EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5854
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Weakly Regular Semigroups Characterized in Terms
of Cubic Bipolar Fuzzy Ideals

P. Khamrod1, N. Deetae2, T. Gaketem3,∗

1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala
University of Technology Lanna of Phitsanulok, Thailand
2 Department of Statistics, Faculty of Science and Technology, Pibulsongkram Rajabhat
University, Phitsanulok, Thailand
3 Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics,
School of Science, University of Phayao, Phayao 56000, Thailand

Abstract. In this paper, we introduce the concept of cubic bipolar fuzzy subsemigroups and cubic
bipolar fuzzy ideals in the context of semigroups. We explore their fundamental properties and
examine how these structures interact within semigroups. The main thing this study adds is a way
to describe weakly regular semigroups using the features of cubic bipolar fuzzy ideals. Through
a detailed analysis, we establish several key results that highlight the role of these fuzzy ideals in
understanding the algebraic structure of weakly regular semigroups.

2020 Mathematics Subject Classifications: 03E72, 18B40

Key Words and Phrases: Cubic bipolar fuzzy ideal, weakly regular semigroup

1. Introduction

The theory of fuzzy sets was conceptualized by Zadeh in 1965 [1]. This researchers is
used in mathematics and logic but also in medical science, theoretical physics, robotics,
computer science, control engineering, information science, etc. After that time, in 1979,
Kuroki [2] defined the fuzzy semigroup and various kinds of fuzzy ideals in semigroups
and characterized them. Later in 1975, Zadeh [3], extended the concept of fuzzy sets by
interval valued fuzzy sets as a generalization of the notion of fuzzy sets. In 1994 Zhang
[4] introduced the notion of bipolar fuzzy sets with the extension of fuzzy sets whose
membership degree range is enlarged from the interval [0, 1] to [−1, 1], and used them for
modeling and decision analysis. In 2000, Lee [5] used the term bipolar valued fuzzy sets
and applied it to algebraic structures. In 2012, Jun et al. [6] introduced a new notion, a
cubic set, and investigated several properties of cubic setes as well as introducting cubic

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5854

Email addresses:
pkg@rmutl.ac.th (P.Khamrot)natthinee@psru.ac.th (N.Deetae), thiti.ga@up.ac.th (T.Gaketem)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 2 of 14

subsemigroups and cubic left (right) ideals of semigroups. In 2018, Wei et al. [7] studied
the concept of interval valued bipolar fuzzy set with the generalization ofa bipolarr fuzzy
set. It is a study of the values of positive and negative function. Riaz and Tehrim [8]
discussed the concept of cubic bipolar fuzzy sets and some properties.

In this paper, we consider the relationship between of cubic bipolar fuzzy ideals and
interior ideals on semigroups. In the goal results, we characterized weak regualr semigroup
by using of cubic bipolar fuzzy ideals on semigroups.

2. Preliminaries

In this section, we will give some basic definitions and results needed for the next
section.

A subsemigroup of a semigroup S is a non-empty subset T of S such that T2 ⊆ T.
A left (right) ideal of a semigroup S is a non-empty subset T of S such that ST ⊆ T
(TS ⊆ T). By an ideal of a semigroup S, we mean a nonempty subset of T which is
both a left and a right ideal of S. A generalized bi-ideal of a semigroup S is a non-empty
subset T of S such that TST ⊆ T. A subsemigroup T of a semigroup S is called a bi-ideal
(interior ideal) of S if TST ⊆ T (STS ⊆ T).

2.1. Fuzzy sets and Interval valued fuzzy sets

In this subsection, we review the concept of important fuzzy sets and fuzzy sets valued
in intervals.

For any hi ∈ [0, 1], i ∈ F, define

∨
i∈F

hi := sup
i∈F

{hi} and ∧
i∈F

hi := inf
i∈F

{hi}.

We see that for any h1, h2 ∈ [0, 1], we have

h1 ∨ h2 = max{h1, h2} and h1 ∧ h2 = min{h1, h2}.

A fuzzy set ω of a non-empty set S is a function such that ω : S → [0, 1].
Now, we review the concept of interval valued fuzzy sets.
We use CS[0, 1] to denote the set of all closed subintervals in [0, 1], i.e.,

CS[0, 1] = {h := [hl, hu] | 0 ≤ hl ≤ hu ≤ 1}.

We note that [h, h] = {h} for all h ∈ [0, 1]. For h = 0 or 1, we shall denote 0 = [0, 0] =
{0} and 1 = [1, 1] = {1}.

For any two interval numbers h1 and h2 in CS[0, 1], define the operations “⪯”, “=”,
“⋏” “⋎” as follows:

(1) h1 ⪯ h2 if and only if hl1 ≤ hl2 and hu1 ≤ hu2

(2) h1 = h2 if and only if hl1 = hl2 and hu1 = hu2



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 3 of 14

(3) h1 ⋏ h2 = [(hl1 ∧ hl2), (h
u
1 ∧ hu2)]

(4) h1 ⋎ h2 = [(hl1 ∨ hl2), (h
u
1 ∨ hu2)].

If h1 ⪰ h2, we mean h2 ⪯ h1.

Proposition 1. [9] For any elements h1, h2 and h3 in CS[0, 1], the following properties
are satisfied:

(1) h1 ⋏ h1 = h1 and h1 ⋎ h1 = h1,

(2) h1 ⋏ h2 = h2 ⋏ h1 and h1 ⋎ h2 = h2 ⋎ h1,

(3) (h1 ⋏ h2)⋏ h3 = h1 ⋏ (h2 ⋏ h3) and (h1 ⋎ h2)⋎ h3 = h1 ⋎ (h2 ⋎ h3),

(4) (h1 ⋏ h2)⋎ h3 = (h1 ⋎ h3)⋏ (h2 ⋎ h3) and (h1 ⋎ h2)⋏ h3 = (h1 ⋏ h3)⋎ (h2 ⋏ h3),

(5) If h1 ⪯ h3, then, h1 ⋏ h3 ⪯ h2 ⋏ h3 and h1 ⋎ h3 ⪯ h2 ⋎ h3.

For each interval {hi := [hli, h
u
i ] | i ∈ F} be a family of closed subintervals of [0, 1].

Define ⋏
i∈F

hi = [ ∧
i∈F

hli, ∧
i∈F

hui ] and ⋎
i∈F

hi = [ ∨
i∈F

hli, ∨
i∈F

hui ].

Definition 1. [10] An interval valued fuzzy subset (shortly, IVF subset) of S is a function such
that µ : S → CS[0, 1].

