EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 3, 2022, 924-937
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

 Lukasiewicz fuzzy BE-algebras and BE-filters

Young Bae Jun1, Sun Shin Ahn2,∗

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

Abstract. By applying the concept of  Lukasiewicz fuzzy set to BE-algebras, the notions of
 Lukasiewicz fuzzy BE-algebra and  Lukasiewicz fuzzy BE-filter are introduced, and their prop-
erties are investigated. Characterizations of  Lukasiewicz fuzzy BE-algebra and  Lukasiewicz fuzzy
BE-filter are discussed, and the relationship between fuzzy BE-algebra (resp., fuzzy BE-filter) and
 Lukasiewicz fuzzy BE-algebra (resp.,  Lukasiewicz fuzzy BE-filter) is established. The conditions for
the ∈-set, q-set and O-set of  Lukasiewicz fuzzy set to be BE-subalgebras are explored. Lukasiewicz
fuzzy BE-filter is created by using BE-filter.

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

Key Words and Phrases:  Lukasiewicz fuzzy BE-algebra,  Lukasiewicz fuzzy BE-filter, ∈-set,
q-set, O-set.

1. Introduction

BCK-algebra and BCI-algebra, introduced by Y. Imai, K. Iséki and S. Tanaka in 1966,
are algebraic structures of universal algebra which describe fragments of propositional
calculus related to implications known as BCK and BCI-logic. After that, various gener-
alizations were attempted, and BCC-algebras, BCH-algebras, BH-algebras, d-algebras etc.
appeared. In 2007, H. S. Kim and Y. H. Kim [3] 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 [7] and [6], S. S. Ahn et al. and A. Rezaei et al. studied fuzzy
BE-algebras. G. Dymek and A. Walendziak [1] developed the theory of fuzzy filters in
BE-algebras. In the website https://plato.stanford.edu/entries/lukasiewicz/, we can see
that Jan  Lukasiewicz (1878–1956) was a Polish logician and philosopher who introduced
mathematical logic into Poland, became the earliest founder of the Warsaw school of logic,
and one of the principal architects and teachers of that school. His most famous achieve-
ment was to give the first rigorous formulation of many-valued logic. He introduced many
improvements in propositional logic, and became the first historian of logic to treat the

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i3.4446

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

https://www.ejpam.com 924 © 2022 EJPAM All rights reserved.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 925

subject’s history from the standpoint of modern formal logic.  Lukasiewicz logic, which
is the logic of the  Lukasiewicz t-norm, is a non-classical and many-valued logic. It was
originally defined in the early 20th century by Jan  Lukasiewicz as a three-valued logic.
Using the idea of  Lukasiewicz t-norm, Y. B. Jun [2] constructed the concept of  Lukasiewicz
fuzzy sets based on a given fuzzy set and applied it to BCK-algebras and BCI-algebras.

In this paper, we apply the concept of  Lukasiewicz fuzzy set to BE-algebras. We
introduce the notion of  Lukasiewicz fuzzy BE-algebra and  Lukasiewicz fuzzy BE-filter,
and investigate several properties. We discuss the characterization of  Lukasiewicz fuzzy
BE-algebra and  Lukasiewicz fuzzy BE-filter. We conside the relationship between fuzzy
BE-algebra (resp., fuzzy BE-filter) and  Lukasiewicz fuzzy BE-algebra (resp.,  Lukasiewicz
fuzzy BE-filter). We explore the conditions for the ∈-set, q-set and O-set of  Lukasiewicz
fuzzy set to be BE-subalgebras. We use BE-filter to create Lukasiewicz fuzzy BE-filter.

2. Preliminary

A BE-algebra (see [3]) 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 [3]):

(∀a, b ∈ X) (a ∗ (b ∗ a) = 1) . (2)

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

A subset A of a BE-algebra X is called

• a BE-subalgebra of X if it satisfies:

(∀a, b ∈ A)(a ∗ b ∈ A), (4)

• a BE-filter of X (see [3]) if it satisfies:

1 ∈ A, (5)

(∀a, b ∈ X)(a ∗ b ∈ A, a ∈ A ⇒ b ∈ A). (6)

A fuzzy set ξ in a BE-algebra X is called

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

(∀a, b ∈ X)(ξ(a ∗ b) ≥ min{ξ(a), ξ(b)}). (7)



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 926

• a fuzzy BE-filter of X (see [1]) if it satisfies:

(∀a ∈ X)(ξ(1) ≥ ξ(a)), (8)

(∀a, b ∈ X)(ξ(b) ≥ min{ξ(a ∗ b), ξ(a)}). (9)

A fuzzy set ξ in a set X of the form

ξ(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 [a/t].
For a fuzzy set ξ in a set X, we say that a fuzzy point [a/t] is

(i) contained in ξ, denoted by [a/t] ∈ ξ, (see [5]) if ξ(a) ≥ t.

