EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 426-434
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

More Results on Intuitionistic Fuzzy Ideals of
BE-algebras

Mohamed E Elnair1,2

1 Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk
71491, Saudi Arabia
2 Department of Mathematics and Physics, Gezira University, P. O. Box 20, Sudan

Abstract. red This paper explores the intuitionistic fuzzy ideals in BE-algebras and establishes
several new results related to their structure. We investigate the fundamental concepts and prop-
erties of intuitionistic fuzzy ideals and provide characterizations of an intuitionistic fuzzy ideal in
BE-algebras. Our study focuses on examining the fundamental concepts and properties of these
ideals and provides characterizations of intuitionistic fuzzy ideals in BE-algebras.

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

Key Words and Phrases: BE-algebra, Fuzzy BE-algebra, Fuzzy ideal, Intuitionistic fuzzy ideal,
Upper set

1. Introduction

Intuitionistic fuzzy sets, introduced by Atanassov [5–7], have become a significant
tool in dealing with uncertainty and vagueness in real-world situations. The concept of
intuitionistic fuzzy sets extends the notion of fuzzy sets by considering a non-membership
degree in addition to the membership degree. The non-membership degree represents
the extent to which an element does not belong to a particular set, and this degree can
reflect human reasoning more accurately. Since their introduction, numerous mathematical
structures inspired by intuitionistic fuzzy sets have been proposed and investigated [14,
15, 17, 18, 27].

One of the recent areas of research in the field of intuitionistic fuzzy sets is the study
of intuitionistic fuzzy subalgebras [2, 11] and ideals [1, 3, 12, 26] in BE-algebras. BE-
algebras, introduced by Kim and Kim [10], are a generalization of Boolean algebras, in
which the complementation operation is replaced by a weaker negation operation that
satisfies weaker versions of the classical De Morgan’s laws. New concepts on BE-algebras,
fuzzy BE-algebras and intuitionistic fuzzy BE-algebras have been given in [8, 9, 23–25].

DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5030

Email addresses: abomunzir124@gmail.com (Mohamed E Elnair)

https://www.ejpam.com 426 © 2024 EJPAM All rights reserved.



Mohamed E Elnair / Eur. J. Pure Appl. Math, 17 (1) (2024), 426-434 427

Recently, many authors have studied more concepts on subalgebras and ideals in various
algebraic structures [4, 13, 16, 19–22, 28], motivating our interest in the present study.

redThe study of intuitionistic fuzzy ideals of BE-algebras has been an active area of
research in recent years, with several existing studies exploring various aspects of this
topic. However, there is still much more to be explored in this field, and this paper aims
to contribute to this area of study by presenting new results that build upon previous
research.

redWhile the existing studies have provided valuable insights into intuitionistic fuzzy
ideals in BE-algebras, the present study offers new results that further deepen our under-
standing of this topic. By considering the present study, researchers and practitioners in
this field can gain a more comprehensive and up-to-date understanding of intuitionistic
fuzzy ideals and their applications in BE-algebras. This can in turn lead to advancements
in various fields where BE-algebras are used, such as computer science, engineering, and
economics.

Motivated by a lot of work in this direction, in this paper, as a generalization of
fuzzy BE-algebra, we discuss intuitionistic fuzzy ideal theory applied to BE-algebras. We
introduce the notion of intuitionistic fuzzy BE-ideals, and investigate several properties.
We organize this paper as follows: In Section 2, some fundamental notions of BE-algebras
are presented. In Section 3, the notion of intuitionistic fuzzy BE-ideal is defined, and
related properties are investigated with many examples.

2. Preliminaries

Let K(τ) be the class of all algebras of type τ = (2, 0). By a BE-algebra we mean a
system (M ; ∗, 1) ∈ K(τ) in which the following axioms hold (see [10]):

(∀m0 ∈ M) (m0 ∗m0 = 1); (1)

(∀m0 ∈ M) (m0 ∗ 1 = 1); (2)

(∀m0 ∈ M) (1 ∗m0 = m0); (3)

(∀m0,m1,m2 ∈ M) (m0 ∗ (m1 ∗m2) = m1 ∗ (m0 ∗m2)). (exchange) (4)

A relation “≤” on a BE-algebra M is defined by

(∀m0,m1 ∈ M) (m0 ≤ m1 ⇐⇒ m0 ∗m1 = 1). (5)