Definition 2. [11] For every subset K of set S, an interval valued characteristic function λK of
K is defined to be a function λK : S → CS[0, 1] by

λK(h) =

{
1 if h ∈ K

0 if h /∈ K

for all h ∈ S.

For two IVF subsets µ and λ of a non-empty set S, define

(1) µ ⊑ λ⇔ µ(h) ⪯ λ(h) for all h ∈ F,

(2) µ = λ⇔ µ ⊑ λ and λ ⊑ µ,

(3) (µ ⊓ λ)(h) = µ(h)⋏ λ(h) for all h ∈ F.

For h ∈ S, define Ah := {(k, o) ∈ F× F | h = ko}.
For two IVF sets µ and λ of F, define the product µ⃝ λ as follows : for all h ∈ S,

(µ⃝ λ)(h) =

 ⋎
(k,o)∈Ah

{µ(k)⋏ λ(o)} if Ah ̸= ∅

0 if Ah = ∅.

Definition 3. [11] An IVF set µ of a semigroup S is said to be an

(1) IVF subsemigroup of S, if µ(h1h2) ⪰ µ(h1)⋏ µ(h2) for all h1, h2 ∈ S,

(2) IVF left (right) ideal of S, if µ(h1h2) ⪰ µ(h2) (µ(h1h2) ⪰ µ(h1))for all h1, h2 ∈ S. An IVF
ideal of S if it is both an IVF left ideal and an IVF right ideal of S,



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 4 of 14

(3) IVF genralized bi-ideal of S, if µ(h1h2h3) ⪰ µ(h1)⋏ µ(h3) for all h1, h2, h3 ∈ S,

(4) IVF bi-ideal of S, if µ is an IVF subsemigroup of S and µ(h1h2h3) ⪰ µ(h1) ⋏ µ(h3) for all
h1, h2, h3 ∈ S,

(5) IVF interior ideal of S, if µ is an IVF subsemigroup of S and µ(h1h2h3) ⪰ µ(h2) for all
h1, h2, h3 ∈ S,

(6) IVF quasi ideal of S, if (S⃝ µ) ⊓ (µ⃝S) ⊑ µ, where S is an IVF set of S mapping every
element of S on 1.

2.2. Bipolar fuzzy set

Definition 4. [4] A bipolar fuzzy set (shortly, BF set) ω on S is an object having the form

ω := {(S, ωp(h), ωn(h)) | h ∈ S},

where ωp : S → [0, 1] and ωn : S → [−1, 0].

Remark 1. For the sake of simplicity, we shall use the symbol ω = (S;ωp, ωn) for the BF set
ω = {(S, ωp(h), ωn(h)) | h ∈ S}.

Define products ωp ∗ ψp and ωn ∗ ψn as follows: For h ∈ S

(ωp ∗ ψp)(h) =


∨

(k,o)∈Ah

{ωp(k) ∧ ψp(o)} if Ah ̸= ∅

0 if Ah = ∅

and

(ωn ∗ ψn)(h) =


∧

(k,o)∈Ah

{ωn(k) ∨ ψn(o)} if Ah ̸= ∅

0 if Ah = ∅.

Definition 5. [12] A positive characteristic function and a negative characteristic function of a
non-empty set K of F defined by

χp
K : S → [0, 1], h 7→ χp

K(h) :=

{
1 h ∈ K
0 h /∈ K

and

χn
K : S → [−1, 0], h 7→ χn

K(h) :=

{
−1 h ∈ K
0 h /∈ K.

respectively.

Definition 6. [12] A BF set ω = (S;ωp, ωn) on a semigroup S is called a

(1) BF subsemigroup on S, if ωp(h1h2) ≥ ωp(h1) ∧ ωp(h2) and ωn(h1h2) ≤ ωn(h1) ∨ ωn(h2) for
all h1, h2 ∈ S,

(2) BF left (right) ideal on S, if ωp(h1h2) ≥ ωp(h2) (ωp(h1h2) ≥ ωp(h1)) and ωn(h1h2) ≤ ωn(h2)
(ωn(h1h2) ≤ ωn(h1)) for all h1, h2 ∈ S,

(3) BF generalized bi-ideal on S, if ωp(h1h2h3) ≥ ωp(h1) ∧ ωp(h3) and ωn(h1h2h3) ≤ ωn(h1) ∨
ωn(h3) for all h1, h2, h3 ∈ S,



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 5 of 14

(4) BF bi-ideal on S, if ω = (S;ωp, ωn) is a BF subsemigroup of S, ωp(h1h2h3) ≥ ωp(h1)∧ωp(h3)
and ωn(h1h2h3) ≤ ωn(h1) ∨ ωn(h3) for all h1, h2, h3 ∈ S,

(5) BF interior ideal on S, if ω = (S;ωp, ωn) is a BF subsemigroup of S, ωp(h1h2h3) ≥ ωp(h2)
and ωn(h1h2h3) ≤ ωn(h2) for all h1, h2, h3 ∈ S,

(6) BF quasi ideal on S, if (Sp ∗ ωp)∩ (ωp ∗Sp) ⊆ ωp and (Sn ∗ ωn)∩ (ωn ∗Sn) ⊇ ωn, where S
is an BF set of S mapping every element of S on [−1, 1].

Definition 7. [7] An interval valued bipolar fuzzy set (shortly, IVBF set)C = (S;µp, µn) of a
non-empty set F if µp : S → CS[0, 1] and µn : S → CS[−1, 0].

2.3. Cubic sets

Definition 8. [6] A cubic set C of a non-empty set S is a structure of the form

C = {⟨h, µ(h), ω(h)⟩ | h ∈ S}

and denoted by C = ⟨µ, ω⟩ where µ is an IVF set and ω is a fuzzy set. In this case, we use

C(h) = ⟨µ(h), ω(h)⟩ = ⟨[µn(h), µp(h)], ω(h)⟩

for all r ∈ S.

