EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1521-1535
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Dokdo BE-subalgebras and BE-filters of BE-algebras

Young Bae Jun1, Sun Shin Ahn2,∗, Eun Hwan Roh3

1 Department of Mathematics Education, Gyeongsang National University, Jinju 52828,
Korea
2 Department of Mathematics Education, Dongguk University, Seoul 04620, Korea
3 Department of Mathematics Education, Chinju National University of Education, Jinju
52673, Korea

Abstract. With the aim of applying the Dokdo structure to BE-algebra, the notions of (weak)
Dokdo BE-subalgebra and Dokdo BE-filter are introduced, and their properties are investigated.
The relationship between weak Dokdo BE-subalgebra, Dokdo BE-subalgebra and Dokdo BE-filter
is established. The conditions under which Dokdo structure can be weak Dokdo BE-subalgebra
and Dokdo BE-filter, and the condition under which weak Dokdo BE-subalgebra can be Dokdo
BE-subalgebra are explored. Characterizations of Dokdo BE-filter are provided.

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

Key Words and Phrases: Weak Dokdo BE-subalgebra, Dokdo BE-subalgebra, Dokdo BE-filter

1. Introduction

Soft sets and fuzzy sets (interval value, bipolar) are useful tools for solving the problem
of maintaining uncertainty in everyday life. Fuzzy sets are an extension of an existing
set using fuzzy logic, and interval-valued fuzzy sets are also an extension of fuzzy sets
whose membership degree range is a subinterval of [0, 1]. As an extension of fuzzy sets,
bipolar fuzzy sets whose membership degree range is [−1, 1] are a very useful tool for
considering positive information and negative information at the same time. Soft set theory
is a generalization of fuzzy set theory. (Bipolar, interval-valued) fuzzy set theory and
soft set theory are good mathematical tools for dealing with uncertainty in a parametric
manner, and have many applications in medical diagnosis and decision making etc. In
the information age, there is a growing need to use hybrid structures in various fields.
It has become necessary to study hybrid structures based on logical algebra to present
the mathematical tools needed to meet these needs. Hybrid structures dealing with two
or more different concepts at the same time have the advantage of reducing the loss of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4510

Email addresses: skywine@gmail.com (Y. B. Jun),
sunshine@dongguk.edu (S. S. Ahn), ehroh9988@gmail.com (E. H. Roh)

https://www.ejpam.com 1521 © 2022 EJPAM All rights reserved.



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1522

information when addressing uncertainty issues. In line with this background and need,
Jun [5] introduced a new type of hybrid structure called Dokdo structure, where “Dokdo”
is the name of the most beautiful island in Korea, using the concepts of bipolar fuzzy set,
soft set and interval-valued fuzzy and first applied it to the algebraic structure BCK/BCI-
algebras (see [5, 6]). In 2007, H. S. Kim and Y. H. Kim [7] introduced the notion of
a BE-algebra as a dualization of a generalization of a BCK-algebra. They defined and
studied the concept of a filter in BE-algebras. In [1, 11], S. S. Ahn et al. and A. Rezaei
et al. studied fuzzy BE-algebras. G. Dymek and A. Walendziak [2] developed the theory
of fuzzy filters in BE-algebras.

For the purpose of applying the Dokdo structure to BE-algebra, we introduce (weak)
Dokdo BE-subalgebra and Dokdo BE-filter and study its characteristics. We investigate
the relationship between weak Dokdo BE-subalgebra, Dokdo BE-subalgebra and Dokdo
BE-filter. We explore the conditions under which Dokdo structure can be weak Dokdo
BE-subalgebra and Dokdo BE-filter, and the condition under which weak Dokdo BE-
subalgebra can be Dokdo BE-subalgebra. We discuss the characterization of Dokdo BE-
filter.

2. Preliminaries

2.1. Basic concepts about BE-algebras

A BE-algebra (see [7]) is defined to be a set X together with a binary operation “ ∗ ”
and a special element “1” satisfying the conditions:

(BE1) (∀a ∈ X) (a ∗ a = 1),
(BE2) (∀a ∈ X) (a ∗ 1 = 1),
(BE3) (∀a ∈ X) (1 ∗ a = a),
(BE4) (∀a, b, c ∈ X) (a ∗ (b ∗ c) = b ∗ (a ∗ c)).
The order relation “ ≤ ” in a BE-algebra X is defined as follows:

(∀a, b ∈ X)(a ≤ b ⇔ a ∗ b = 1). (1)

Every BE-algebra X satisfies the following conditions (see [7]):

(∀a, b ∈ X) (a ∗ (b ∗ a) = 1) , (2)

(∀a, b ∈ X) (a ∗ ((a ∗ b) ∗ b) = 1) . (3)

A BE-algebra X is said to be self-distributive (see [7]) if it satisfies:

(∀x, b, c ∈ X) (x ∗ (b ∗ c) = (x ∗ b) ∗ (x ∗ c)) . (4)

A subset A of a BE-algebra X is called

• a BE-subalgebra of X (see [7]) if it satisfies:

(∀a, b ∈ A)(a ∗ b ∈ A), (5)



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1523

• an n-fold weak BE-subalgebra of X (see [4]) if it satisfies:

(∀a, b ∈ A)(an ∗ b ∈ A), (6)

where n is a natural number with n ≥ 2 and an ∗ b = a ∗ (a ∗ (· · · (a ∗ b) · · · )) in
which a appears n times. The n-fold weak BE-subalgebra with n = 2 is called a
weak BE-subalgebra.