(ii) quasi-coincident with ξ, denoted by [a/t] q ξ, (see [5]) if ξ(a) + t > 1.

If [a/t]α ξ is not established for α ∈ {∈, q}, it is denoted by [a/t]α ξ.
Let ξ be a fuzzy set in a set X and let ε ∈ (0, 1). A function

 Lε
ξ : X → [0, 1], x 7→ max{0, ξ(x) + ε− 1}

is called the  Lukasiewicz fuzzy set of ξ in X.
For the  Lukasiewicz fuzzy set  Lε

ξ of ξ in X and t ∈ (0, 1], consider the sets

( Lε
ξ, t)∈ := {x ∈ X | [x/t] ∈  Lε

ξ},

( Lε
ξ, t)q := {x ∈ X | [x/t] q  Lε

ξ},

which are called the ∈-set and q-set, respectively, of  Lε
ξ (with value t). Also, consider a

set:

O( Lε
ξ) := {x ∈ X |  Lε

ξ(x) > 0} (10)

which is called an O-set of  Lε
ξ. It is observed that

O( Lε
ξ) = {x ∈ X | ξ(x) + ε− 1 > 0}.

3.  Lukasiewicz fuzzy BE-algebras

In what follows, let X and ξ be a BE-algebra and a fuzzy set in X respectively, and ε
is an element of (0, 1) unless otherwise specified.

Definition 1. The  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is called a  Lukasiewicz fuzzy BE-

algebra of X if it satisfies:

[x/ta] ∈  Lε
ξ, [y/tb] ∈  Lε

ξ ⇒ [(x ∗ y)/min{ta, tb}] ∈  Lε
ξ (11)

for all x, y ∈ X and ta, tb ∈ (0, 1].



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 927

Example 1. Consider a set X = {1, b1, b2, b3, b4, b5} with a binary operation “∗” given in
the table below.

∗ 1 b1 b2 b3 b4 b5
1 1 b1 b2 b3 b4 b5
b1 1 1 b1 b3 b3 b4
b2 1 1 1 b3 b3 b3
b3 1 b1 b2 1 b1 b2
b4 1 1 b1 1 1 b1
b5 1 1 1 1 1 1

Then (X, ∗, 1) is a BE-algebra (see [7]). Define a fuzzy set ξ in X as follows:

ξ : X → [0, 1], x 7→
{

0.76 if x ∈ {1, b1, b2},
0.52 otherwise.

Given ε := 0.67, the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is given as follows:

 Lε
ξ : X → [0, 1], x 7→

{
0.43 if x ∈ {1, b1, b2},
0.19 otherwise.

It is routine to verify that  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra of X.

We provide a characterization of  Lukasiewicz fuzzy BE-algebra.

Theorem 1. Given the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X, the following assertions are

equivalent.

(i)  Lε
ξ satisfies  Lε

ξ(x ∗ y) ≥ min{ Lε
ξ(x),  Lε

ξ(y)} for all x, y ∈ X.

(ii)  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra of X.

Proof. (i) ⇒ (ii). Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈  Lε
ξ and

[y/tb] ∈  Lε
ξ. Then  Lε

ξ(x) ≥ ta and  Lε
ξ(y) ≥ tb, which implis that

 Lε
ξ(x ∗ y) ≥ min{ Lε

ξ(x),  Lε
ξ(y)} ≥ min{ta, tb}.

Therefore [(x ∗ y)min{tb,tb}] ∈  Lε
ξ, and consequently  Lε

ξ is a  Lukasiewicz fuzzy BE-algebra
of X.

(ii) ⇒ (i). Let x, y ∈ X. It is clear that [x/ Lε
ξ(x)] ∈  Lε

ξ and [y/ Lε
ξ(y)] ∈  Lε

ξ. Hence
[(x ∗ y)/min{ Lε

ξ(x),  Lε
ξ(y)}] ∈  Lε

ξ by (11), that is,  Lε
ξ(x ∗ y) ≥ min{ Lε

ξ(x),  Lε
ξ(y)}.

Proposition 1. If ξ is order preserving or order reversing in X, then its  Lukasiewicz
fuzzy set  Lε

ξ is also order preserving or order reversing in X.

Proof. Straightforward.

In Poposition 1, the converse may not be true as seen in the following example.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 928

Example 2. Consider the BE-algebra X given in Example 1. Its Hasse diagram is given
as follows:

r
b5
JJ 


rb2 rZ
Z

b4

rJJ b3r

b1

r1

(1) Let ξ be a fuzzy set in X defined as follows:

ξ : X → [0, 1], x 7→



0.88 if x = 1,
0.78 if x = b1,
0.63 if x = b2,
0.48 if x = b3,
0.55 if x = b4,
0.47 if x = b5.