Definition 9. [6] A cubic set C = ⟨µ, ω⟩ of S is called

(1) a cubic subsemigroup of S, if µ(h1h2) ⪰ µ(h1) ⋏ µ(h2) and ω(h1h2) ≤ ω(h1) ∨ (h2) for all
h1, h2 ∈ S,

(2) a cubic left(right)ideal of S, if µ(h1h2) ⪰ µ(h2)(µ(h1h2) ⪰ µ(h1)) and ω(h1h2) ≤ ω(h2)(ω(h1h2) ≤
ω(h1)) for all h1, h2 ∈ S. A cubic ideal of S, if it is a cubic left ideal and a cubic right ideal
of S,

(3) a cubic genralized bi ideal of S, if µ(h1h2h3) ⪰ µ(h1)⋏ µ(h3) and ω(h1h2h3) ≤ ω(h1) ∨ ω(h3)
for all h1, h2, h3 ∈ S,

(4) a cubic bi ideal of S, if µ is a cubic subsemigorup of S and µ(h1h2h3) ⪰ µ(h1) ⋏ µ(h3) and
ω(h1h2h3) ≤ ω(h1) ∨ ω(h3) for all h1, h2, h3 ∈ S,

(5) a cubic interior ideal of S, if µ is a cubic subsemigorup of S, µ(h1h2h3) ⪰ µ(h2) and
ω(h1h2h3) ≤ ω(h2) for all h1, h2, h3 ∈ S,

(6) a cubic quasi ideal of S, if (S⃝ µ) ∩ (µ⃝S) ⊑ µ and (S ∗ ω) ∩ (ω ∗S) ⊇ ω.

Riaz and Tehrim [8] discussed the concept of cubic bipolar fuzzy sets and some properties.
In this paper, we consider the concepts of cubic bipolar fuzzy subsemigroups and types of cubic
bipolar fuzzy ideals. We provide properties of cubic bipolar fuzzy subsemigroups and quasi ideals.
In the important results, regular and intra-regular semigroups are characterized in terms of cubic
bipolar fuzzy quasi ideals are provided.



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 6 of 14

3. Cubic bipolar fuzzy subsemigroup and ideals in semigroups

In this part, we give the concepts of cubic bipolar fuzzy subsemigroups and ideals in semigroup.
And we study important properties for reference in the next part.

Definition 10. A cubic bipolar set (shortly, CB set) C̈ of a set S if

C̈ = {⟨h, (µp(h), µn(h)), ω(h)⟩ | h ∈ S}

and denoted by C̈ = ⟨µ, ω⟩ where µ is an IVBF set and ω is a BF set.

Definition 11. A CB set C̈ = ⟨µ, ω⟩ of a semigroup S is called a cubic bipolar fuzzy subsemigroup
(shortly, CBF subsemigroup) of S if
µp(h1h2) ⪰ µp(h1)⋏ µp(h2), µ

n(h1h2) ⪯ µn(h1)⋎ µn(h2) and
ωp(h1h2) ≥ ωp(h1) ∧ ωn(h2), ω

n(h1h2) ≤ ωn(h1) ∨ ωn(h2) for all h1, h2 ∈ S.

Example 1. Let S be a semigroup defined by the following table:

· a b c
a a b c
b b b c
c c c b

Thus, a CB set C̈ = ⟨µ, ω⟩ in F as follows: µp(a) = [0.6, 0.7], µp(b) = [0.4, 0.5], µp(c) = [0.1, 0.2],
µn(a) = [−0.9,−0.8], µn(b) = [−0.7,−0.6], µp(c) = [−0.3,−0.2] and ωp(a) = 0.7, ωp(b) =
0.4, ωp(c) = 0.2, ωn(a) = −0.7, ωn(b) = −0.3, ωn(c) = −0.2. Thus, C̈ = ⟨µ, ω⟩ is a CBF
subsemigroup of S.

Definition 12. A CB set C̈ = ⟨µ, ω⟩ of a semigroup S is called

(1) a cubic bipolar fuzzy left ideal (shortly, CBF left ideal) of S, if
µp(h1h2) ⪰ µp(h2), µ

n(h1h2) ⪯ µn(h2) and ωp(h1h2) ≥ ωp(h2), ω
n(h1h2) ≤ ωn(h2) for all

h1, h2 ∈ S,

(2) a cubic bipolar fuzzy right ideal (shortly, CBF rihgt ideal) of S, if
µp(h1h2) ⪰ µp(h1), µ

n(h1h2) ⪯ µn(h1) and ωp(h1h2) ≥ ωp(h1), ω
n(h1h2) ≤ ωn(h1) for all

h1, h2 ∈ S,

(3) a cubic bipolar fuzzy ideal (shortly, CBF ideal) of S, if it is a CBF left ideal and a CBF right
ideal of S,

(4) a cubic bipolar fuzzy generalized bi-ideal (shortly, CBF generalized bi-ideal) of S, if µp(h1h2h3) ⪰
µp(h1)⋏µp(h3), µ

n(h1h2h3) ⪯ µn(h1)⋎µn(h3) and ω
p(h1h2h3) ≥ ωp(h1)∧ωn(h3), ω

n(h1h2h3) ≤
ωn(h1) ∨ ωn(h3) for all h1, h2, h3 ∈ S,

(5) a cubic bipolar fuzzy bi-ideal (shortly, CBF bi-ideal) of S, if C̈ = ⟨µ, ω⟩ is a CBF subsemigroup
of S, µp(h1h2h3) ⪰ µp(h1)⋏µp(h3) µ

n(h1h2h3) ⪯ µn(h1)⋎µn(h3) and ω
p(h1h2h3) ≥ ωp(h1)∧

ωn(h3), ω
n(h1h2h3) ≤ ωn(h1) ∨ ωn(h3) for all h1, h2, h3 ∈ S.

Next, we study the subset and product of the CBF set as defined.
Let C̈1 = ⟨µ, ω⟩ and C̈2 = ⟨λ, ψ⟩ are CBF sets of a semigroup S. Define

(i) C̈1⊏C̈2 if and only if µp(h) ⪯ λ
p
(h), µn(h) ⪰ λ

n
(h) and ωp(h) ≤ ψp(h), ωn(h) ≥ ψn(h), for

all h ∈ S.



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 7 of 14

(ii) C̈1⊓C̈2 = ⟨µ ⊓ λ, ω ∩ ψ⟩ if and only if (µp ⊓ λp)(h) = (µp(h)⋏ λ
p
(h)),

(µn ⊓ λn)(h) = (µn(h)⋎ λ
n
(h)) and (ωp ∩ ψp)(h) = (ωp(h) ∧ ψp(h)),

(ωn ∩ ψn)(h) = (ωn(h) ∨ ψn(h) for all h ∈ S.

(iii) C̈1 ⊛ C̈2 = ⟨µ⃝ λ, ω ∗ ψ⟩ and define µ⃝ λ as follows. For h ∈ S

(µp ⃝ λ
p
)(h) =

 ⋎
(k,o)∈Ah

{µp(k)⋏ λ
p
(o)} if Ah ̸= ∅

0 if Ah = ∅,

(µn ⃝ λ
n
)(h) =

 ⋏
(k,o)∈Ah

{µn(k)⋎ λ
n
(o)} if Ah ̸= ∅

0 if Ah = ∅,

and ω ∗ ψ is a product of a BF set.

