

EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 253-260
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Multipolar fuzzy KU-ideals in KU-algebras

Halimah Alshehri

Department of Computer Science and Engineering, Faculty Applied Studies and
Community Service, King Saud University, Riyadh, Saudi Arabia

Abstract. This article presents the idea of an m-polar fuzzy KU-ideal and investigates its charac-
teristics. The relationship between an m-polar fuzzy KU-subalgebra and an m-polar fuzzy KU-ideal
is presented in the discussion

2020 Mathematics Subject Classifications: 03B52, 03E72, 08A72

Key Words and Phrases: KU-algebras, m-Polar fuzzy set, m-Polar fuzzy KU-subalgebra, m-
Polar fuzzy KU-ideal

1. Introduction

A fuzzy set is a useful tool developed by Zadeh [11] for dealing with probabilistic
uncertainty related to perceptions, state inaccuracies, and preferences. Since that time,
fuzzy set theory has gained much attention in a variety of disciplines, including graph
theory, statistics, life and medical sciences, engineering, social sciences, decision-making,
computer networks, robotics automata theory, artificial intelligence, pattern recognition,
and many others. In [8] and [9], a new algebraic structure called KU-algebras was con-
structed. Mostafa et al. [7] introduced the notion of fuzzy KU-ideals of KU-algebras and
then investigated several basic properties related to fuzzy KU-ideals. Recently, Akram
and Sarwar [3] applied the notion of m-polar fuzzy set theory to the graph theory. Also,
Al-Masarwah and Ahmad [4] discussed the notion of m-polar fuzzy sets with an application
to BCK/BCI-algebras. In this study, we will introduce the notions of m-polar fuzzy subal-
gebras and m-polar fuzzy (closed, commutative) ideals and investigate several properties.
This manuscript aims to apply the notion of an m-polar fuzzy set to fuzzy KU-ideal in
KU-algebras. The notions of an m-polar fuzzy KU-ideal were introduced and their prop-
erties were investigated. The relationship between an m-polar fuzzy KU-subalgebra and
an m-polar fuzzy KU-ideal was examined. Moreover, the relationship between an m-polar
fuzzy KU-ideal and the ideal of BCK/BCI-algebras was presented.

DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4618

Email address: haalshehri@ksu.edu.sa (H. Alshehri)

https://www.ejpam.com 253 © 2023 EJPAM All rights reserved.


H. Alshehri / Eur. J. Pure Appl. Math, 16 (1) (2023), 253-260 254

2. Preliminaries

We first recall some elementary aspects which are used in the present paper.
Throughout this paper, X always denotes a KU-algebra without any specifications.

Definition 1 (8). Let X be a nonempty set with a binary operation ∗ and a constant 0,
then (X, ∗, 0) is called a KU -algebra, if for all u, v, w ∈ X the following axioms are hold:
(ku 1) (u ∗ v) ∗ [(v ∗ w)) ∗ (u ∗ w)] = 0 ,
(ku 2) u ∗ 0 = 0,
(ku 3) 0 ∗ u = u,
(ku 4) u ∗ v = 0 and v ∗ u = 0 implies u = v,
(ku 5) u ∗ u = 0

On a KU-algebra (X, ∗, 0) we can define a binary relation ≤ on X by putting: u ≤
v ⇐⇒ v ∗ u = 0. Then (X,≤) is a partially ordered set and 0 is its smallest element.
Thus (X, ∗, 0) satisfies the following conditions: for all u, v, w ∈ X.
(1) : (v ∗ w) ∗ (u ∗ w) ≤ (u ∗ v)
(2): 0 ≤ u
(3): u ≤ v, v ≤ u implies u = v ,
(4): v ∗ u ≤ u.

A subset S of a KU-algebra X is called KU-subalgebra of X, if u, v ∈ S , implies
(u ∗ v) ∈ S. A non- empty subset I of a KU-algebra X is said to be a KU-ideal of X if it
satisfies:
(K1) 0 ∈ I,
(K2) u ∗ (v ∗ w) ∈ I and v ∈ I imply u ∗ w ∈ I for all u, v and w ∈ X.

Theorem 1 (7). In a KU-algebra (X, ∗, 0) , the following axioms are satisfied: for all
u, v, w ∈ X ,
(1) u ≤ v imply v ∗ w ≤ u ∗ w ,
(2) u ∗ (v ∗ w) = v ∗ (u ∗ w) ,for all u, v, w ∈ X ,
(3) ((v ∗ u) ∗ u) ≤ v

Definition 2 (7). Let µ be a fuzzy set on a KU-algebra X, then µ is called a fuzzy KU-
subalgebra of X if µ(u ∗ v) ≥ min{µ(u), µ(v)}, for all u, v ∈ X.