Given ε := 0.43, the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is given as follows:

 Lε
ξ : X → [0, 1], x 7→



0.31 if x = 1,
0.21 if x = b1,
0.06 if x = b2,
0.00 if x = b3,
0.00 if x = b4,
0.00 if x = b5.

Then  Lε
ξ is order preversing in X, but ξ is not order preserving in X since b4 ≤ b3 and

ξ(b4) ≥ ξ(b3).
(2) Let ζ be a fuzzy set in X defined as follows:

ζ : X → [0, 1], x 7→



0.34 if x = 1,
0.31 if x = b1,
0.55 if x = b2,
0.48 if x = b3,
0.53 if x = b4,
0.63 if x = b5.

Given δ := 0.62, the  Lukasiewicz fuzzy set  Lδ
ζ of ζ in X is given as follows:

 Lδ
ζ : X → [0, 1], x 7→



0.00 if x = 1,
0.00 if x = b1,
0.17 if x = b2,
0.10 if x = b3,
0.15 if x = b4,
0.25 if x = b5.

Then  Lε
ξ is order reversing in X, but ζ is not order reversing in X since b1 ≤ 1 and

ζ(b1) ≤ ζ(1).



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 929

Lemma 1. If  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra of X, then  Lε

ξ(1) ≥  Lε
ξ(x) for all

x ∈ X.

Proof. It can be induced by (BE1) and Theorem 1.

Proposition 2. If a  Lukasiewicz fuzzy BE-algebra  Lε
ξ of ξ is order reversing in X, then

it is constant.

Proof. Let  Lε
ξ be a  Lukasiewicz fuzzy BE-algebra of X which is order reversing. Since

x ≤ 1 for all x ∈ X, we have  Lε
ξ(x) ≥  Lε

ξ(1) for all x ∈ X. The combination of this and
Lemma 1 induces  Lε

ξ(x) =  Lε
ξ(1) for all x ∈ X. Hence  Lε

ξ is a constant on X.

Theorem 2. If ξ is a fuzzy BE-algebra of X, then its  Lukasiewicz fuzzy set  Lε
ξ is a

 Lukasiewicz fuzzy BE-algebra of X.

Proof. Assume that ξ is a fuzzy BE-algebra of X. Let x, y ∈ X and ta, tb ∈ (0, 1] be
such that [x/ta] ∈  Lε

ξ and [y/tb] ∈  Lε
ξ. Then  Lε

ξ(x) ≥ ta and  Lε
ξ(y) ≥ tb, so

 Lε
ξ(x ∗ y) = max{0, ξ(x ∗ y) + ε− 1}

≥ max{0,min{ξ(x), ξ(y)} + ε− 1}
= max{0,min{ξ(x) + ε− 1, ξ(y) + ε− 1}}
= min{max{0, ξ(x) + ε− 1},max{0, ξ(y) + ε− 1}}
= min{ Lε

ξ(x),  Lε
ξ(y)} ≥ min{ta, tb}.

Hence [(x∗y)/min{ta, tb}] ∈  Lε
ξ, and therefore  Lε

ξ is a  Lukasiewicz fuzzy BE-algebra of X.

The converse of Theorem 2 may not be true as shown in the following example.

Example 3. Consider a set X = {1, b1, b2, b3, b4} with a binary operation “∗” given in
the table below.

∗ 1 b1 b2 b3 b4
1 1 b1 b2 b3 b4
b1 1 1 b2 b3 b4
b2 1 b1 1 b3 b3
b3 1 1 b2 1 b2
b4 1 1 1 1 1

Then (X, ∗, 1) is a BE-algebra (see [7]). Define a fuzzy set ξ in X as follows:

ξ : X → [0, 1], x 7→


0.73 if x = 1,
0.42 if x = b1,
0.59 if x = b2,
0.46 if x = b3,
0.68 if x = b4.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 930

Given ε := 0.41, the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is given as follows:

 Lε
ξ : X → [0, 1], x 7→


0.14 if x = 1,
0.00 if x = b1,
0.00 if x = b2,
0.00 if x = b3,
0.09 if x = b4.

It is routine to verify that  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra of X. But ξ is not a fuzzy

BE-algebra of X because of ξ(b2 ∗ b4) = ξ(b3) = 0.46 ≱ 0.59 = min{ξ(b2), ξ(b4)}.

Theorem 3. Given a BE-subalgebra F of X, define a fuzzy set ξ in X as follows:

ξ : X → [0, 1], x 7→
{

t0 if x ∈ F,
t1 if x /∈ F

(12)

where t0 > t1 in [0, 1]. Then the  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy

BE-algebra of X.