• a BE-filter of X (see [7]) if it satisfies:

1 ∈ A, (7)

(∀a, b ∈ X)(a ∗ b ∈ A, a ∈ A ⇒ b ∈ A). (8)

2.2. Basic concepts about Dokdo structures

Let X be a set. A bipolar fuzzy set in X (see [8]) is an object having the form

φ̊ = {(a, φ−(a), φ+(a)) | a ∈ X} (9)

where φ− : X → [−1, 0] and φ+ : X → [0, 1] are mappings. The bipolar fuzzy set which is
described in (9) is simply denoted by φ̊ := (X;φ−, φ+).

A bipolar fuzzy set can be reinterpreted as a function:

φ̊ : X → [−1, 0]× [0, 1], x 7→ (φ−(x), φ+(x)).

Let U be an initial universe set and X be a set of parameters. For any subset A of
X, a pair (φs, A) is called a soft set over U (see [9]), where φs is a mapping described as
follows:

φs : A → 2U

where 2U is the power set of U . If A = X, the soft set (φs, A) over U is simply denoted
by φs only.

A mapping φ̃ : X → [[0, 1]] is called an interval-valued fuzzy set (briefly, an IVF set)
in X (see [3, 12]) where [[0, 1]] is the set of all closed subintervals of [0, 1], and members
of [[0, 1]] are called interval numbers and are denoted by ã, b̃, c̃, etc., where ã = [a−, a+]
with 0 ≤ a− ≤ a+ ≤ 1.

For every two interval numbers ã and b̃, we define

ã ⪯ b̃ (or b̃ ⪰ ã) ⇔ a− ≤ b−, a+ ≤ b+, (10)

ã = b̃ ⇔ ã ⪯ b̃, b̃ ⪯ ã, (11)

rmin{ã, b̃} = [min{a−, b−},min{a+, b+}]. (12)

Let U be an initial universe set and X a set of parameters. A triple Dokφ := (φ̊, φs,
φ̃) is called a Dokdo structure (see [5]) in (X,U) if φ̊ : X → [−1, 0] × [0, 1] is a bipolar



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1524

fuzzy set in X, φs : X → 2U is a soft set over U and φ̃ : X → [[0, 1]] is an interval-valued
fuzzy set in X.

The Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) can be represented as follows:

Dokφ := (φ̊, φs, φ̃) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→ (φ̊(x), φs(x), φ̃(x))
(13)

where φ̊(x) = (φ̊−(x), φ̊+(x)) and φ̃(x) = [φ̃L(x), φ̃R(x)].
Given a Dokdo structure Dokφ := (φ̊, φs, φ̃) in a Dokdo universe (X,U), we consider

the following sets:

φ̊(max,min) :=

{
x

(y,z) ∈
X

X×X

∣∣∣ φ̊−(x) ≤ max{φ̊−(y), φ̊−(z)}
φ̊+(x) ≥ min{φ̊+(y), φ̊+(z)}

}
,

φ̊(s,−) := {x ∈ X | φ̊−(x) ≤ s}, φ̊(t,+) := {x ∈ X | φ̊+(x) ≥ t},
φ̊(s, t) := φ̊(s,−) ∩ φ̊(t,+),

φs
α := {x ∈ X | φs(x) ⊇ α}, φ̃ã := {x ∈ X | φ̃(x) ⪰ ã},

where (s, t) ∈ [−1, 0]× [0, 1], α ∈ 2U and ã = [aL, aR].

3. Dokdo BE-subalgebras

Let U be an initial universe set and X a set of parameters. We say that the pair (X,U)
is called a Dokdo BE-universe if X is a BE-algebra. In what follows, let (X,U) denote
the Dokdo BE-universe unless otherwise specified.

Definition 1. A Dokdo structure Dokφ := (φ̊, φs, φ̃) is called a Dokdo BE-subalgebra of
(X,U) if it satisfies:

(∀x, y ∈ X)
(

x∗y
(x,y) ∈ φ̊(max,min)

)
, (14)

(∀x, y ∈ X) (φs(x ∗ y) ⊇ φs(x) ∩ φs(y)) , (15)

(∀x, y ∈ X) (φ̃(x ∗ y) ⪰ rmin{φ̃(x), φ̃(y)}) . (16)

Example 1. Let (X,U) be a BE-Dokdo universe in which U = Z and X = {1, 2, 3, 4, 5, 6}
is a BE-algebra (see [1]) with a binary operation “∗” given in the table below.

∗ 1 2 3 4 5 6

1 1 2 3 4 5 6
2 1 1 2 4 4 5
3 1 1 1 4 4 4
4 1 2 3 1 2 3
5 1 1 2 1 1 2
6 1 1 1 1 1 1



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1525

Let Dokφ := (φ̊, φs, φ̃) be a Dokdo structure in (X,U = Z) which is defined as follows:

X φ̊(x) φs(x) φ̃(x)

1 (−0.7, 0.8) 2Z [0.4, 0.8]
2 (−0.6, 0.5) 8Z [0.3, 0.6]
3 (−0.3, 0.2) 4Z [0.1, 0.5]
4 (−0.5, 0.4) 8N [0.3, 0.7]
5 (−0.4, 0.3) 16N [0.2, 0.6]
6 (−0.3, 0.2) 16N [0.1, 0.5]

It is routine to verify that Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra of (X,U = Z).