Definition 13. A cubic biopolar set C̈ = ⟨µ, ω⟩ of S is called a cubic biopolar fuzzy quasi ideal
(shortly, CBF quasi ideal) of S, if (S⃝ µ) ⊓ (µ⃝S)⊏µ and (S ∗ ω) ∩ (ω ∗S) ⊆ ω.

The follows theorem are basic propeties of CBF ideal of a semigroup S.

Theorem 1. Every CBF ideal of a semigroup S is a CBF interior ideal of S.

Proof. Let C̈ = ⟨µ, ω⟩ be a CBF ideal of S and Let h1, h2 ∈ S. Then, C̈ = ⟨µ, ω⟩ is a
CBF left ideal and CBF right ideal of S. Thus, µp(h1h2) ⪰ µp(h2), µ

n(h1h2) ⪯ µp(h2) and
ωp(h1h2) ≥ ωn(h2), ω

n(h1h2) ≤ ωn(h2).
Hence, µp(h1h2) ⪰ µp(h1)⋏ µp(h2), µ

n(h1h2) ⪯ µp(h1)⋏ µp(h2) and
ωp(h1h2) ≥ ωn(h2) ∧ ωn(h2), ω

n(h1h2) ≤ ωn(h1) ∨ ωn(h2).
This shows that C̈ = ⟨µ, ω⟩ is a CBF subsemigroup of S. Let h1, h2, h3 ∈ S. Then,

µp(h1h2h3) = µp(h1(h2h3)) ⪰ µp(h2h3) ⪰ µp(h2),

µn(h1h2h3) = µn(h1(h2h3)) ⪯ µn(h2h3) ⪯ µn(h2)

and
ωp(h1h2h3) = ωp(h1(h2h3)) ≥ ωp(h2h3) ≥ ωp(h2),

ωn(h1h2h3) = ωn(h1(h2h3)) ≤ ωn(h2h3) ≥ ωnωn(h2).

Thus, C̈ = ⟨µ, ω⟩ is a CBF interior ideal of S.

Theorem 2. Every CBF quasi ideal of a semigroup S is a CBF bi-ideal of S.

Proof. Assume that C̈ = ⟨µ, ω⟩ is a CBF quasi ideal of S and h1, h2 ∈ S. Then,

µp(h1h2) ⪰ (µp ⃝S
p
)(h1h2)⋏ (S

p ⃝ µp)(h1h2)

= ⋎
(i,j)∈Ah1h2

{µp(i)⋏S
p
(j)}⋏ ⋎

(k,o)∈Ah1h2

{Sp
(k)⋏ µp(o)}

⪰ µp(h1)⋏S
p
(h2)⋏S

p
(h1)⋏ µp(h2)

= µp(h1)⋏ 1⋏ 1⋏ µp(h2) = µp(h1)⋏ µp(h2),

µn(h1h2) ⪯ (µn ⃝S
n
)(h1h2)⋎ (S

n ⃝ µn)(h1h2)

= ⋏
(i,j)∈Ah1h2

{µn(i)⋎S
n
(j)}⋎ ⋏

(k,o)∈Ah1h2

{Sn
(k)⋎ µn(o)}

⪯ µn(h1)⋎S
n
(h2)⋏S

n
(h1)⋏ µn(h2)

= µn(h1)⋎−1⋏−1⋎ µp(h2) = µp(h1)⋎ µp(h2).



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 8 of 14

And
ωp(h1h2) ≥ (ωp ∗Sp)(h1h2) ∧ (Sp ∗ ωp)(h1h2)

=
∨

(i,j)∈Ah1h2

{ωp(i) ∧ Fp(j)} ∧
∨

(k,o)∈Ah1h2

{Sp(k) ∧ ωp(o)}

≥ ωp(h1) ∧Sp(h2) ∧Sp(h1) ∧ ωp(h2)
= ωp(h1) ∧ 1 ∧ 1 ∧ ωp(h2) = ωp(h1) ∧ ωp(h2),

ωn(h1h2) ≤ (ωn ∗Sn)(h1h2) ∨ (Sn ∗ ωn)(h1h2)
=

∧
(i,j)∈Ah1h2

{ωp(i) ∨Sn(j)} ∨
∧

(k,o)∈Ah1h2

{Sn(k) ∨ ωn(o)}

≤ ωn(h1) ∨Sn(h2) ∨Sn(h1) ∨ ωn(h2)
= ωn(h1) ∨ −1 ∨ −1 ∨ ωn(h2) = ωn(h1) ∧ ωn(h2).

Thus, µp(h1h2) ⪰ µp(h1)⋏ µp(h2), µ
n(h1h2) ⪯ µn(h1)⋎ µn(h2) and

ωp(h1h2) ≥ ωp(h1) ∧ ωp(h2), ω
n(h1h2) ≤ ωn(h1) ∨ ωn(h2).

Hence, C̈ = ⟨µ, ω⟩ is a CBF subsemigroup of S.
In a similar way, let h1, h2, h3 ∈ S we get that

µp(h1h2h3) ⪰ (µp ⃝S
p
)(h1h2h3)⋏ (S

p ⃝ µp)(h1h2h3)

= ⋎
(i,j)∈Ah1h2h3

{µp(i)⋏S
p
(j)}⋏ ⋎

(k,o)∈Ah1h2h3

{Sp
(k)⋏ µp(o)}

⪰ µp(h1)⋏G
p
(h2h3)⋏G

p
(h1h2)⋏ µp(h3)

= µp(h1)⋏ 1⋏ 1⋏ µp(h3) = µp(h1)⋏ µp(h3),

µn(h1h2h3) ⪯ (µn ⃝S
n
)(h1h2h3)⋎ (S

n ⃝ µn)(h1h2h3)

= ⋏
(i,j)∈Ah1h2h3

{µn(i)⋎ F
n
(j)}⋎ ⋏

(k,o)∈Ah1h2h3

{Fn
(k)⋎ µn(o)}

⪯ µn(h1)⋎S
n
(h2r3)⋏S

n
(h1h2)⋏ µn(h3)

= µn(h1)⋎−1⋏−1⋎ µp(h3) = µp(h1)⋎ µp(h3).