Proof. It is easy to verify that the fuzzy set ξ given in (12) is a fuzzy BE-algebra of
X. Hence the  Lukasiewicz fuzzy set  Lε

ξ of ξ is a  Lukasiewicz fuzzy BE-algebra of X by
Theorem 2.

Proposition 3. If ξ is a fuzzy BE-algebra of X, then its  Lukasiewicz fuzzy set  Lε
ξ satisfies:

(∀x, y ∈ X)
(

 Lε
ξ(y) =  Lε

ξ(1) ⇔  Lε
ξ(x) ≤  Lε

ξ(x ∗ y)
)
. (13)

Proof. If ξ is a fuzzy BE-algebra of X, then its  Lukasiewicz fuzzy set  Lε
ξ is a  Lukasiewicz

fuzzy BE-algebra of X (see Theoem 2). Assume that  Lε
ξ(y) =  Lε

ξ(1) for all y ∈ X. Then

 Lε
ξ(x) = min{ Lε

ξ(x),  Lε
ξ(1)} = min{ Lε

ξ(x),  Lε
ξ(y)} ≤  Lε

ξ(x ∗ y)

for all x, y ∈ X by Theorem 1 and Lemma 1.
Conversely, suppose that  Lε

ξ(x) ≤  Lε
ξ(x ∗ y) for all x, y ∈ X. Then  Lε

ξ(y) =  Lε
ξ(1 ∗ y) ≥

 Lε
ξ(1) by (BE3), and so  Lε

ξ(y) =  Lε
ξ(1) for all y ∈ X.

Theorem 4. If the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X satisfies:

[x/ta] ∈  Lε
ξ, [z/tc] ∈  Lε

ξ ⇒ [(x ∗ y)/min{ta, tc}] ∈  Lε
ξ (14)

for all ta, tc ∈ (0, 1] and x, y, z ∈ X with z ≤ y, then  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra

of X.

Proof. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that [x/ta] ∈  Lε
ξ and [y/tb] ∈  Lε

ξ. Since
y ≤ y for all y ∈ X, it follows from (14) that [(x ∗ y)/min{ta, tb}] ∈  Lε

ξ. Hence  Lε
ξ is a

 Lukasiewicz fuzzy BE-algebra of X.

We consider the conditions for the ∈-set and q-set of  Lukasiewicz fuzzy set to be
BE-subalgebras.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 931

Theorem 5. If  Lε
ξ is the  Lukasiewicz fuzzy set of ξ in X which satisfies:

(∀x, y ∈ X)
(
min{ Lε

ξ(x),  Lε
ξ(y)} ≤ max{ Lε

ξ(x ∗ y), 0.5}
)
, (15)

then the ∈-set ( Lε
ξ, t)∈ of  Lε

ξ is a BE-subalgebra of X for the value t ∈ (0.5, 1].

Proof. Assume that  Lε
ξ satisfies the condition (15) and let x, y ∈ X be such that

x, y ∈ ( Lε
ξ, t)∈ for t ∈ (0.5, 1]. Then  Lε

ξ(x) ≥ t and  Lε
ξ(y) ≥ t, which imply from (15) that

max{ Lε
ξ(x ∗ y), 0.5} ≥ min{ Lε

ξ(x),  Lε
ξ(y)} ≥ t > 0.5.

Hense [(x ∗ y)/t] ∈  Lε
ξ, i.e., x ∗ y ∈ ( Lε

ξ, t)∈, and therefore ( Lε
ξ, t)∈ is a BE-subalgebra of X

for t ∈ (0.5, 1].

Theorem 6. For the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X, if its ∈-set ( Lε

ξ, t)∈ is a BE-
subalgebra of X for the value t ∈ (0.5, 1], then  Lε

ξ satisfies the condition (15).

Proof. Assume that  Lε
ξ does not satisfy the condition (15). Then

min{ Lε
ξ(a),  Lε

ξ(b)} > max{ Lε
ξ(a ∗ b), 0.5}

for some a, b ∈ X, and so s ∈ (0.5, 1] and [a/s], [b/s] ∈  Lε
ξ, i.e., a, b ∈ ( Lε

ξ, s)∈ where
s := min{ Lε

ξ(a),  Lε
ξ(b)}. Since ( Lε

ξ, s)∈ is a BE-subalgebra of X by assumption, we have
a ∗ b ∈ ( Lε

ξ, s)∈ and hence [(a ∗ b)//s] ∈  Lε
ξ, i.e.,

 Lε
ξ(a ∗ b) ≥ s = min{ Lε

ξ(a),  Lε
ξ(b)}.

This is a contradiction. Hence min{ Lε
ξ(x),  Lε

ξ(y)} ≤ max{ Lε
ξ(x ∗ y), 0.5} for all x, y ∈ X,

that is,  Lε
ξ satisfies the condition (15).