Definition 3 (7). Let X be a KU-algebra. A fuzzy set µ in X is called a fuzzy KU-ideal
of X if it satisfies:
(FK1) µ(0) ≥ µ(u), (FK2) µ(u ∗ w) ≥ min{µ(u ∗ (v ∗ w)), µ(v)}, for all u, v and w ∈ X.

Lemma 1 (7). If A fuzzy KU-subalgebra of X, then µ(0) ≥ µ(u), for all u ∈ X.

Proposition 1 (7). If A fuzzy KU-ideal of X and u ≤ v, then µ(u) ≥ µ(v), for all
u, v ∈ X.


H. Alshehri / Eur. J. Pure Appl. Math, 16 (1) (2023), 253-260 255

Theorem 2 (7). A fuzzy KU-ideal of X is a fuzzy KU-subalgebra of X.

By an m-polar fuzzy set of a set X (see [5]), we mean a function Ô : X → [0, 1]m. The
membership value of every element u ∈ X is denoted by

Ô(u) := {(π1 ◦ Ô)(u), (π2 ◦ Ô)(u), ..., (πm ◦ Ô)(u)},

Where πi : [0, 1]
m → [0, 1] is the i-th projection for all i = 1, 2, ...,m.

Given an m-polar fuzzy set on a set X, we consider the set

U(Ô; r̂) := {u ∈ X|Ô(u) ≥ r̂}

that is,

U(Ô; r̂) := {u ∈ X|(πi ◦ Ô)(u) ≥ ri, i = 1, 2, ...,m},

which is called an m-polar r̂-level cut set of Ô.

Ô(v) =

{
t̂ = (t1, t2, ..., tm) ∈ (0, 1]m ;u = v

0̂ = (0, 0, ....0) ;u ̸= v

and it is denoted by ut̂. We say that u is the support of ut̂, and t̂ is the value of ut̂.

We say that an m-polar fuzzy point ut̂ is contained in an m-polar fuzzy set Ô denoted

by ut̂ ∈ Ô, if Ô(u) ≥ t̂, that is, (πi ◦ Ô)(u) ≥ ti for all i = 1, 2, ...,m.

Definition 4 (4). An m-polar fuzzy set Ô of BCK/BCI-algebra X is called an m-polar
fuzzy subalgebra if the following assertion is valid:

Ô(u ∗ v) ≥ min{Ô(u), Ô(v)}

that is,

(πi ◦ Ô)(u ∗ v) ≥ min{(πi ◦ Ô)(u), (πi ◦ Ô)(v)}

for all u, v ∈ X, i = 1, 2, ...,m.

Definition 5 (4). An m-polar fuzzy set Ô of BCK/BCI-algebra X is called an m-polar
fuzzy ideal if the following assertion is valid:

Ô(0) ≥ Ô(u) ≥ min{Ô(u ∗ v), Ô(v)}

that is,

(πi ◦ Ô)(0) ≥ (πi ◦ Ô)(u) ≥ min{(πi ◦ Ô)(u ∗ v), (πi ◦ Ô)(v)}

for all u, v ∈ X, i = 1, 2, ...,m.


H. Alshehri / Eur. J. Pure Appl. Math, 16 (1) (2023), 253-260 256

3. m-Polar fuzzy KU-subalgebras and KU-ideals

In this section, we introduce the notions of an m-polar fuzzy KU-subalgebras, an m-
polar fuzzy KU-ideals in KU-algebras and investigate some of their related properties.

Definition 6. An m-polar fuzzy set Ô of X is called an m-polar fuzzy KU-subalgebra if
the following assertion is valid for all u, v ∈ X.

Ô(u ∗ v) ≥ min{Ô(u), Ô(v)} (1)

that is,

(πi ◦ Ô)(u ∗ v) ≥ min{(πi ◦ Ô)(u), (πi ◦ Ô)(v)}
for all u, v ∈ X, i = 1, 2, ...,m.

Example 1. Let X = {0, 1, 2, 3, 4} be KU-algebra with a binary operation ∗ defined by the
following table

∗ 0 1 2 3 4

0 0 1 2 3 4

1 0 0 2 3 3

2 0 0 0 1 4

3 0 0 0 0 3

4 0 0 0 0 0

Define a 3-polar fuzzy set Ô = X → [0, 1]3 by:

Ô(u)=


(0.3, 0.4, 0.6) ;u = 0
(0.2, 0.3, 0.2) ;u = 1
(0.1, 0.2, 0.3) ;u = 2
(0.2, 0.3, 0.4) ;u = 3
(0.2, 0.3, 0.5) ;u = 4

It is routine to verify that Ô is a 3-polar fuzzy KU-subalgebra of X.