Definition 2. A Dokdo structure Dokφ := (φ̊, φs, φ̃) is called a weak Dokdo BE-subalgebra
of (X,U) if it satisfies:

(∀x, y ∈ X)
(
x∗(x∗y)
(x,y) ∈ φ̊(max,min)

)
, (17)

(∀x, y ∈ X) (φs(x ∗ (x ∗ y)) ⊇ φs(x) ∩ φs(y)) , (18)

(∀x, y ∈ X) (φ̃(x ∗ (x ∗ y)) ⪰ rmin{φ̃(x), φ̃(y)}) . (19)

Example 2. Let (X,U) be a BE-Dokdo universe in which U = Z and X = {1, 2, 3, 4} is
a BE-algebra (see [10]) with a binary operation “∗” given in the table below.

∗ 1 2 3 4

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

Define a Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) as follows:

Dokφ := (φ̊, φs, φ̃) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→
{

((−0.46, 0.73),Z, [0.41, 0.73]) if x = 1,
((−0.36, 0.63),N, [0.32, 0.64]) otherwise.

It is routine to check that Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of (X,U).

Lemma 1. Every Dokdo BE-subalgebra is a weak Dokdo BE-subalgebra.

Proof. The proof is straightforward.

The converse of Lemma 1 may not be true as seen in the following example.

Example 3. Let (X,U) be a BE-Dokdo universe in which U = Z and X = {1, 2, 3, 4} is
a BE-algebra (see [10]) with a binary operation “∗” given in the table below.



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1526

∗ 1 2 3 4

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

Define a Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) as follows:

Dokφ := (φ̊, φs, φ̃) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→


((−0.46, 0.73),Z, [0.41, 0.73]) if x = 1,
((−0.36, 0.63), 2Z, [0.32, 0.64]) if x ∈ {2, 4},
((−0.27, 0.58), 2N, [0.29, 0.59]) if x = 3.

It is routine to check that Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of (X,U).
But it is not a Dokdo BE-subalgebra of (X,U) since 2∗4

(2,4) =
3

(2,4) /∈ φ̊(max,min), φs(2∗4) =
φs(3) = 2N ⊉ 2Z = φs(2) ∩ φs(4), or φ̃(2 ∗ 4) = φ̃(3) = [0.29, 0.59] ⪰̸ [0.32, 0.64] =
rmin{φ̃(2), φ̃(4)}.

We explore the conditions under which the converse of Lemma 1 becomes true.

Theorem 1. If a weak Dokdo BE-subalgebra Dokφ := (φ̊, φs, φ̃) of (X,U) satisfies:

(∀x, y ∈ X)

 x∗y
(x∗(x∗y), x∗(x∗y)) ∈ φ̊(max,min),

φs(x ∗ y) ⊇ φs(x ∗ (x ∗ y)),
φ̃(x ∗ y) ⪰ φ̃(x ∗ (x ∗ y))

 , (20)

then Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra of (X,U).

Proof. For every x, y ∈ X, we have

φ̊−(x ∗ y) ≤ φ̊−(x ∗ (x ∗ y)) ≤ max{φ̊−(x), φ̊−(y)}

and φ̊+(x ∗ y) ≥ φ̊+(x ∗ (x ∗ y)) ≥ min{φ̊+(x), φ̊+(y)}. Hence x∗y
(x, y) ∈ φ̊(max,min). Also,

φs(x∗y) ⊇ φs(x∗ (x∗y)) ⊇ φs(x)∩φs(y) and φ̃(x∗y) ⪰ φ̃(x∗ (x∗y)) ⪰ rmin{φ̃(x), φ̃(y)}.
Therefore Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra of (X,U).

Proposition 1. If Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of (X,U), then

(i) φ−(1) is a lower bound of {φ−(x) | x ∈ X},

(ii) φ+(1) is an upper bound of {φ−(x) | x ∈ X},

(iii) (∀x ∈ X) (φs(1) ⊇ φs(x), φ̃(1) ⪰ φ̃(x)).



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1527

Proof. Let Dokφ := (φ̊, φs, φ̃) be a weak Dokdo BE-subalgebra of (X,U). For every

x ∈ X, if we use (BE1) and (BE2), then 1
(x,x) =

x∗(x∗x)
(x,x) ∈ φ̊(max,min) which implies that

φ−(1) ≤ max{φ−(x), φ−(x)} = φ−(x)

and
φ+(1) ≥ min{φ+(x), φ+(x)} = φ+(x).

Hence (i) and (ii) are valid. Also, φs(1) = φs(x ∗ (x ∗ x)) ⊇ φs(x) ∩ φs(x) = φs(x) and
φ̃(1) = φ̃(x ∗ (x ∗ x)) ⪰ rmin{φ̃(x), φ̃(x)} = φ̃(x).

The combination of Lemma 1 and Proposition 1 leads to the following corollary.

Corollary 1. If Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra of (X,U), then the results
(i), (ii), and (iii) in Proposition 1 are valid.

