideal of (X, ∗, 0).

Theorem 9. A Y ε
J -fuzzy ideal ζ of (X, ∗, 0) is closed if and only if it satisfies:

(∀x ∈ X)(YJ(ε, ζ(0 ∗ x)) ≤ YJ(ε, ζ(x))). (18)

Proof. Let ζ be a closed Y ε
J -fuzzy ideal of (X, ∗, 0). Then the nonempty Y-level set

ε(ζ)t of ε(ζ) is a closed ideal of (X, ∗, 0) for all t ∈ I \ {0, 1} by Lemma 1. Then

YJ(ε, ζ(0 ∗ x))
(13)

≤ YJ(ε, ζ(0)) ∨ YJ(ε, ζ(x))
(14)

≤ YJ(ε, ζ(x))

for all x ∈ X.
Conversely, let ζ be a Y ε

J -fuzzy ideal of (X, ∗, 0) that satisfies the condition (18). Then

YJ(ε, ζ((x ∗ y) ∗ x))
(4)
= YJ(ε, ζ((x ∗ x) ∗ y))

(III)
= YJ(ε, ζ(0 ∗ y))

(18)

≤ YJ(ε, ζ(y))



E. H. Roh, E. Yang, Y. B. Jun / Eur. J. Pure Appl. Math, 16 (4) (2023), 2009-2024 2022

for all x, y ∈ X, and so

YJ(ε, ζ(x ∗ y)
(15)

≤ YJ(ε, ζ((x ∗ y) ∗ x)) ∨ YJ(ε, ζ(x))

≤ YJ(ε, ζ(x)) ∨ YJ(ε, ζ(y))

for all x, y ∈ X. Hence ζ is a closed Y ε
J -fuzzy ideal of (X, ∗, 0).

Theorem 10. Given an element a ∈ X, let ζa be the fuzzy set in X defined by

ζa : X → [0, 1], y 7→
{

s1 if y ∈ Xa,
s2 otherwise,

where s1 > s2 in (0, 1) and Xa := {x ∈ X | a ∗ x = a}. Then ζa is a closed Y ε
J -fuzzy ideal

of (X, ∗, 0).

Proof. The Y-level set ε(ζa)t is calculated as follows:

ε(ζa)t =


∅ if 0 < t < 1 − s1,
Xa if 1 − s1 ≤ t < 1 − s2,
X if 1 − s2 ≤ t < 1.

It is clear that 0 ∈ Xa by (2). For every x ∈ Xa, we have

0 ∗ x (III)
= (a ∗ a) ∗ x (4)

= (a ∗ x) ∗ a x∈Xa= a ∗ a (III)
= 0 ∈ Xa (19)

Let x, y ∈ X be such that x ∗ y ∈ Xa and y ∈ Xa. Then

0 ∗ (x ∗ y) = 0 and 0 ∗ y = 0

by (19). It follows that

(a ∗ x) ∗ a (4)
= (a ∗ a) ∗ x (III)

= 0 ∗ x (2)
= (0 ∗ x) ∗ (0 ∗ y)

(6)
= 0 ∗ (x ∗ y) = 0,

i.e., a ∗ x ≤X a. Since x ∗ y ∈ Xa and y ∈ Xa, we get

a = a ∗ (x ∗ y) = (a ∗ y) ∗ (x ∗ y) ≤ a ∗ x.

Hence a ∗ x = a, i.e., x ∈ Xa. This shows that Xa is a closed ideal of (X, ∗, 0). Thus we
know that the nonempty Y-level set ε(ζa)t is a closed ideal of (X, ∗, 0) for all t ∈ I \{0, 1}.
Therefore ζa is a closed Y ε

J -fuzzy ideal of (X, ∗, 0) by Lemma 1.

We explore the conditions under which a Y ε
J -fuzzy subalgebra becomes a Y ε

J -fuzzy
ideal.

Theorem 11. In a p-semisimple BCI-algebra (X, ∗, 0), every Y ε
J -fuzzy subalgebra is a

Y ε
J -fuzzy ideal.



REFERENCES 2023

Proof. Let ζ be a Y ε
J -fuzzy subalgebra of a p-semisimple BCI-algebra (X, ∗, 0), and let

t ∈ I \ {0, 1} be such that ε(ζ)t ̸= ∅. Then ε(ζ)t is a subalgebra of (X, ∗, 0) by Theorem
1. It is clear that 0 ∈ ε(ζ)t. Let x, y ∈ X be such that x ∗ y ∈ ε(ζ)t and y ∈ ε(ζ)t. Then
0 ∗ y ∈ ε(ζ)t and (x ∗ y) ∗ (0 ∗ y) ∈ ε(ζ)t. On the other hand, we have

((x ∗ y) ∗ (0 ∗ y)) ∗ x (4)
= ((x ∗ y) ∗ x) ∗ (0 ∗ y)

(4)
= ((x ∗ x) ∗ y) ∗ (0 ∗ y)

(III)
= (0 ∗ y) ∗ (0 ∗ y)

(III)
= 0,

that is, (x ∗ y) ∗ (0 ∗ y) ≤X x. Since (X, ∗, 0) is p-semisimple, x is a minimal element of
X. It follows that x = (x ∗ y) ∗ (0 ∗ y) ∈ ε(ζ)t. Hence ε(ζ)t is an ideal of (X, ∗, 0), and
therefore ζ is a Y ε

J -fuzzy ideal of (X, ∗, 0) by Theorem 6.

Corollary 2. If a BCI-algebra (X, ∗, 0) satisfies:

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

or

(∀x ∈ X)(0 ∗ x = 0 ⇒ x = 0),

then every Y ε
J -fuzzy subalgebra is a Y ε

J -fuzzy ideal.

Acknowledgements

This paper was supported by the research fund in Chinju National University of Edu-
cation, 2022.

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