Theorem 3. Let Ô be an m-polar fuzzy set of X. Then Ô is an m-polar fuzzy KU-
subalgebra of X if and only if U(Ô; r) ̸= ϕ is a KU-subalgebra of X for all r̂ = (r1, r2, ..., rm) ∈
[0, 1]m.

Proof. Assume that Ô is an m-polar fuzzy subalgebra of X and let r̂ ∈ [0, 1]m be
such that U(Ô; r) ̸= ϕ. Let u, v ∈ U(Ô; r̂). Then Ô(u) ≥ r̂ and Ô(v) ≥ r̂. It follows
from definition 6 that Ô(u ∗ v) ≥ min{Ô(u), Ô(v)} ≥ r̂, so that (u ∗ v) ∈ U(Ô; r̂). Hence
U(Ô; r̂) is a subalgebra of X.

Conversely, assume that U(Ô; r̂) is a subalgebra of X. Suppose that there exist u, v ∈ X
such that Ô(u∗v) < min{Ô(u), Ô(v)}. Then there exists r̂ = (r1, r2, ..., rm) ∈ [0, 1]m such
taht Ô(u ∗ v) < r̂ ≤ min{Ô(u), Ô(v)}. It follows that u, v ∈ U(Ô; r̂), but u ∗ v /∈ U(Ô; r̂).
This is a contradiction, and so Ô(u ∗ v) ≥ min{Ô(u), Ô(v)}, ∀u, v ∈ X. Therefore Ô is an
m-polar fuzzy KU-subalgebra of X.


H. Alshehri / Eur. J. Pure Appl. Math, 16 (1) (2023), 253-260 257

Lemma 2. Every m-polar fuzzy subalgebra Ô of X satisfies the following inequality:

(∀u ∈ X)(Ô(0) ≥ Ô(u)) (2)

that is,

(πi ◦ Ô)(0) ≥ (πi ◦ Ô)(u)

for all u ∈ X, i = 1, 2, ...,m.

Proof. Note that u ∗ u = 0 for all u ∈ X. Using definition 6, we have

Ô(0) = Ô(u ∗ u) ≥ min{Ô(u), Ô(u)} = Ô(u).

for all u ∈ X.

Proposition 2. If every an m-polar fuzzy subalgebra Ô of X satisfies the following in-
equality:

(∀u, v ∈ X)(Ô(u ∗ v) ≥ Ô(v)) (3)

Then, Ô(0) = Ô(u)
that is,

(πi ◦ Ô)(u ∗ v) ≥ (πi ◦ Ô)(v)

Then, (πi ◦ Ô)(0) = (πi ◦ Ô)(u), for all u, v ∈ X, i = 1, 2, ...,m.

Proof. Let u ∈ X. Using (ku 2) and (3), we have Ô(u) = Ô(u ∗ 0) ≥ Ô(0).
It follows from Lemma 2 that Ô(0) = Ô(u).

Definition 7. An m-polar fuzzy set Ô of X is called an m-polar fuzzy KU-ideal if the
following conditions are valid:

(∀u ∈ X)(Ô(0) ≥ Ô(u))

(∀u, v, w ∈ X)(Ô(u ∗ w) ≥ min{Ô(u ∗ (v ∗ w)), Ô(v)} (4)

that is,

(∀u ∈ X)((πi ◦ Ô)(0) ≥ (πi ◦ Ô)(u))

(∀u, v, w ∈ X)((πi ◦ Ô)(u ∗ w) ≥ min{(πi ◦ Ô)(u ∗ (v ∗ w)), (πi ◦ Ô)(v)}

for all i = 1, 2, ...,m.

Proposition 3. If Ô is an m-polar fuzzy KU-ideal of X and u ≤ v, then

(Ô(u) ≥ Ô(v))(∀u, v ∈ X) (5)

that is,


H. Alshehri / Eur. J. Pure Appl. Math, 16 (1) (2023), 253-260 258

((πi ◦ Ô)(u) ≥ (πi ◦ Ô)(v))(∀u, v ∈ X, i = 1, 2, ...,m)

Proof. If u ≤ v, then v ∗ u = 0 and (ku 3) 0 ∗ u = u. Since Ô is an m-polar fuzzy
KU-ideal of X, we get

Ô(0 ∗ u) = Ô(u) ≥ min{Ô(0 ∗ (u ∗ v)), O(v)} = min{Ô(0 ∗ 0), O(v)} =
min{Ô(0), O(v)} = Ô(v).

for all u, v ∈ X.