And
ωp(h1h2h3) ≥ (ωp ∗ Fp)(h1h2h3) ∧ (Sp ∗ ωp)(h1h2h3)

=
∨

(i,j)∈Ah1h2h3

{ωp(i) ∧Sp(j)} ∧
∨

(k,o)∈Ah1h2h3

{Fp(k) ∧ ωp(o)}

≥ ωp(h1) ∧Sp(h2h3) ∧Sp(h1h2) ∧ ωp(h3)
= ωp(h1) ∧ 1 ∧ 1 ∧ ωp(h3) = ωp(h1) ∧ ωp(h3),

ωn(h1h2h3) ≤ (ωn ∗Sn)(h1h2h3) ∨ (Sn ∗ ωn)(h1h2h3)
=

∧
(i,j)∈Ah1h2h3

{ωp(i) ∨ Fn(j)} ∨
∧

(k,o)∈Ah1h2h3

{Sn(k) ∨ ωn(o)}

≤ ωn(h1) ∨ Fn(h2h3) ∨Sn(h1h2) ∨ ωn(h3)
= ωn(h1) ∨ −1 ∨ −1 ∨ ωn(h3) = ωn(h1) ∧ ωn(h3).

Thus, µp(h1h2h3) ⪰ µp(h1)⋏ µp(h3), µ
n(h1r2h3) ⪯ µn(h1)⋎ µn(h3) and

ωp(h1h2) ≥ ωp(h1) ∧ ωp(h3), ω
n(h1h2) ≤ ωn(h1) ∨ ωn(h3).

Hence, C̈ = ⟨µ, ω⟩ is a CBF bi-ideal of S.

Next, we review the definition of the characteristic cubic bipolar fuzzy function. Let T be a non-
empty subset of S. The characteristic cubic bipolar fuzzy set (shortly, CCBF set) χT = ⟨µχT

, ωχT
⟩

is defined as follows:

µp
χT

(h) =

{
1 if h ∈ T

0 if h /∈ T
, µn

χT
(h) =

{
−1 if h ∈ T

0 if h /∈ T



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 9 of 14

for all h ∈ S and ωχT
is a characteristic bipolar fuzzy set.

In the following theorems, we give a relationship between a left ideal (right ideal, generalized
bi-ideal, bi-ideal, interior ideal, quasi-ideal) and the CCBF function.

Theorem 3. Let T be a non-empty subset of a semigroup S. Then, T is a left ideal (right ideal,
generalized bi-ideal, bi-ideal, interior ideal, quasi-ideal) of S if and only if χT = ⟨µχT

, ωχT
⟩ is a

CBF left ideal (right ideal, generalized bi-ideal, bi-ideal, interior ideal, quasi-ideal) of S.

Proof. (⇒) Suppose that T is a left ideal of S and let h1, h2 ∈ S.
If h2 ∈ T, then, h1h2 ∈ T. Thus, 1 = µp

χT
(h2) = µp

χT
(h1h2), −1 = µn

χT
(h2) = µn

χT
(h1h2) and

1 = ωp
χT

(h2) = ωp
χT

(h1h2), −1 = ωn
χT

(h2) = ωn
χT

(h1h2).
Hence, µp

χT
(h1h2) ⪰ µp

χT
(h2), µ

n
χT

(h1h2) ⪯ µn
χT

(h2) and ωp
χT

(h1h2) ≥ ωp
χT

(h2), ω
n
χT

(h1h2) ≤
∨ωn

χT
(h2).

If h2 /∈ T, then, µp
χT

(h1h2) ⪰ µp
χT

(h2), µ
n
χT

(h1h2) ⪯ µn
χT

(h2) and ωp
χT

(h1h2) ≥ ωp
χT

(h2),
ωn
χT

(h1h2) ≤ ωn
χT

(h2).
Thus, χT = ⟨µχT

, ωχT
⟩ is a CBF left ideal of S.

⇐ Suppose that χT = ⟨µχT
, ωχT

⟩ is a CBF left ideal of S and let h2 ∈ T. Then, µp
χT

(h2) = 1,

µn
χT

(h2) = −1 and ωp
χT

(h2) = 1, ωn
χT

(h2) = −1. If h1h2 /∈ T, then, µp
χT

(h1h2) = 0 = µn
χT

(h1h2) and
ωp
χT

(h1h2) = 0 = ωn
χT

(h1h2). Thus,

0 = µp
χT

(h1h2) ⪰ µp
χT

(h2) = 1, 0 = µn
χT

(h1h2) ⪯ µn
χT

(h2) = −1

and
0 = ωp

χT
(h1h2) ≥ ωp

χT
(h2) = 1, 0 = ωn

χT
(h1h2) ≤ ωn

χT
(h2) = −1.

It is a contradiction. Hence, h1h2 ∈ T. Therefore T is a left ideal of S.

4. Characterizations of weakly regular semigroups in terms of cubic
bipolar fuzzy ideals.

In this section, we will characterize weakly regular semigroups in terms of CBF subsemigroups.

Theorem 4. Let M and N be a non-empty subsets of a semigroup S. Then,

(1) χM ⊛ χN = χMN i.e. ⟨µχM
⃝ µχN

, ωχM
∗ ωχN

⟩ = ⟨µχMN
, ωχMN

⟩

(2) χM⊓χN = χM⊓N i.e. ⟨µχM
⊓ µχN

, ωχM
∩ ωχN

⟩ = ⟨µχM⊓N
, ωχM∩N

⟩

On the basis of Lemma 1, we can prove Theorem 5.

Lemma 1. If C̈1 = ⟨µ, ω⟩ is a CBF right ideal and C̈2 = ⟨λ, ψ⟩ is a CBF left ideal of S, then,
C̈1 ⊛ C̈2⊏C̈1⊓C̈2.

Proof. Assume that C̈1 = ⟨µ, ω⟩ is a CBF right ideal and C̈2 = ⟨λ, ψ⟩ is a CBF left ideal of S
and let h ∈ S.

If Ah = ∅, then, it is easy to verify that, (µp ⃝ λ
p
)(h) ⪯ (µp ⊓ λp)(h),

(µn ⃝ λ
n
)(h) ⪰ (µn ⊓ λn)(h) and (ωp ∗ ψp)(h) ≤ (ωp ∩ ψp)(h), (ωn ∗ ψn)(h) ≥ (ωn ∩ ψn)(h).