Proposition 2. Every weak Dokdo BE-subalgebra Dokφ := (φ̊, φs, φ̃) of (X,U) satisfies:

(∀x, y ∈ X)

(
y

(y∗(y∗x),y∗(y∗x)) ∈ φ̊(max,min) ⇒
{

φ−(y) = φ−(1)
φ+(y) = φ+(1)

)
, (21)

(∀x, y ∈ X) (φs(y) ⊇ φs(y ∗ (y ∗ x)) ⇒ φs(y) = φs(1)) , (22)

(∀x, y ∈ X) (φ̃(y) ⪰ φ̃(y ∗ (y ∗ x)) ⇒ φ̃(y) = φ̃(1)) . (23)

Proof. Assume that y
(y∗(y∗x),y∗(y∗x)) ∈ φ̊(max,min), φs(y) ⊇ φs(y ∗ (y ∗ x)) and φ̃(y) ⪰

φ̃(y∗(y∗x)) for all x, y ∈ X. If we take x = 1 and use (BE2), then y
(1,1) =

y
(y∗(y∗1),y∗(y∗1)) ∈

φ̊(max,min), φs(y) ⊇ φs(y ∗ (y ∗ 1)) = φs(1) and φ̃(y) ⪰ φ̃(y ∗ (y ∗ 1)) = φ̃(1). The
combination of these and Proposition 1 leads to φ−(y) = φ−(1), φ+(y) = φ+(1), φs(y) =
φs(1) and φ̃(y) = φ̃(1).

Corollary 2. Every Dokdo BE-subalgebra Dokφ := (φ̊, φs, φ̃) of (X,U) satisfies (21),
(22) and (23).

Theorem 2. If Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of (X,U), then the
nonempty sets φ̊(s, t), φs

α and φ̃ã are weak BE-subalgebras of X for all (s, t) ∈ [−1, 0] ×
[0, 1], α ∈ 2U and ã = [a−, a+].

Proof. Let (s, t) ∈ [−1, 0]× [0, 1], α ∈ 2U and ã = [a−, a+] be such that φ̊(s, t), φs
α and

φ̃ã are nonempty. Let x, y ∈ φ̊(s, t) ∩ φs
α ∩ φ̃ã. Then φ−(x) ≤ s, φ−(y) ≤ s, φ+(x) ≥ t,

φ+(y) ≥ t, φs(x) ⊇ α, φs(y) ⊇ α, φ̃(x) ⪰ ã and φ̃(y) ⪰ ã. Hence

φ−(x ∗ (x ∗ y)) ≤ max{φ−(x), φ−(y)} ≤ s,

φ+(x ∗ (x ∗ y)) ≥ min{φ+(x), φ+(y)} ≥ t,

and so x ∗ (x ∗ y) ∈ φ̊(s, t). Also we have φs(x ∗ (x ∗ y)) ⊇ φs(x) ∩ φs(y) ⊇ α and
φ̃(x ∗ (x ∗ y)) ⪰ rmin{φ̃(x), φ̃(y)} ⪰ ã, that is, x ∗ (x ∗ y) ∈ φs

α and x ∗ (x ∗ y) ∈ φ̃ã.
Therefore φ̊(s, t), φs

α and φ̃ã are weak BE-subalgebras of X.



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1528

Corollary 3. If Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra of (X,U), then the
nonempty sets φ̊(s, t), φs

α and φ̃ã are weak BE-subalgebras of X for all (s, t) ∈ [−1, 0] ×
[0, 1], α ∈ 2U and ã = [a−, a+].

The following example shows that the converse of Theorem 2 may not be true.

Example 4. Let (X,U) be the BE-Dokdo universe in Example 1. Define a Dokdo structure
Dokφ := (φ̊, φs, φ̃) in (X,U) as follows:

Dokφ := (φ̊, φs, φ̃) : X → ([−1, 0]× [0, 1])× 2U × [[0, 1]],

x 7→



((−0.85, 0.71), Z, [0.42, 0.76]) if x = 1,
((−0.66, 0.53), 4Z, [0.29, 0.58]) if x = 2,
((−0.44, 0.57), 4Z, [0.29, 0.58]) if x = 3,
((−0.44, 0.53), 4Z, [0.29, 0.58]) if x = 4,
((−0.44, 0.53), 4Z, [0.29, 0.58]) if x = 5,
((−0.72, 0.68), 2Z, [0.33, 0.72]) if x = 6.

It is routine to verify that the nonempty sets φ̊(s, t), φs
α and φ̃ã are weak BE-subalgebras

of X for all (s, t) ∈ [−1, 0] × [0, 1], α ∈ 2U and ã = [a−, a+]. But Dokφ := (φ̊, φs, φ̃) is

not a weak Dokdo BE-subalgebra of (X,U) because of 2∗(2∗6)
(2,6) = 4

(2,6) /∈ φ̊(max,min).

We provide conditions for a Dokdo structure to be a weak Dokdo BE-subalgebra.

Theorem 3. Given a Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U), if the nonempty sets
φ̊(s,−), φ̊(t,+), φs

α and φ̃ã are weak BE-subalgebras of X for all (s, t) ∈ [−1, 0] × [0, 1],
α ∈ 2U and ã = [a−, a+], then Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of
(X,U).

Proof. Assume that φ̊(s,−), φ̊(t,+), φs
α and φ̃ã are nonempty weak BE-subalgebras

of X for all (s, t) ∈ [−1, 0] × [0, 1], α ∈ 2U and ã = [a−, a+]. If there exist x, y ∈ X such

that x∗(x∗y)
(x,y) /∈ φ̊(max,min), then

φ−(x ∗ (x ∗ y)) > max{φ−(x), φ−(y)} or φ+(x ∗ (x ∗ y)) < min{φ+(x), φ+(y)}.