Theorem 7. If the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is a  Lukasiewicz fuzzy BE-algebra

of X, then its q-set ( Lε
ξ, t)q is a BE-subalgebra of X for the value t ∈ (0, 1].

Proof. Let t ∈ (0, 1] and x, y ∈ ( Lε
ξ, t)q. Then [x/t] q  Lε

ξ and [y/t] q  Lε
ξ, that is,  Lε

ξ(x) +
t > 1 and  Lε

ξ(y) + t > 1. It follows from Theorem 1 that

 Lε
ξ(x ∗ y) + t ≥ min{ Lε

ξ(x),  Lε
ξ(y)} + t = min{ Lε

ξ(x) + t,  Lε
ξ(y) + t} > 1.

Thus [(x ∗ y)/t] q  Lε
ξ, i.e., x ∗ y ∈ ( Lε

ξ, t)q. Hence ( Lε
ξ, t)q is a BE-subalgebra of X.

Corollary 1. If ξ is a fuzzy BE-algebra of X, then the q-set ( Lε
ξ, t)q of  Lε

ξ is a BE-
subalgebra of X for the value t ∈ (0, 1].

Theorem 8. For the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X, if the q-set ( Lε

ξ, t)q is a BE-
subalgebra of X, then  Lε

ξ satisfies:

x ∈ ( Lε
ξ, ta)q, y ∈ ( Lε

ξ, tb)q ⇒ x ∗ y ∈ ( Lε
ξ,max{ta, tb})∈ (16)

for all x, y ∈ X and ta, tb ∈ (0, 0.5].



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 932

Proof. Assume that the q-set ( Lε
ξ, t)q is a BE-subalgebra of X. Let x, y ∈ X and

ta, tb ∈ (0, 0.5] be such that x ∈ ( Lε
ξ, ta)q and y ∈ ( Lε

ξ, tb)q. Then x, y ∈ ( Lε
ξ,max{ta, tb})q,

and hence x ∗ y ∈ ( Lε
ξ,max{ta, tb})q by hypothesis. It follows that

 Lε
ξ(x ∗ y) > 1 − max{ta, tb} ≥ max{ta, tb}

since max{ta, tb} ≤ 0.5. Therefore [(x∗y)/max{ta, tb}] ∈  Lε
ξ, that is, x∗y ∈ ( Lε

ξ,max{ta, tb})∈.

Theorem 9. If the  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy BE-algebra of X,

then its O-set O( Lε
ξ) is a BE-subalgebra of X.

Proof. Assume that  Lε
ξ is a  Lukasiewicz fuzzy BE-algebra of X and let x, y ∈ O( Lε

ξ).
Then ξ(x) + ε− 1 > 0 and ξ(y) + ε− 1 > 0 which implies that

 Lε
ξ(x ∗ y) ≥ min{ Lε

ξ(x),  Lε
ξ(y)} = min{ξ(x) + ε− 1, ξ(y) + ε− 1} > 0

by Theoem 1. Hence x ∗ y ∈ O( Lε
ξ), and therefore O( Lε

ξ) is a BE-subalgebra of X.

Corollary 2. If ξ is a fuzzy BE-algebra of X, then the O-set O( Lε
ξ) of  Lε

ξ is a BE-
subalgebra of X.

Theorem 10. If the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X satisfies:

x ∈ ( Lε
ξ, ta)∈, y ∈ ( Lε

ξ, tb)∈ ⇒ x ∗ y ∈ ( Lε
ξ,max{ta, tb})q (17)

for all x, y ∈ X and ta, tb ∈ (0, 1], then its O-set O( Lε
ξ) is a BE-subalgebra of X.

Proof. Assume that  Lε
ξ satisfies the condition (17) for all x, y ∈ X and ta, tb ∈ (0, 1].

Let x, y ∈ O( Lε
ξ). Then ξ(x) + ε− 1 > 0 and ξ(y) + ε− 1 > 0. Since x ∈ ( Lε

ξ,  Lε
ξ(x))∈ and

y ∈ ( Lε
ξ,  Lε

ξ(y))∈, it follows from (17) that

x ∗ y ∈ ( Lε
ξ,max{ Lε

ξ(x),  Lε
ξ(y)})q. (18)

If x ∗ y /∈ O( Lε
ξ), then  Lε

ξ(x ∗ y) = 0 and so

 Lε
ξ(x ∗ y) + max{ Lε

ξ(x),  Lε
ξ(y)} = max{ Lε

ξ(x),  Lε
ξ(y)}

= max{max{0, ξ(x) + ε− 1},max{0, ξ(y) + ε− 1}}
= max{ξ(x) + ε− 1, ξ(y) + ε− 1}
= max{ξ(x), ξ(y)} + ε− 1

≤ 1 + ε− 1 = ε ≤ 1,

that is, [(x ∗ y)/max{ Lε
ξ(x),  Lε

ξ(y)}] q  Lε
ξ which shows that (18) is not valid. This is a

contradiction, and thus x ∗ y ∈ O( Lε
ξ). Hence O( Lε

ξ) is a BE-subalgebra of X.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 933

Theorem 11. If the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X satisfies the condition (16) for

all x, y ∈ X and ta, tb ∈ (0, 1], then its O-set O( Lε
ξ) is a BE-subalgebra of X.