If Ah ̸= ∅, then,

(µp ⃝ λ
p
)(h) = ⋎

(k,o)∈Ah

{µp(k)⋏ λ
p
(o)} ⪯ ⋎

(k,o)∈Ah

{µp(ko)⋏ λ
p
(ko)}

= µp(h)⋏ λ
p
(h) = (µp ⊓ λp)(h),



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 10 of 14

(µn ⃝ λ
n
)(h) = ⋏

(k,o)∈Ah

{µn(k)⋎ λ
n
(o)} ⪰ ⋏

(k,o)∈Ah

{µn(ko)⋎ λ
n
(ko)}

= µn(h)⋎ λ
n
(h) = (µn ⊓ λn)(h),

and
(ωp ∗ ψp)(h) =

∨
(k,o)∈Ah

{ωp(k) ∧ ψp(o)} ≤
∨

(k,o)∈Ah

{ωp(ko) ∧ ψp(ko)}

= ωp(h) ∧ ψp(h) = (ωp ∩ ψp)(h),

(ωn ∗ ψn)(h) =
∧

(k,o)∈Ah

{ωn(k) ∨ ψn(o)} ≥
∧

(k,o)∈Ah

{ωp(ko) ∨ ψn(ko)}

= ωn(h) ∨ ψn(h) = (ωn ∩ ψn)(h).

Thus, (µp ⃝ λ
p
)(h) ⪯ (µp ⊓ λp)(h), (µn ⃝ λ

n
)(h) ⪰ (µn ⊓ λn)(h) and

(ωp ∗ ψp)(h) ≤ (ωp ∩ ψp)(h), (ωn ∗ ψn)(h) ≥ (ωn ∩ ψn)(h). Hence, C̈1 ⊛ C̈2⊏C̈1⊓C̈2.

Any way, in the proof of Theorem 5, these are used.

Definition 14. [13] A semigroup S is called weakly regular if for every h ∈ S, h ∈ (hS)2.

Lemma 2. [13] A monoid S is weakly regular if and only if R ∩ J = RJ for every right ideal R
and every ideal J of S.

Now we characterize weakly regular semigroups in terms of generalized IVF ideals.

Theorem 5. A monoid S is weakly regular if and only if C̈1 ⊛ C̈2 = C̈1⊓C̈2 for every CBF right
ideal C̈1 = ⟨µ, ω⟩ and every CBF ideal C̈2 = ⟨λ, ψ⟩ of S.

Proof. Assume that C̈1 = ⟨µ, ω⟩ is a CBF right ideal and C̈2 = ⟨λ, ψ⟩ is a CBF ideal of S.
Let h ∈ S. Since S is weakly regular, there exist p, q ∈ S such that h = hphq. Thus,

(µp ⃝ λ
p
)(h) = ⋎

(k,o)∈Ah

{µp(k)⋏ λ
p
(o)} = ⋎

(k,o)∈Ahphq

{µp(k)⋏ λ
p
(o)}

⪰ µp(hp)⋏ λ
p
(hq) ⪰ µp(h)⋏ λ

p
(h) = (µp ⊓ λp)(h),

(µn ⃝ λ
n
)(h) = ⋏

(k,o)∈Ah

{µn(k)⋎ λ
n
(o)} = ⋏

(k,o)∈Ahphq

{µn(k)⋎ λ
n
(o)}

⪯ µn(hp)⋎ λ
n
(hq) ⪯ µn(h)⋎ λ

n
(h) = (µn ⊓ λn)(h),

and
(ωp ∗ ψp)(h) =

∨
(k,o)∈Ah

{ωp(k) ∧ ψp(o)} =
∨

(k,o)∈Ahphq

{ωp(k) ∧ ψp(k)}

≥ ωp(hp) ∧ ψp(hq) ≥ ωp(h) ∧ ψp(h) = (ωp ∩ ψp)(h),

(ωn ∗ ψn)(h) =
∧

(k,o)∈Ah

{ωn(k) ∨ ψn(o)} =
∧

(k,o)∈Ahphq

{ωn(ko) ∨ ψn(ko)}

≤ ωn(hp) ∨ ψn(hq) ≤ ωn(h) ∨ ψn(h) = (ωn ∩ ψn)(h).

Hence, (µp ⊓λp)(h) ⪯ (µp ⃝λ
p
)(h), (µn ⊓λn)(h) ⪰ (µn ⃝λ

n
)(h) and (ωp ∩ψp)(h) ≤ (ωp ∗ψp)(h),

(ωn ∩ ψn)(h) ≥ (ωn ∗ ψn)(h). Therefore, C̈1⊓C̈2⊏C̈1 ⊛ C̈2.

On the other hand, since C̈2 is a CBF ideal of S we see that C̈2 is a CBF left ideal of S. Thus,
by Lemma 1, C̈1 ⊛ C̈2⊏C̈1⊓C̈2. Hence, C̈1 ⊛ C̈2 = C̈1⊓C̈2.

Conversely, let R and J be a right ideal and ideal of G. Then, by Theorem 3, χR and χJ CBF
right ideal and CBF ideal of S. By supposition and Lemma 4, we have



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 11 of 14

µp
χRJ

(h) = (µp
χR

⃝ µp
χJ

)(h) = (µp
χR

⊓ µp
χJ

)(h) = µp
χR⊓J

(h) = 1,

µn
χRJ

(h) = (µp
χR

⃝ µp
χJ

)(h) = (µn
χR

⊓ µp
χJ

)(h) = µn
χR⊓J

(h) = −1,

and
ωp
χRJ

(h) = (ωp
χR

∗ ωp
χJ

)(h) = (ωp
χR

∩ ωp
χJ

)(h) = ωp
χR∩J

(h) = 1,

ωn
χRJ

(h) = (ωn
χR

∗ ωn
χJ

)(h) = (ωn
χR

∩ ωn
χJ

)(h) = ωn
χR∩J

(h) = −1.

Thus, h ∈ RJ. Hence, R ∩ J = RJ. Therefore, by Lemma 2, S is weakly regular.

Lemma 3. [13] Let S be a monoid. Then, the following statements are equivalent:

(1) S is weakly regular.

(2) Q ∩ J ⊆ QJ for every quasi-ideal Q and every ideal J of S.

On the basis of Lemma 3, we can prove Theorem 6.

Theorem 6. For a monoid S, the following statements are equivalent.

(1) S is weakly regular.

(2) C̈1⊓C̈2⊏C̈1 ⊛ C̈2 for every CBF quasi-ideal C̈1 = ⟨µ, ω⟩ and every CBF ideal C̈2 = ⟨λ, ψ⟩ of S.

(3) C̈1⊓C̈2⊏C̈1 ⊛ C̈2 for every CBF bi-ideal C̈1 = ⟨µ, ω⟩ and every CBVF ideal C̈2 = ⟨λ, ψ⟩ of S.