It follows that x, y ∈ φ̊(s,−) ∩ φ̊(t,+), x ∗ (x ∗ y) /∈ φ̊(s,−) and x ∗ (x ∗ y) /∈ φ̊(t,+) for
s := max{φ−(x), φ−(y)} and t := min{φ+(x), φ+(y)}. This is a contradiction, and thus
x∗(x∗y)
(x,y) ∈ φ̊(max,min) for all x, y ∈ X. For every x, y ∈ X, let φs(x) = αx, φ

s(y) = αy,

φ̃(x) = ã and φ̃(y) = b̃. If we take α := αx ∩ αy and c̃ := rmin{ã, b̃}, then x, y ∈ φs
α ∩ φ̃c̃

and so x ∗ (x ∗ y) ∈ φs
α ∩ φ̃c̃. Hence

φs(x ∗ (x ∗ y)) ⊇ α = αx ∩ αy = φs(x) ∩ φs(y)

and
φ̃(x ∗ (x ∗ y)) ⪰ c̃ = rmin{ã, b̃} = rmin{φ̃(x), φ̃(y)}.

Therefore Dokφ := (φ̊, φs, φ̃) is a weak Dokdo BE-subalgebra of (X,U).



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1529

4. Dokdo BE-filters

Definition 3. A Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) is called a Dokdo BE-filter
of (X,U) if it satisfies:

(∀x ∈ X)

( 1
(x,x) ∈ φ̊(max,min),

φs(1) ⊇ φs(x), φ̃(1) ⪰ φ̃(x)

)
, (24)

(∀x, y ∈ X)

 y
(x,x∗y) ∈ φ̊(max,min),

φs(y) ⊇ φs(x) ∩ φs(x ∗ y),
φ̃(y) ⪰ rmin{φ̃(x), φ̃(x ∗ y)}

 . (25)

Example 5. Let (X,U) be a BE-Dokdo universe in which U = N and X = {1, 2, 3, 4, 5}
is a BE-algebra (see [7]) with a binary operation “∗” given in the table below.

∗ 1 2 3 4 5

1 1 2 3 4 5
2 1 1 3 4 5
3 1 2 1 4 4
4 1 1 3 1 3
5 1 1 1 1 1

Let Dokφ := (φ̊, φs, φ̃) be a Dokdo structure in (X,U = N) given in the table below.

X φ̊(x) φs(x) φ̃(x)

1 (−0.6, 0.8) 2N [0.4, 0.9]
2 (−0.6, 0.8) 2N [0.4, 0.9]
3 (−0.3, 0.6) 4N [0.2, 0.5]
4 (−0.5, 0.4) 8N [0.3, 0.7]
5 (−0.3, 0.4) 8N [0.2, 0.5]

Through routine calculations, we can confirm that Dokφ := (φ̊, φs, φ̃) in (X,U) is a
Dokdo BE-filter of (X,U = N).

Proposition 3. Every Dokdo BE-filter Dokφ := (φ̊, φs, φ̃) of (X,U) satisfies:

(∀x, y ∈ X)

(
x ≤ y ⇒

{ y
(x,x) ∈ φ̊(max,min)

φs(y) ⊇ φs(x), φ̃(y) ⪰ φ̃(x)

)
, (26)

(∀x, y, z ∈ X)

x ≤ y ∗ z ⇒


z

(x,y) ∈ φ̊(max,min)

φs(z) ⊇ φs(x) ∩ φs(y)
φ̃(z) ⪰ rmin{φ̃(x), φ̃(y)}

 , (27)

(∀x, y ∈ X)

 y∗x
(1, 1) ∈ φ̊(max,min),

φs(y ∗ x) = φs(1),
φ̃(y ∗ x) = φ̃(1)

 ⇒


x

(y,y) ∈ φ̊(max,min)

φs(x) ⊇ φs(y),
φ̃(x) ⪰ φ̃(y)

 . (28)



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1530

Proof. If x ≤ y, then x ∗ y = 1 and so (26) is derived from the definition of Dokdo
BE-filter. Let x, y, z ∈ X be such that x ≤ y ∗ z. Then x ∗ (y ∗ z) = 1, and so

φ−(z) ≤ max{φ−(y), φ−(y ∗ z)}
≤ max{φ−(y),max{φ−(x), φ−(x ∗ (y ∗ z))}}
= max{φ−(y),max{φ−(x), φ−(1)}}
= max{φ−(x), φ−(y)}

and

φ+(z) ≥ min{φ+(y), φ+(y ∗ z)}
≥ min{φ+(y),min{φ+(x), φ+(x ∗ (y ∗ z))}}
= min{φ+(y),min{φ+(x), φ+(1)}}
= min{φ+(x), φ+(y)},

that is, z
(x,y) ∈ φ̊(max,min). Also, we have

φs(z) ⊇ φs(y) ∩ φs(y ∗ z) ⊇ φs(y) ∩ (φs(x) ∩ φs(x ∗ (y ∗ z)))
= φs(y) ∩ (φs(x) ∩ φs(1)) = φs(y) ∩ φs(x)

and

φ̃(z) ⪰ rmin{φ̃(y), φ̃(y ∗ z)} ⪰ rmin{φ̃(y), rmin{φ̃(x), φ̃(x ∗ (y ∗ z))}}
= rmin{φ̃(y), rmin{φ̃(x), φ̃(1)}} = rmin{φ̃(y), φ̃(x)}.