A BE-algebra (M ; ∗, 1) is said to be transitive (see [1]) if it satisfies:

(∀m0,m1,m2 ∈ M) (m1 ∗m2 ≤ (m0 ∗m1) ∗ (m0 ∗m2)). (6)

A BE-algebra (M ; ∗, 1) is said to be self distributive (see [10]) if it satisfies:

(∀m0,m1,m2 ∈ M) (m0 ∗ (m1 ∗m2) = (m0 ∗m1) ∗ (m0 ∗m2)). (7)

Note that every self distributive BE-algebra is transitive, but the converse is not true
in general (see [1]).



Mohamed E Elnair / Eur. J. Pure Appl. Math, 17 (1) (2024), 426-434 428

A nonempty subset I of a BE-algebra M is called an ideal of M (see [1]) if it satisfies:

(∀m0 ∈ M)(∀α ∈ I)(m0 ∗ α ∈ I); (8)

(∀m0 ∈ M) (∀α, β ∈ I) (α ∗ (β ∗m0)) ∗m0 ∈ I). (9)

A mapping µ : M → [0, 1], where M is an arbitrary nonempty set, is called a fuzzy set
in M . For any fuzzy set µ in M and any t ∈ [0, 1] we define two sets

U(µ; t) = {m0 ∈ M | µ(m0) ≥ t} and L(µ; t) = {m0 ∈ M | µ(m0) ≤ t},

which are called an upper and lower t-level cut of µ and can be used to the characterization
of µ.

Definition 1. A fuzzy set µ in M is called a fuzzy ideal of M if it satisfies:

(∀m0,m1 ∈ M) (µ(m0 ∗m1) ≥ µ(m1)); (10)

(∀m0,m1,m2 ∈ M) (µ((m0 ∗ (m1 ∗m2)) ∗m2) ≥ min{µ(m0), µ(m1)}). (11)

An intuitionistic fuzzy set (IFS) A in M (see [5]) is an object having the form

A = {⟨m0, µA(m0), γA(m0)⟩ | m0 ∈ M} (12)

where the functions µA : M → [0, 1] and γA : M → [0, 1] denote the degree of membership
(namely µA(m0)) and the degree of nonmembership (namely γA(m0)) of each element
m0 ∈ M to the set A, respectively, and

0 ≤ µA(m0) + γA(m0) ≤ 1 (13)

for each m0 ∈ M . For the sake of simplicity, we shall use the symbol A = ⟨M,µA, γA⟩ for
the intuitionistic fuzzy set A = {⟨m0, µA(m0), γA(m0)⟩ | m0 ∈ M}. Obviously, every fuzzy
set A′ corresponds to the following intuitionistic fuzzy set:

A′ = {⟨m0, αA′(m0), 1− αA′(m0)⟩ | m0 ∈ M}. (14)

Obviously, for an IFS A = ⟨M,µA, γA⟩ in M, when

γA(m0) = 1− µ(m0)thatis, µ(m0) + γA(m0) = 1 (15)

for every m0 ∈ M, the IFS A is a fuzzy set. Hence the notion of intuitionistic fuzzy set
theory is a generalization of fuzzy set theory. Let A be an IFS in M and let s, t ∈ [0, 1]

be such that s+ t ≤ 1. Then the set X
(s,t)
A := m0 ∈ M |µ(m0) ≥ s, γA(m0) ≤ t is called an

it (s, t)-level subset of A = M,µAγ(A) Note that

M
(s,t)
A = m0 ∈ M | µ(M) ≥ s, γA(m0) ≤ t

= m0 ∈ M | µ(m0) ≥ s ∩m0 ∈ M | γA(m0) ≤ t

= U(µA; s) ∩ L(γA; t).



Mohamed E Elnair / Eur. J. Pure Appl. Math, 17 (1) (2024), 426-434 429

3. Intuitionistic fuzzy ideals

In what follows, let M denote a BE-algebra unless otherwise specified.

Definition 2. An IFS A in M is called an intuitionistic fuzzy ideal of M if it satisfies:

µ(m0 ∗m1) ≥ µ(m1), γA(m0 ∗m1) ≤ γA(m1), (16)

µ((m0 ∗ (m1 ∗m2)) ∗m2) ≥ min{µ(m0), µ(m1)},

γA((m0 ∗ (m1 ∗m2)) ∗m2) ≤ max{γA(m0), γA(m1)}
(17)

for all m0,m1,m2 ∈ M.