(4) C̈1⊓C̈2⊏C̈1 ⊛ C̈2 for every CBF generalized bi-ideal C̈1 = ⟨µ, ω⟩ and every CBF interior ideal
C̈2 = ⟨λ, ψ⟩ of S.

Proof. (1) ⇒ (4) Assume that C̈1 = ⟨µ, ω⟩ is a CBF generalized bi-ideal and C̈2 = ⟨λ, ψ⟩ is a
CBF interior ideal of S. Let h ∈ S. Then, there exist p, q ∈ S such that h = hphq. Thus,

(µp ⃝ λ
p
)(h) = ⋎

(k,o)∈Ah

{µp(k)⋏ λ
p
(o)} = ⋎

(k,o)∈Ahphq

{µp(k)⋏ λ
p
(o)}

⪰ µp(h)⋏ λ
p
(phq) ⪰ µp(h)⋏ λ

p
(h) = (µp ⊓ λp)(h),

(µn ⃝ λ
n
)(h) = ⋏

(k,o)∈Ah

{µn(k)⋎ λ
n
(o)} = ⋏

(k,o)∈Ahphq

{µn(k)⋎ λ
n
(o)}

⪯ µn(h)⋎ λ
n
(phq) ⪯ µn(h)⋎ λ

n
(h) = (µn ⊓ λn)(h),

and
(ωp ∗ ψp)(h) =

∨
(k,o)∈Ah

{ωp(k) ∧ ψp(o)} =
∨

(k,o)∈Ahphq

{ωp(k) ∧ ψp(k)}

≥ ωp(h) ∧ ψp(phq) ≥ ωp(h) ∧ ψp(h) = (ωp ∩ ψp)(h),

(ωn ∗ ψn)(h) =
∧

(k,o)∈Ah

{ωn(k) ∨ ψn(o)} =
∧

(k,o)∈Ahphq

{ωn(k) ∨ ψn(k)}

≤ ωn(h) ∨ ψn(phq) ≤ ωn(h) ∨ ψn(h) = (ωn ∩ ψn)(h).

Hence, (µp ⊓λp)(h) ⪯ (µp ⃝λ
p
)(h), (µn ⊓λn)(h) ⪰ (µn ⃝λ

n
)(h) and (ωp ∩ψp)(h) ≤ (ωp ∗ψp)(h),

(ωn ∩ ψn)(h) ≥ (ωn ∗ ψn)(h). Therefore, C̈1⊓C̈2⊏C̈1 ⊛ C̈2.

(4) ⇒ (3) ⇒ (2) This is obvious because every CBF bi-ideal is a CBF generalized bi-ideal of
S, every CBF ideal is a CBF interior ideal of S and every CBF quasi-ideal is a CBF bi-ideal of S.



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 12 of 14

(2) ⇒ (1) Let Q be a quasi-ideal and J be an ideal of S. Then, by Theorem 3, χQ is a CBF
quasi-ideal and χJ is a CBF ideal of S. By supposition and Lemma 4, we have

µp
χQJ

(h) = (µp
χQ

⃝ µp
χJ

)(h) ⪰ (µp
χQ

⊓ µp
χJ

)(h) = µp
χQ⊓J

(h) = 1,

µn
χQJ

(h) = (µn
χQ

⃝ µn
χJ

)(h) ⪯ (µn
χQ

⊓ µn
χJ

)(h) = µn
χQ⊓J

(h) = −1,

and
ωp
χQJ

(h) = (ωp
χQ

∗ ωp
χJ

)(h) ≥ (ωp
χQ

∩ ωp
χJ

)(h) = ωp
χQ∩J

(h) = 1,

ωn
χQ

(h) = (ωn
χQ

∗ ωn
χJ

)(h) ≤ (ωn
χQ

∩ ωn
χJ

)(h) = ωn
χQ∩J

(h) = −1.

Thus, h ∈ QJ. Hence, Q ∩ J ⊑ QJ. Therefore, by Lemma 3, S is weakly regular.

Lemma 4. [13] Let S be a monoid. Then, the following statements are equivalent:

(1) S is weakly regular.

(2) Q ∩ J ∩R ⊆ QJR for every quasi-ideal Q, every ideal J and every right ideal R of S.

On the basis of Lemma 4, we can prove Theorem 7.

Theorem 7. Let S be a monoid. Then, the following statements are equivalent:

(1) S is weakly regular.

(2) C̈1⊓C̈2⊓C̈3⊏C̈1 ⊛ C̈2 ⊛ C̈3, for every CBF quasi-ideal C̈1 = ⟨µ, ω⟩, every CBF ideal C̈2 = ⟨λ, ψ⟩
and every CBF right ideal C̈3 = ⟨ν, κ⟩ of S.

(3) C̈1⊓C̈2⊓C̈3⊏C̈1 ⊛ C̈2 ⊛ C̈3, for every CBF bi-ideal C̈1 = ⟨µ, ω⟩, every CBF ideal C̈2 = ⟨λ, ψ⟩
and every CBF right ideal C̈3 = ⟨ν, κ⟩ of S.

(4) C̈1⊓C̈2⊓C̈3⊏C̈1 ⊛ C̈2 ⊛ C̈3, for every CBF generalized bi-ideal C̈1 = ⟨µ, ω⟩, every CBF interior

ideal C̈2 = ⟨λ, ψ⟩ and every CBF right ideal C̈3 = ⟨ν, κ⟩ of S.

Proof. (1) ⇒ (4) Assume that C̈1 = ⟨µ, ω⟩ is a CBF generalized bi-ideal, C̈2 = ⟨λ, ψ⟩ is a CBF
interior ideal and C̈3 = ⟨ν, κ⟩ is a CBF right ideal of S. Let h ∈ S. Then, there exist p, q ∈ S such
that h = hphq. Thus,

(µp ⃝ λ
p ⃝ νp)(h) = µp ⃝ (λ

p ⃝ νp)(h) = ⋎
(k,o)∈Ah

{µp(k)⋏ (λ
p ⃝ νp)(o)}

= ⋎
(k,o)∈Ahphq

{µp(k)⋏ (λ
p ⃝ νp)(o)} ⪰ µp(h)⋏ (λ

p ⃝ νp)(phq)

= µp(h)⋏ ⋎
(y,z)∈Aphq

(λ
p
(y)⋏ νp(z))

= µp(h)⋏ ⋎
(y,z)∈Aphphqq

(λ
p
(y)⋏ νp(z)) = µp(h)⋏ (λ

p
(php)⋏ νp(hqq))

= µp(h)⋏ (λ
p
(php)⋏ νp(hq2)) ⪰ µp(h)⋏ (λ

p
(h)⋏ νp(h))