Proof. Let x, y ∈ O( Lε
ξ). Then ξ(x) + ε− 1 > 0 and ξ(y) + ε− 1 > 0. Hence

 Lε
ξ(x) + 1 = max{0, ξ(x) + ε− 1} + 1 = ξ(x) + ε− 1 + 1 = ξ(x) + ε > 1

and
 Lε
ξ(y) + 1 = max{0, ξ(y) + ε− 1} + 1 = ξ(y) + ε− 1 + 1 = ξ(y) + ε > 1,

that is, x ∈ ( Lε
ξ, 1)q and y ∈ ( Lε

ξ, 1)q. It follows from (16) that

x ∗ y ∈ ( Lε
ξ,max{1, 1})∈ = ( Lε

ξ, 1)∈.

Thus  Lε
ξ(x ∗ y) + 1 > 1, and so  Lε

ξ(x ∗ y) > 0, i.e., x ∗ y ∈ O( Lε
ξ). Therefore O( Lε

ξ) is a
BE-subalgebra of X.

4.  Lukasiewicz fuzzy BE-filters

Definition 2. The  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is called a  Lukasiewicz fuzzy BE-

filter of X if it satisfies:

x ∈ ( Lε
ξ, ta)∈ ⇒ 1 ∈ ( Lε

ξ, ta)∈, (19)

x ∗ y ∈ ( Lε
ξ, ta)∈, x ∈ ( Lε

ξ, tb)∈ ⇒ y ∈ ( Lε
ξ,min{ta, tb})∈ (20)

for all x, y ∈ X and ta, tb ∈ (0, 1].

Example 4. Consider a set X = {1, b1, b2, b3} with a binary operation “∗” given in the
table below.

∗ 1 b1 b2 b3
1 1 b1 b2 b3
b1 1 1 b2 b2
b2 1 b1 1 b1
b3 1 1 1 1

Then (X, ∗, 1) is a BE-algebra (see [4]). Define a fuzzy set ξ in X as follows:

ξ : X → [0, 1], x 7→


0.73 if x = 1,
0.62 if x = b1,
0.48 if x ∈ {b2, b3}.

Given ε := 0.62, the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is given as follows:

 Lε
ξ : X → [0, 1], x 7→


0.35 if x = 1,
0.24 if x = b1,
0.10 if x ∈ {b2, b3}.

It is routine to verify that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X.



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 934

We discuss relationship between fuzzy BE-filter and  Lukasiewicz fuzzy BE-filter.

Theorem 12. If ξ is a fuzzy BE-filter of X, then its  Lukasiewicz fuzzy set  Lε
ξ is a

 Lukasiewicz fuzzy BE-filter of X.

Proof. Assume that ξ is a fuzzy BE-filter of X and let  Lε
ξ be its  Lukasiewicz fuzzy set

in X. Let x ∈ X and ta ∈ (0, 1] be such that x ∈ ( Lε
ξ, ta)∈. Then

 Lε
ξ(1) = max{0, ξ(1) + ε− 1} ≥ max{0, ξ(x) + ε− 1} =  Lε

ξ(x) ≥ ta,

and so 1 ∈ ( Lε
ξ, ta)∈. Let x, y ∈ X and ta, tb ∈ (0, 1] be such that x ∗ y ∈ ( Lε

ξ, ta)∈ and
x ∈ ( Lε

ξ, tb)∈. Then  Lε
ξ(x ∗ y) ≥ ta and  Lε

ξ(x) ≥ tb, which imply that

 Lε
ξ(y) = max{0, ξ(y) + ε− 1} ≥ max{0,min{ξ(x ∗ y), ξ(x)} + ε− 1}

= max{0,min{ξ(x ∗ y) + ε− 1, ξ(x) + ε− 1}}
= min{max{0, ξ(x ∗ y) + ε− 1},max{0, ξ(x) + ε− 1}}}
= min{ Lε

ξ(x ∗ y),  Lε
ξ(x)} ≥ min{ta, tb}.

Hence [y/min{ta, tb}] ∈  Lε
ξ, that is, y ∈ ( Lε

ξ,min{ta, tb})∈. Therefore  Lε
ξ is a  Lukasiewicz

fuzzy BE-filter of X.

In Theorem 12, the converse may not be true as shown in the following example.