Example 1. red Let M = {1, α, β, γ, λ, 0} be a set with the following Cayley Table1.

Table 1: Cayley Table of the binary operation ∗

∗ 1 α β γ λ 0

1 1 α β γ λ 0
α 1 1 α γ γ λ
β 1 1 1 γ γ γ
γ 1 α β 1 α β
λ 1 1 α 1 1 α
0 1 1 1 1 1 1

Then (M ; ∗, 1) is a BE-algebra (see [10]). Let A be an IFS in M given by

A = ⟨M,
(

1
0.7 ,

α
0.7 ,

β
0.7 ,

γ
0.2 ,

λ
0.2 ,

0
0.2

)
,
(

1
0.1 ,

α
0.1 ,

β
0.1 ,

γ
0.3 ,

λ
0.3 ,

0
0.3⟩.

Then A is an intuitionistic fuzzy ideal of M.

Example 2. Let M = {1, α, β, γ, λ, 0} be the BE-algebra which is given in Example 1.
Let B be an IFS in M given by

B = ⟨⟨M,
(

1
0.6 ,

α
0.6 ,

β
0.3 ,

γ
0.3 ,

λ
0.3 ,

0
0.3

)
,
(

1
0.2 ,

α
0.2 ,

β
0.5 ,

γ
0.5 ,

λ
0.5 ,

0
0.5

)
⟩.

Then B is not an intuitionistic fuzzy ideal of M since

µ((α ∗ (α ∗ β)) ∗ β) < µ(α) = min{µ(α), µ(β)}

and/or
γA((α ∗ (α ∗ β)) ∗ β) > γA(α) = max{γA(α), γA(α)}.

Lemma 1. Every intuitionistic fuzzy ideal A of M satisfies the following inequality:

(∀m0 ∈ M)(µ(1) ≥ µ(m0), γA(1) ≤ γA(m0)). (18)



Mohamed E Elnair / Eur. J. Pure Appl. Math, 17 (1) (2024), 426-434 430

Proof. Using (1) and (16), we have

µ(1) = µ(m0 ∗m0) ≥ µ(m0), γA(1) = γA(m0 ∗m0) ≤ γA(m0)

for all m0 ∈ M.

Proposition 1. If A is an intuitionistic fuzzy ideal of M, then

(∀m0,m1 ∈ M) (µ((m0 ∗m1) ∗m1) ≥ µ(m0), γA((m0 ∗m1) ∗m1) ≤ γA(m0)). (19)

Proof. Taking m1 = 1 and m2 = m1 in (17) and using (3) and Lemma 1, we get

µ((m0 ∗m1) ∗m1) = µ((m0 ∗ (1 ∗m1)) ∗m1) ≥ min{µ(m0), µ(1)} = µ(m0)

and

γA((m0 ∗m1) ∗m1) = γA((m0 ∗ (1 ∗m1)) ∗m1) ≤ max{γA(m0), γA(1)} = γA(m0)

for all m0,m1 ∈ M.

Corollary 1. Every intuitionistic fuzzy ideal A of M is intuitionistic order preserving,
that is, A satisfies:

(∀m0,m1 ∈ M) (m0 ≤ y ⇒ µ(m0) ≤ µ(m1), γA(m0) ≥ γA(m1)). (20)

Proof. Let m0,m1 ∈ M be such that m0 ≤ m1. Then m0 ∗m1 = 1, and so

µ(m1) = µ(1 ∗m1) = µ((m0 ∗m1) ∗m1) ≥ µ(m0)

and
γA(m1) = γA(1 ∗m1) = γA((m0 ∗m1) ∗m1) ≥ µ(m0)

by (3) and (19).

Proposition 2. Let A be an IFS in M which satisfies (18) and

µ(m0 ∗m2) ≥ min{µ(m0 ∗ (m1 ∗m2)), µ(m1)}),

γA(m0 ∗m2) ≤ max{γA(m0 ∗ (m1 ∗m2)), γA(m1)})
(21)

for all m0,m1,m2 ∈ M. Then A is intuitionistic order preserving.