Let x, y ∈ X be such that y∗x
(1, 1) ∈ φ̊(max,min), φs(y ∗ x) = φs(1) and φ̃(y ∗ x) = φ̃(1). It

follows that

φ−(x) ≤ max{φ−(y), φ−(y ∗ x)} ≤ max{φ−(y), φ−(1)} = φ−(y)

and
φ+(x) ≥ min{φ+(y), φ+(y ∗ x)} ≥ min{φ+(y), φ+(1)} = φ+(y),

that is, x
(y,y) ∈ φ̊(max,min). Also, we get

φs(x) ⊇ φs(y) ∩ φs(y ∗ x) = φs(y) ∩ φs(1) = φs(y)

and
φ̃(x) ⪰ rmin{φ̃(y), φ̃(y ∗ x)} = rmin{φ̃(y), φ̃(1)} = φ̃(y).

This completes the proof.

Theorem 4. Every Dokdo BE-filter is a (weak) Dokdo BE-subalgebra.



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1531

Proof. Let Dokφ := (φ̊, φs, φ̃) be a Dokdo BE-filter of (X,U). Since x ≤ y ∗ x for all
x, y ∈ X, we have y∗x

(x,x) ∈ φ̊(max,min), φs(y ∗ x) ⊇ φs(x), and φ̃(y ∗ x) ⪰ φ̃(x) by (26). It

follows from (25) that

φ−(y ∗ x) ≤ φ−(x) ≤ max{φ−(y), φ−(y ∗ x)} ≤ max{φ−(x), φ−(y)},
φ+(y ∗ x) ≥ φ+(x) ≥ min{φ+(y), φ+(y ∗ x)} ≥ min{φ+(x), φ+(y)},
φs(y ∗ x) ⊇ φs(x) ⊇ φs(y) ∩ φs(y ∗ x) ⊇ φs(x) ∩ φs(y),

φ̃(y ∗ x) ⪰ φ̃(x) ⪰ rmin{φ̃(y), φ̃(y ∗ x)} ⪰ rmin{φ̃(x), φ̃(y)}.

Therefore Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-subalgebra, and hence a weak Dokdo BE-
subalgebra of (X,U).

The converse of Theorem 4 may not be true as seen in the following example.

Example 6. (i) Let Dokφ := (φ̊, φs, φ̃) be the Dokdo BE-subalgebra of (X,U) which is
described in Example 1. It is not a Dokdo BE-filter of (X,U) since φs(5) = 16N ⊉ 8N =
φs(2) ∩ φs(2 ∗ 5) or 3

(5,5∗3) =
3

(5,2) /∈ φ̊(max,min).

(ii) Let Dokφ := (φ̊, φs, φ̃) be the weak Dokdo BE-subalgebra of (X,U) which is
described in Example 3. It is not a Dokdo BE-filter of (X,U) since 3

(2,2∗3) = 3
(2,4) /∈

φ̊(max,min) or φ̃(3) = [0.29, 0.59] ⪰̸ [0.32, 0.64] = rmin{φ̃(2), φ̃(2 ∗ 3)}.

Proposition 4. Let (X,U) be a Dokdo BE-universe in which X is a self-distributive BE-
algebra. If Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U), then the next assertions
are equivalent.

(∀x, y ∈ X)

 y∗x
(y∗(y∗x), y∗(y∗x)) ∈ φ̊(max,min),

φs(y ∗ x) ⊇ φs(y ∗ (y ∗ x)),
φ̃(y ∗ x) ⪰ φ̃(y ∗ (y ∗ x))

 . (29)

(∀x, y, z ∈ X)

 (z∗y)∗(z∗x)
(z∗(y∗x), z∗(y∗x)) ∈ φ̊(max,min),

φs((z ∗ y) ∗ (z ∗ x)) ⊇ φs(z ∗ (y ∗ x)),
φ̃((z ∗ y) ∗ (z ∗ x)) ⪰ φ̃(z ∗ (y ∗ x))

 . (30)

Proof. Let x, y, z ∈ X. Since X is self-distributive, we have

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

Assume that (29) is valid. Using (BE4), (26) and (29), we have

φ−((z ∗ y) ∗ (z ∗ x)) = φ−(z ∗ ((z ∗ y) ∗ x))
≤ φ−(z ∗ (z ∗ ((z ∗ y) ∗ x))) ≤ φ−(z ∗ (y ∗ x))

and

φ+((z ∗ y) ∗ (z ∗ x)) = φ+(z ∗ ((z ∗ y) ∗ x))



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1532

≥ φ+(z ∗ (z ∗ ((z ∗ y) ∗ x))) ≥ φ+(z ∗ (y ∗ x)),

that is, (z∗y)∗(z∗x)
(z∗(y∗x), z∗(y∗x)) ∈ φ̊(max,min). Also, we have

φs((z ∗ y) ∗ (z ∗ x)) = φs(z ∗ ((z ∗ y) ∗ x))
⊇ φs(z ∗ (z ∗ ((z ∗ y) ∗ x))) ⊇ φs(z ∗ (y ∗ x)),

and

φ̃((z ∗ y) ∗ (z ∗ x)) = φ̃(z ∗ ((z ∗ y) ∗ x))
⪰ φ̃(z ∗ (z ∗ ((z ∗ y) ∗ x))) ⪰ φ̃(z ∗ (y ∗ x)).