Example 5. Consider the BE-algebra (X, ∗, 1) in Example 4 and let ξ be a fuzzy set in
X defined by

ξ : X → [0, 1], x 7→


0.73 if x = 1,
0.51 if x = b1,
0.62 if x = b2,
0.47 if x = b3.

Then ξ is not a fuzzy BE-filter of X since

ξ(b3) = 0.47 ≱ 0.51 = min{ξ(b1 ∗ b3), ξ(b1)}.

Given ε := 0.49, the  Lukasiewicz fuzzy set  Lε
ξ of ξ in X is calculated as follows:

 Lε
ξ : X → [0, 1], x 7→


0.22 if x = 1,
0.00 if x = b1,
0.11 if x = b2,
0.00 if x = b3,

and it is a  Lukasiewicz fuzzy BE-filter of X.

Theorem 13. The  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy BE-filter of X if

and only if it satisfies:

 Lε
ξ(1) is an upper bound of { Lε

ξ(x) | x ∈ X}, (21)

(∀x, y ∈ X)( Lε
ξ(y) ≥ min{ Lε

ξ(x ∗ y),  Lε
ξ(x)}). (22)



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 935

Proof. Assume that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X. Since x ∈ ( Lε

ξ,  Lε
ξ(x))∈ for

all x ∈ X, it follows from (19) that 1 ∈ ( Lε
ξ,  Lε

ξ(x))∈. Hence  Lε
ξ(1) ≥  Lε

ξ(x) for all x ∈ X,
and thus (21) is valid. Since x ∗ y ∈ ( Lε

ξ,  Lε
ξ(x ∗ y))∈ and x ∈ ( Lε

ξ,  Lε
ξ(x))∈ for all x, y ∈ X,

we have y ∈ ( Lε
ξ,min{ Lε

ξ(x ∗ y),  Lε
ξ(x)})∈ by (20). Hence  Lε

ξ(y) ≥ min{ Lε
ξ(x ∗ y),  Lε

ξ(x)} for
all x, y ∈ X.

Conversely, suppose that  Lε
ξ satisfies (21) and (22). Let x, y ∈ X and ta, tb ∈ (0, 1]. If

x ∈ ( Lε
ξ, ta)∈, then  Lε

ξ(1) ≥  Lε
ξ(x) ≥ ta and so 1 ∈ ( Lε

ξ, ta)∈. Assume that x ∗ y ∈ ( Lε
ξ, ta)∈

and x ∈ ( Lε
ξ, tb)∈. Then  Lε

ξ(x ∗ y) ≥ ta and  Lε
ξ(x) ≥ tb It follows from (22) that  Lε

ξ(y) ≥
min{ Lε

ξ(x∗y),  Lε
ξ(x)} ≥ min{ta, tb}, i.e., [y/min{ta, tb}] ∈  Lε

ξ. Hence y ∈ ( Lε
ξ,min{ta, tb})∈.

Therefore  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X.

Corollary 3. If ξ is a fuzzy BE-filter of X, then its  Lukasiewicz fuzzy set  Lε
ξ satisfies

(21) and (22).

Theorem 14. The  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy BE-filter of X if

and only if it satisfies the condition (21) and

(∀x, y, z ∈ X)( Lε
ξ(x ∗ z) ≥ min{ Lε

ξ(x ∗ (y ∗ z)),  Lε
ξ(y)}). (23)

Proof. Assume that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X. The condition (21) was

verified by the proof of Theorem 13. Using (BE4) and (22), we get

 Lε
ξ(x ∗ z) ≥ min{ Lε

ξ(y ∗ (x ∗ z)),  Lε
ξ(y)} = min{ Lε

ξ(x ∗ (y ∗ z)),  Lε
ξ(y)}.

Conversely, suppose that  Lε
ξ satisfies the conditions (21) and (23). If we take x := 1 in

(23) and use (BE3), then

 Lε
ξ(z) =  Lε

ξ(1 ∗ z) ≥ min{ Lε
ξ(1 ∗ (y ∗ z)),  Lε

ξ(y)} = min{ Lε
ξ(y ∗ z),  Lε

ξ(y)}

for all y, z ∈ X. Therefore  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X by Theorem 13.

Corollary 4. If ξ is a fuzzy BE-filter of X, then its  Lukasiewicz fuzzy set  Lε
ξ satisfies

(23).