Proof. Let m0,m1 ∈ M be such that m0 ≤ m1. Then m0 ∗m1 = 1, and so

µ(m1) = µ(1 ∗m1) ≥ min{µ(1 ∗ (m0 ∗m1)), µ(m0)}
= min{µ(1 ∗ 1), µ(m0)} = µ(m0)

and

γA(m1) = γA(1 ∗m1) ≤ max{γA(1 ∗ (m0 ∗m1)), γA(m0)}
= max{γA(1 ∗ 1), γA(m0)} = γA(m0)

by (1), (3), (21) and (18).

We give a characterization of fuzzy ideals.



Mohamed E Elnair / Eur. J. Pure Appl. Math, 17 (1) (2024), 426-434 431

Theorem 1. Let M be a transitive BE-algebra. An IFS A in M is an intuitionistic fuzzy
ideal of M if and only if it satisfies conditions (18) and (21).

Proof. Assume that A is an intuitionistic fuzzy ideal of M. By Lemma 1, A satisfies
(18). Since M is transitive, we have

(m1 ∗m2) ∗m2 ≤ (m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2), (22)

i.e., ((m1 ∗m2)∗m2)∗ ((m0 ∗ (m1 ∗m2))∗ (m0 ∗m2)) = 1 for all m0,m1,m2 ∈ M. It follows
from (3), (17) and Proposition 1 that

µ(m0 ∗m2) = µ(1 ∗ (m0 ∗m2))

= µ((((ym1 ∗m2) ∗m2) ∗ ((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2))) ∗ (m0 ∗m2))

≥ min{µ((m1 ∗m2) ∗m2), µ(m0 ∗ (m1 ∗m2))}
≥ min{µ(m0 ∗ (m1 ∗m2)), µ(m1)}

and

γA(m0 ∗m2) = γA(1 ∗ (m0 ∗m2))

= γA((((m1 ∗m2) ∗m2) ∗ ((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2))) ∗ (m0 ∗m2))

≤ max{γA((m1 ∗m2) ∗m2), γA(m0 ∗ (m1 ∗m2))}
≤ max{γA(m0 ∗ (m1 ∗m2)), γA(m1)}.

Hence A satisfies (21). Conversely suppose that A satisfies two conditions (18) and (21).
Using (21), (1), (2) and (18), we have

µ(m0 ∗m1) ≥ min{µ(m0 ∗ (m1 ∗m1)), µ(m1)}
= min{µ(m0 ∗ 1), µ(m1)}
= min{µ(1), µ(m1)} = µ(m1),

(23)

γA(m0 ∗m1) ≤ max{γA(m0 ∗ (m1 ∗m1)), γA(m1)}
= max{γA(m0 ∗ 1), γA(m1)}
= max{γA(1), γA(m1)} = γA(m1),

(24)

µ((m0 ∗m1) ∗m1) ≥ min{µ((m0 ∗m1) ∗ (m0 ∗m1)), µ(m0)}
= min{µ(1), µ(m0)} = µ(m0),

(25)

γA((m0 ∗m1) ∗m1) ≤ max{γA((m0 ∗m1) ∗ (m0 ∗m1)), γA(m0)}
= max{γA(1), γA(m0)} = γA(m0)

(26)

for all m0,m1 ∈ M. Since A is intuitionistic order preserving by Proposition 2, it follows
from (22) that

µ((m1 ∗m2) ∗m2) ≤ µ((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2))

and
γA((m1 ∗m2) ∗m2) ≥ γA((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2))



REFERENCES 432

so from (21), (25) and (26) that

µ((m0 ∗ (m1 ∗m2)) ∗m2) ≥ min{µ(((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2)), µ(m0)}
≥ min{µ((m1 ∗m2) ∗m2), µ(m0)}
≥ min{µ(m0), µ(m1)}

and

γA((m0 ∗ (m1 ∗m2)) ∗m2) ≤ max{γA(((m0 ∗ (m1 ∗m2)) ∗ (m0 ∗m2)), γA(m0)}
≤ max{γA((m1 ∗m2) ∗m2), γA(m0)}
≤ max{γA(m0), γA(m1)}

for all m0,m1,m2 ∈ M. Hence A is an intuitionistic fuzzy ideal of M.

Acknowledgements

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

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