= µp(h)⋏ (λ
p ⊓ νp)(h) = (µp ⊓ λp ⊓ νp)(h),

(µn ⃝ λ
n ⃝ νn)(h) = µn ⃝ (λ

n ⃝ νn)(h) = ⋏
(k,o)∈Ah

{µn(k)⋎ (λ
n ⃝ νn)(o)}

= ⋏
(k,o)∈Ahphq

{µn(k)⋎ (λ
n ⃝ νn)(o)} ⪯ µn(h)⋎ (λ

n ⃝ νn)(phq)

= µn(h)⋎ ⋏
(y,z)∈Aphq

(λ
n
(y)⋎ νn(z))

= µn(h)⋎ ⋎
(y,z)∈Aphphqq

(λ
n
(y)⋎ νn(z)) = µn(h)⋎ (λ

n
(php)⋎ νn(hqq))

= µn(h)⋎ (λ
p
(php)⋎ νn(hq2)) ⪯ µn(h)⋎ (λ

n
(h)⋎ νn(h))

= µn(h)⋎ (λ
n ⊓ νn)(h) = (µn ⊓ λn ⊓ νn)(h),



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 13 of 14

and

(ωp ∗ ψp ∗ κp)(h) = ωp ∗ (ψp ∗ κp)(h) =
∨

(k,o)∈Ah

{ωp(k) ∧ (ψp ∗ κp)(o)}

=
∨

(k,o)∈Ahphq

{ω(k) ∧ (ψp ∗ κp)(o)} ≥ ωp(h) ∧ (ψp ∗ κp)(phq)

= ωp(h) ∧
∨

(y,z)∈Aphq

(ψp(y) ∧ κp(z))

= ωp(h) ∧
∨

(y,z)∈Aphphqq

(ψp(y) ∧ κp(z)) = ωp(h) ∧ (ψp(php) ∧ κp(hqq))

= ωp(h) ∧ (ψp(php) ∧ κp(hq2)) ≥ ωp(h) ∧ (ψp(h) ∧ κp(h))
= ωp(h) ∧ (ψp ∩ κp)(h) = (ωp ∩ ψp ∩ κp)(h),

(ωn ∗ ψn ∗ κn)(h) = ωn ∗ (ψn ∗ κn)(h) =
∧

(k,o)∈Ah

{ωn(k) ∨ (ψn ∗ κn)(o)}

=
∧

(k,o)∈Ahphq

{ω(k) ∧ (ψn ∗ κn)(o)} ≤ ωn(h) ∨ (ψn ∗ κn)(phq)

= ωn(h) ∨
∧

(y,z)∈Aphq

(ψn(y) ∨ κn(z))

= ωn(h) ∨
∧

(y,z)∈Aphphqq

(ψn(y) ∨ κn(z)) = ωn(h) ∨ (ψp(php) ∨ κn(hqq))

= ωn(h) ∨ (ψn(php) ∨ κn(hq2)) ≤ ωn(h) ∨ (ψn(h) ∨ κn(h))
= ωn(h) ∨ (ψn ∩ κn)(h) = (ωn ∩ ψn ∩ κn)(h).

Hence, (µp⊓λp⊓νp)(h) ⪯ (µp⃝λ
p⃝νp)(h), (µn⊓λn⊓νn)(h) ⪰ (µn⃝λ

n⃝νn)(h) and (ωp∩ψp∩
κp)(h) ≤ (ωp∗ψp∗κp)(h), (ωn∩ψn∩κn)(h) ≥ (ωn∗ψn∗κn)(h) Therefore, C̈1⊓C̈2⊓C̈3⊏C̈1⊛C̈2⊛C̈3.

It is obvious that (4) ⇒ (3) ⇒ (2).
(2) ⇒ (1) Let Q be a quasi-ideal, H be an ideal and R be a right ideal of S. Then, by Theorem

3, χQ is a CBF quasi-ideal, χJ is a CBF ideal and χR is a CBF right ideal of S. By supposition
and Lemma 4, we have

µp
χQJR

(h) = (µp
χQ

⃝ µp
χJ

⃝ µp
χR

)(h) ⪰ (µp
χQ

⊓ µp
χJ

⊓ µp
χR

)(h) = µp
χQ⊓J⊓R

(h) = 1,

µn
χQJR

(h) = (µn
χQ

⃝ µn
χJ

⃝ µn
χR

)(h) ⪯ (µn
χQ

⊓ µn
χJ

⊓ µn
χR

)(h) = µn
χQ⊓J⊓R

(h) = −1,

and
ωp
χQJR

(h) = (ωp
χQ

∗ ωp
χJ

∗ ωp
χR

)(h) ≥ (ωp
χQ

∩ ωp
χJ

∩ ωn
χR

)(h) = ωp
χQ∩J∩R

(h) = 1,

ωn
χQJR

(h) = (ωn
χQ

∗ ωn
χJ

∗ ωn
χR

)(h) ≤ (ωn
χQ

∩ ωn
χJ

∩ ωp
χR

)(h) = ωn
χQ∩J∩R

(h) = −1.

Thus, h ∈ QJR. Hence, Q ∩ J ∩R ⊑ QJR. Therefore, by Lemma 4, S is weakly regular.

5. Conclusion

In this article, we extend the concept of cubic fuzzy sets and bipolar fuzzy sets by introducing the
notion of cubic bipolar fuzzy sets, which serve as a more generalized framework for dealing with
uncertainty in algebraic structures. This extended concept provides a powerful tool for analyzing
and characterizing various subsemigroups, offering a new perspective on their structural properties.
One of the main contributions of this paper is the characterization of weakly regular semigroups
in terms of cubic bipolar fuzzy ideals. By exploring the fundamental properties and interactions
of these fuzzy ideals within semigroups, we establish key results that enhance our understanding
of weakly regular semigroups and their algebraic behavior. For future research, we aim to extend
our findings by characterizing certain classes of subsemigroups using cubic bipolar fuzzy ideals.



P. Khamrot, N. Deetae, T. Gaketem / Eur. J. Pure Appl. Math, 18 (2) (2025), 5854 14 of 14

This will further enrich the theoretical framework and provide deeper insights into the role of
fuzzy structures in semigroup theory. Additionally, we plan to investigate potential applications
of cubic bipolar fuzzy sets in other mathematical and computational domains, particularly in
decision-making processes and algebraic systems with uncertainty.

Acknowledgements

This research was supported by the Rajamangala University Technology Lanna , Phitsanulok,
Thailand (Fundamental Fund 2025, Grant No. FF2568P089).

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