Conversely, suppose that (30) is valid. If we put y := z in (30) and use (BE1) and
(BE3), then

z∗x
(z∗(z∗x), z∗(z∗x)) =

1∗(z∗x)
(z∗(z∗x), z∗(z∗x)) =

(z∗z)∗(z∗x)
(z∗(z∗x), z∗(z∗x)) ∈ φ̊(max,min),

φs(z ∗ x) = φs(1 ∗ (z ∗ x)) = φs((z ∗ z) ∗ (z ∗ x)) ⊇ φs(z ∗ (z ∗ x))

and φ̃(z ∗ x) = φ̃(1 ∗ (z ∗ x)) = φ̃((z ∗ z) ∗ (z ∗ x)) ⪰ φ̃(z ∗ (z ∗ x)). This proves (29).

Proposition 5. Let (X,U) be a Dokdo BE-universe in which X is a self-distributive
BE-algebra. Then every Dokdo BE-filter Dokφ := (φ̊, φs, φ̃) of (X,U) satisfies:

(∀x, y, z ∈ X)

 y∗x
(y∗z, z∗x) ∈ φ̊(max,min),

φs(y ∗ x) ⊇ φs(y ∗ z) ∩ φs(z ∗ x),
φ̃(y ∗ x) ⪰ rmin{φ̃(y ∗ z), φ̃(z ∗ x)}

 . (31)

Proof. Using (BE1), (BE2), (BE4) and (4), we have y ∗ z ≤ (z ∗ x) ∗ (y ∗ x) for all
x, y, z ∈ X. Hence (31) is derived from (27).

Theorem 5. If a Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) satisfies (27), then it is
a Dokdo BE-filter of (X,U).

Proof. Since x ≤ x ∗ 1 for all x ∈ X, we have 1
(x,x) ∈ φ̊(max,min), φs(1) ⊇ φs(x),

and φ̃(1) ⪰ φ̃(x) by (27). Since x ∗ y ≤ x ∗ y for all x, y ∈ X, it follows from (27) that
y

(x,x∗y) ∈ φ̊(max,min), φs(y) ⊇ φs(x) ∩ φs(x ∗ y), and φ̃(y) ⪰ rmin{φ̃(x), φ̃(x ∗ y)}. So,
Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U).

Corollary 4. If a Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) satisfies (27), then it is
a (weak) Dokdo BE-subalgebra of (X,U).



Y. B. Jun, S. S. Ahn and E. H. Roh / Eur. J. Pure Appl. Math, 15 (4) (2022), 1521-1535 1533

Theorem 6. A Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) is a Dokdo BE-filter of
(X,U) if and only if it satisfies (24) and

(∀x, y, z ∈ X)

 x∗z
(x∗(y∗z), y) ∈ φ̊(max,min),

φs(x ∗ z) ⊇ φs(x ∗ (y ∗ z)) ∩ φs(y),
φ̃(x ∗ z) ⪰ rmin{φ̃(x ∗ (y ∗ z)), φ̃(y)}

 . (32)

Proof. Assume that Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U) and let x, y, z ∈
X. Then φ−(x ∗ z) ≤ max{φ−(y), φ−(y ∗ (x ∗ z))} = max{φ−(y), φ−(x ∗ (y ∗ z))} and
φ+(x∗z) ≥ min{φ+(y), φ+(y ∗ (x∗z))} = min{φ+(y), φ+(x∗ (y ∗z))}, that is, x∗z

(x∗(y∗z), y) ∈
φ̊(max,min). Also, we have

φs(x ∗ z) ⊇ φs(y) ∩ φs(y ∗ (x ∗ z)) = φs(y) ∩ φs(x ∗ (y ∗ z))

and
φ̃(x ∗ z) ⪰ rmin{φ̃(y), φ̃(y ∗ (x ∗ z))} = rmin{φ̃(y), φ̃(x ∗ (y ∗ z))}.

Conversely, suppose that Dokφ := (φ̊, φs, φ̃) satisfies (24) and (32). If we put x = 1 in
(32) and use (BE3), then we get z

(y∗z, y) = 1∗z
(1∗(y∗z), y) ∈ φ̊(max,min), φs(z) = φs(1 ∗ z) ⊇

φs(1 ∗ (y ∗ z)) ∩ φs(y) = φs(y ∗ z) ∩ φs(y) and

φ̃(z) = φ̃(1 ∗ z) ⪰ rmin{φ̃(1 ∗ (y ∗ z)), φ̃(y)} = rmin{φ̃(y ∗ z), φ̃(y)}

for all y, z ∈ X. Therefore Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U).

Theorem 7. A Dokdo structure Dokφ := (φ̊, φs, φ̃) in (X,U) is a Dokdo BE-filter of
(X,U) if and only if it satisfies:

(∀x, y ∈ X)

 y∗x
(x, x) ∈ φ̊(max,min),

φs(y ∗ x) ⊇ φs(x),
φ̃(y ∗ x) ⪰ φ̃(x)

 , (33)

(∀x, y, a, b ∈ X)

 (a∗(b∗x))∗x
(a, b) ∈ φ̊(max,min),

φs((a ∗ (b ∗ x)) ∗ x) ⊇ φs(a) ∩ φs(b),
φ̃((a ∗ (b ∗ x)) ∗ x) ⪰ rmin{φ̃(a), φ̃(b)}

 . (34)

Proof. Assume thatDokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U) and let x, y, a, b ∈
X. Then

φ−(y ∗ x) ≤ max{φ−(x), φ−(x ∗ (y ∗ x))} = max{φ−(x), φ−(1)} = φ−(x)

and
φ+(y ∗ x) ≥ min{φ+(x), φ+(x ∗ (y ∗ x))} = min{φ+(x), φ+(1)} = φ+(x),

that is, y∗x
(x, x) ∈ φ̊(max,min). Also, we obtain

φs(y ∗ x) ⊇ φs(x) ∩ φs(x ∗ (y ∗ x)) = φs(x) ∩ φs(1) = φs(x)



REFERENCES 1534

and
φ̃(y ∗ x) ⪰ rmin{φ̃(x), φ̃(x ∗ (y ∗ x))} = rmin{φ̃(x), φ̃(1)} = φ̃(x).