Proposition 4. Let Ô be an m-polar fuzzy KU-ideal of X. If u ∗ v ≤ w, holds in X then,

(Ô(v) ≥ min{Ô(u), Ô(w)})(∀u, v, w ∈ X) (6)

that is,

((πi ◦ Ô)(v) ≥ min{(πi ◦ Ô)(u), (πi ◦ Ô)(w)})(∀u, v, w ∈ X, i = 1, 2, ...,m)

Proof. Assume that the inequality u ∗ v ≤ w, holds in X. Then w ∗ (u ∗ v) = 0 and (4)

Ô(u∗v) ≥ min{Ô(u∗(w∗v)), Ô(w)} = min{Ô(w∗(u∗v)), Ô(w)} = min{Ô(0), Ô(w)} = Ô(w)
(7)

Now,

Ô(0 ∗ v) = Ô(v) = min{Ô(0 ∗ (u ∗ v)), Ô(u)} = min{Ô(u ∗ v), Ô(u)} ≥ min{Ô(w), Ô(u)}

(by using (7)), i.e. Ô(v) ≥ min{Ô(u), Ô(w)}. This completes the proof.

Theorem 4. If Ô is an m-polar fuzzy KU-subalgebra of X satisfies the condition in propo-
sition 4, then Ô is an m-polar fuzzy KU-ideal of X.

Proof. Let Ô be an m-polar fuzzy KU-subalgebra of X satisfies the condition in propo-
sition 4 and by lemma 2. We have Ô(0) ≥ Ô(u) for all u ∈ X. By theorem 1(3), we
have (u ∗ (v ∗ w)) ∗ (u ∗ w) ≤ v, for all u, v, w ∈ X. it follows from proposition 4, that
ô(u ∗w) ≥ min{Ô(u ∗ (v ∗w)), Ô(v)} for all u, v, w ∈ X. Therefore, Ô is an m-polar fuzzy
KU-ideal of X.

Proposition 5. Every m-polar fuzzy KU-ideal of X is an m-polar fuzzy ideal.

Proof. Straightforward.

Proposition 6. If Ô is an m-polar fuzzy KU-ideal of X, then

(Ô(u ∗ (u ∗ v)) ≥ Ô(v))(∀u, v ∈ X) (8)

that is,

((πi ◦ Ô)(u ∗ (u ∗ v)) ≥ (πi ◦ Ô)(v))(∀u, v ∈ X, i = 1, 2, ...,m)


REFERENCES 259

Proof. Let Ô be an m-polar fuzzy KU-ideal of a KU-algebra X and let u, v, w ∈ X.
Taking w = u ∗ v in (4) and using (ku 2), we get

Ô(u ∗ (u ∗ v) ≥ min{Ô(u ∗ (v ∗ (u ∗ v))), Ô(v)}
= min{Ô(u ∗ (u ∗ (v ∗ v))), Ô(v)}

= min{Ô(u ∗ (u ∗ 0)), Ô(v)}
= min{Ô(0)), Ô(v)} = Ô(v)

Theorem 5. If Ô is an m-polar fuzzy KU-ideal of X, then the set B = {u ∈ X : Ô(u) =
Ô(0)} is an m-polar KU-ideal.

Proof. Since 0 ∈ X, then Ô(0) = Ô(0) implies 0 ∈ B, so B ̸= ϕ. Let u ∗ (v ∗ w) ∈ B
and v ∈ B implies Ô(u ∗ (v ∗ w)) = Ô(0) and Ô(v) = Ô(0). Since Ô is an m-polar fuzzy
KU-ideal of X, then

(Ô(u ∗ w)) ≥ min{Ô(u ∗ (v ∗ w)), Ô(v)} = Ô(0).

But Ô(0) ≥ Ô(u∗w). Then Ô(0) = Ô(u∗w), it follows that u∗w ∈ B, for all u, v, w ∈ X.
Hence, the set B is an m-polar KU-ideal.

4. Conclusion

An m-polar fuzzy model is a generalized form of a bipolar fuzzy model. The m-polar
fuzzy models provide more precision, flexibility and compatibility to the system when
more than one agreement is to be dealt with. This article discussed the KU-ideal of KU-
algebras based on m-polar fuzzy sets. The notions of m-polar fuzzy KU-subalgebras and
an m-polar fuzzy KU-ideals were introduced, and several properties were investigated.

5. Compliance with ethical standards

Conflict of interest: The author declares that there is no conflict of interest regarding
the publication of this paper.

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