Theorem 15. The  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy BE-filter of X if

and only if it satisfies:

(∀x, y ∈ X)
(

 Lε
ξ(x ∗ y) ≥  Lε

ξ(y)
)
, (24)

(∀x, y, z ∈ X)
(

 Lε
ξ((x ∗ (y ∗ z)) ∗ z) ≥ min{ Lε

ξ(x),  Lε
ξ(y)}

)
. (25)

Proof. Assume that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X and let x, y, z ∈ X. Then

 Lε
ξ(x ∗ y) ≥ min{ Lε

ξ(y ∗ (x ∗ y)),  Lε
ξ(y)} = min{ Lε

ξ(x ∗ (y ∗ y)),  Lε
ξ(y)}

= min{ Lε
ξ(x ∗ 1),  Lε

ξ(y)} = min{ Lε
ξ(1),  Lε

ξ(y)} =  Lε
ξ(y)



Y. B. Jun, S. S. Ahn / Eur. J. Pure Appl. Math, 15 (3) (2022), 924-937 936

by (BE1), (BE2), (BE4) and Theorem 13. Also, we have

 Lε
ξ((x ∗ (y ∗ z)) ∗ z) ≥ min{ Lε

ξ((x ∗ (y ∗ z)) ∗ (y ∗ z)),  Lε
ξ(y)}

≥ min{min{ Lε
ξ(x ∗ ((x ∗ (y ∗ z)) ∗ (y ∗ z)),  Lε

ξ(x))},  Lε
ξ(y)}

= min{min{ Lε
ξ(1),  Lε

ξ(x)},  Lε
ξ(y)}

= min{ Lε
ξ(x),  Lε

ξ(y)}

by (3), Theorem 13 and Theorem 14.
Conversely, suppose that  Lε

ξ satisfies (24) and (25). If we take y = x in (24) and use
(BE1), then  Lε

ξ(1) =  Lε
ξ(x ∗ x) ≥  Lε

ξ(x) for all x ∈ X, that is,  Lε
ξ(1) is an upper bound of

{ Lε
ξ(x) | x ∈ X}. The combination of (BE1), (BE3) and (25) induces

 Lε
ξ(y) =  Lε

ξ(1 ∗ y) =  Lε
ξ(((x ∗ y) ∗ (x ∗ y)) ∗ y) ≥ min{ Lε

ξ(x ∗ y),  Lε
ξ(x)}

for all x, y ∈ X. It follows from Theorem 13 that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X.

Corollary 5. If ξ is a fuzzy BE-filter of X, then its  Lukasiewicz fuzzy set  Lε
ξ satisfies

(24) and (25).

Theorem 16. The  Lukasiewicz fuzzy set  Lε
ξ of ξ is a  Lukasiewicz fuzzy BE-filter of X if

and only if it satisfies the condition (21) and

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

ξ(z) ≥ min{ Lε
ξ(x),  Lε

ξ(y)}
)
. (26)

Proof. Assume that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X. The condition (21) was

verified by the proof of Theorem 13. Let x, y, z ∈ X be such that x ∗ (y ∗ z) = 1. Using
Theorem 13, we have

 Lε
ξ(y ∗ z) ≥ min{ Lε

ξ(x ∗ (y ∗ z)),  Lε
ξ(x)} = min{ Lε

ξ(1),  Lε
ξ(x)} =  Lε

ξ(x)

and so  Lε
ξ(z) ≥ min{ Lε

ξ(y ∗ z),  Lε
ξ(y)} ≥ min{ Lε

ξ(x),  Lε
ξ(y)}.

Conversely, suppose that  Lε
ξ satisfies the condition (21) and (26). Since (x∗y)∗(x∗y) =

1 for all x, y ∈ X, we have  Lε
ξ(y) ≥ min{ Lε

ξ(x ∗ y),  Lε
ξ(x)} for all x, y ∈ X. It follows from

Theorem 13 that  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X.

Corollary 6. If ξ is a fuzzy BE-filter of X, then its  Lukasiewicz fuzzy set  Lε
ξ satisfies

(26).

We use BE-filter to create a  Lukasiewicz fuzzy BE-filter.

Theorem 17. Let F be a BE-filter of X and let α, β ∈ (0, 1] with α ≥ β. For every ε,
define the  Lukasiewicz fuzzy set  Lε

ξ of ξ in X as follows:

 Lε
ξ : X → [0, 1], x 7→

{
α if x ∈ F,
β otherwise.

Then  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X.



REFERENCES 937

Proof. Since 1 ∈ F , we have  Lε
ξ(1) = α ≥  Lε

ξ(x) for all x ∈ X. Hence  Lε
ξ(1) is an upper

bound of { Lε
ξ(x) | x ∈ X}. Let x, y ∈ X. If y ∈ F , then  Lε

ξ(y) = α ≥ min{ Lε
ξ(x∗y),  Lε

ξ(x)}.
If y /∈ F , then x ∗ y /∈ F or x /∈ F . Hence

min{ Lε
ξ(x ∗ y),  Lε

ξ(x)} = β =  Lε
ξ(y).

Therefore  Lε
ξ is a  Lukasiewicz fuzzy BE-filter of X by Theorem 13.

Acknowledgements

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

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