Hence (33) is valid. The following facts can be obtained by using (3), (26), and Theorem
6.

φ−((a ∗ (b ∗ x)) ∗ x) ≤ max{φ−((a ∗ (b ∗ x)) ∗ (b ∗ x)), φ−(b)} ≤ max{φ−(a), φ−(b)},
φ+((a ∗ (b ∗ x)) ∗ x) ≥ min{φ+((a ∗ (b ∗ x)) ∗ (b ∗ x)), φ+(b)} ≥ min{φ+(a), φ+(b)},
φs((a ∗ (b ∗ x)) ∗ x) ⊇ φs((a ∗ (b ∗ x)) ∗ (b ∗ x)) ∩ φs(b) ⊇ φs(a) ∩ φs(b),

φ̃((a ∗ (b ∗ x)) ∗ x) ⪰ rmin{φ̃((a ∗ (b ∗ x)) ∗ (b ∗ x)), φ̃(b)} ⪰ rmin{φ̃(a), φ̃(b)}.

Thus (34) is valid.
Conversely, suppose that Dokφ := (φ̊, φs, φ̃) satisfies (33) and (34). If we take y = x

in (33) and use (BE1), then 1
(x, x) =

x∗x
(x, x) ∈ φ̊(max,min), φs(1) = φs(x ∗ x) ⊇ φs(x), and

φ̃(1) = φ̃(x ∗ x) ⪰ φ̃(x) for all x ∈ X. Using (BE1), (BE3) and (34), we have

φ−(y) = φ−(1 ∗ y) = φ−(((x ∗ y) ∗ (x ∗ y)) ∗ y) ≤ max{φ−(x ∗ y), φ−(x)},
φ+(y) = φ+(1 ∗ y) = φ+(((x ∗ y) ∗ (x ∗ y)) ∗ y) ≥ min{φ+(x ∗ y), φ+(x)},
φs(y) = φs(1 ∗ y) = φs(((x ∗ y) ∗ (x ∗ y)) ∗ y) ⊇ φs(x ∗ y) ∩ φs(x),

φ̃(y) = φ̃(1 ∗ y) = φ̃(((x ∗ y) ∗ (x ∗ y)) ∗ y) ⪰ rmin{φ̃(x ∗ y), φ̃(x)}.

Consequently, Dokφ := (φ̊, φs, φ̃) is a Dokdo BE-filter of (X,U).

5. Conclusion

To apply the Dokdo structure to BE-algebra, we introduced (weak) Dokdo BE-subalgebra
and Dokdo BE-filter and study its characteristics. We investigated the relationship be-
tween weak Dokdo BE-subalgebra, Dokdo BE-subalgebra and Dokdo BE-filter. We ex-
plored the conditions under which Dokdo structure can be weak Dokdo BE-subalgebra
and Dokdo BE-filter, and the condition under which weak Dokdo BE-subalgebra can be
Dokdo BE-subalgebra. We discussed the characterization of Dokdo BE-filter.

Acknowledgements

The authors wish to thank the anonymous reviewers for their valuable suggestions.

References

[1] S. S. Ahn, Y. H. Kim, and K. S. So. Fuzzy BE-algebras. J. Appl. Math. Informatics,
29:1049–1057, 2011.

[2] G.Dymek and A. Walendiziak. Fuzzy filters of BE-algebras. Math. Slovaca, 63:935–
946, 2013.



REFERENCES 1535

[3] M. B. Gorzalczany. A method of inference in approximate reasoning based on interval-
valued fuzzy sets. Fuzzy sets and Systems, 21:1–17, 1987.

[4] Y. Huang. BCI-algebras. Science Press, Beijing, China, 2006.

[5] Y. B. Jun. Dokdo structure and its application in BCK/BCI-algebras. TWMS J.
Pure Appl. Math.(submitted).

[6] Y. B. Jun. Implicative ideals of bck-algebras based on dokdo structure. Journal of
Algebraic Hyperstructures and Logical Algebras, 3(3):33–43, 2022.

[7] H. S. Kim and Y. H. Kim. On BE-algebras. Sci. Math. Jpn., 66:113–116, 2007.

[8] K. M. Lee. Bipolar-valued fuzzy sets and their operations. Proc. Int. Conf. on
Intelligent Technologies, Bangkok, Thailand, 2000.

[9] D. Molodtsov. Soft set theory – first results. Comput. Math. Appl., 37:19–31, 1999.

[10] S. R. Mukkamala. A course in BE-algebras. Springer Nature Singapor Pte Ltd.,
2018.

[11] A. Rezaei and A. Borumand Saeid. On fuzzy subalgebras of BE-algebras. Afr. Mat.,
22:115–127, 2011.

[12] L. A. Zadeh. The concept of a linguistic variable and its application to approximate
reasoning-i. Inform. Sci., 8:199–249, 1975.