Int. J. Anal. Appl. (2023), 21:1

Essential Bipolar Fuzzy Ideals in Semigroups

Pannawit Khamrot1, Thiti Gaketem2,∗

1Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala

University of Technology Lanna of Phitsanulok, Phitsanulok, Thailand
2Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics School

of Science, University of Phayao, Phayao 56000, Thailand

∗Corresponding author: thiti.ga@up.ac.th

Abstract. In this paper, we give the concepts of essential bipolar fuzzy ideals in semigroups. We

discuss the basic properties and relationships between essential bipolar fuzzy ideals and essential ideals

in semigroups Finally, we extend to 0-essential bipolar fuzzy ideals in semigroups.

1. Introduction

The theory for dealing with uncertainty, fuzzy set theory, was discovered by Zadeh in 1965 [12],

which it has applied in many areas such as medical science, robotics, computer science, information

science, control engineering, measure theory, logic, set theory, topology etc. In 2000, Lee [6] developed

theory of fuzzy set to theory of bipolar fuzzy set which function from intrval [−1, 0]∪[0, 1]. The theory
of bipolar set applied in information affects the effectiveness and efficiency of decision making. It

is used in decision-making problems, organization problems, economic problems, and evaluation, risk

management, environmental and social impact assessments. Later in 2012, S.K. Majumder [2] studies

bipolar fuzzy set in Γ-semigroups and integration properties of bipolar fuzzy ideals in Γ-semigroups In

1971 U. Medhi et al. [7] was introduced Essential fuzzy ideals of ring. In 2013, U. Medhi and

H.K. Saikia [8] discussed the concept of T-fuzzy essential ideals and studied the properties of T-fuzzy

essential ideals. In 2017 S. Wani and K. Pawar [11] extend the concept of essential ideals in semigroups

go to ternary semiring and studied essential ideals in ternary semiring. In 2020, S. Baupradist et al. [1]

Received: Nov. 14, 2022.

2020 Mathematics Subject Classification. 20M12, 06F05.

Key words and phrases. bipolar fuzzy ideals; bipolar fuzzy ideals; 0-essential bipolar fuzzy ideals.

https://doi.org/10.28924/2291-8639-21-2023-1
ISSN: 2291-8639

© 2023 the author(s).

https://doi.org/10.28924/2291-8639-21-2023-1


2 Int. J. Anal. Appl. (2023), 21:1

studied essential ideals and essential fuzzy ideals in semigroups. Together with 0-essential ideals and

0-essential fuzzy ideals in semigroups. In 2022 T. Gaketem et al. [10] studied essential bi-ideals and

fuzzy essential bi-ideals in semigroups. Moreover, T. Gaketem and A. Iampan [3,4] used knowledge

of essential ideals in semigroups go to studied essential ideals in UP-algebra. In this papar, we used

knowledge of essential fuzzy ideals in semigroups go study in bipolar valued fuzzy ideal in semigroup

and we investigate it properties. Moreover, we characterize essential bipolar valued fuzzy ideals and

0-essential bipolar valued fuzzy ideals of semigroups.

2. Preliminaries

In this section, we review concepts basic definitions and the theorem used to prove all result in the

next section.

A non-empty subset I of a semigroup S is called a subsemigroup of S if I2 ⊆ I.
A non-empty subset I of a semigroup S is called a left (right) ideal of S if SI ⊆ I (IS ⊆ I). An
ideal I of a semigroup S is a non-empty subset which is both a left ideal and a right ideal of S. An

essential ideal I of a semigroup S if I is an ideal of S and I∩J 6= ∅ for every ideal J of S.
We see that for any ζ1,ζ2 ∈ [0, 1], we have

ζ1 ∨ζ2 = max{ζ1,ζ2} and ζ1 ∧ζ2 = min{ζ1,ζ2}.

A fuzzy set ζ of a non-empty set T is function from T into unit closed interval [0, 1] of real numbers,

i.e., ζ : T → [0, 1].
For any two fuzzy sets ζ and % of a non-empty set T, define ≥, =,∧, and ∨ as follows:

(1) ζ ≥ % ⇔ ζ(k) ≥ %(k) for all k ∈ T,
(2) ζ = % ⇔ ζ ≥ % and % ≥ ζ,
(3) (ζ ∧%)(k) = min{ζ(k),%(k)} = ζ(k) ∧%(k) for all k ∈ T,
(4) (ζ ∨%)(k) = max{ζ(k),%(k)} = ζ(k) ∨%(k) for all k ∈ T.

For the symbol ζ ≤ %, we mean % ≥ ζ.

For any element k in a semigroup S, define the set Fk by

Fk := {(y,z) ∈ S×S | k = yz}.

For two fuzzy sets ζ and % on a semigroup S, define the product ζ ◦% as follows: for all k ∈ S,

(ζ ◦%)(k) =




∨
(y,z)∈Fk

{ζ(y) ∧%(z) if Fk 6= ∅,

0 if Fk = ∅.

The following definitions are types of fuzzy subsemigroups on semigroups.

Definition 2.1. [9] A fuzzy set ζ of a semigroup S is said to be a fuzzy ideal of S if ζ(uv) ≥ ζ(u)∨ζ(v)
for all u,v ∈ S.



Int. J. Anal. Appl. (2023), 21:1 3

Definition 2.2. [1] An essential fuzzy ideal ζ of a semigroup S if ζ is a nonzero fuzzy ideal of S and

ζ ∧% 6= 0 for every nonzero fuzzy ideal % of S.

Now, we reivew definition of bipolar valued fuzzy set and basic properties used in next section.

Definition 2.3. [6] Let S be a non-empty set. A bipolar fuzzy set (BF set)

ζ on S is an object having the form

ζ := {(k,ζp(k),ζn(k)) | k ∈ S},

where ζp : S → [0, 1] and ζn : S → [−1, 0].

Remark 2.1. For the sake of simplicity we shall use the symbol ζ = (S; ζp,ζn) for the BF set

ζ = {(k,ζp(k),ζn(k)) | k ∈ S}.

The following example of a BF set.

Example 2.1. Let S = {21, 22, 23...}. Define ζp : S → [0, 1] is a function

ζp(u) =


0 if u is old number

1 if u is even number

and ζn : S → [−1, 0] is a function

ζn(u) =


−1 if u is old number

0 if u is even number.

Then ζ = (S; ζp,ζn) is a BF set.

For BF sets ζ = (S; ζp,ζn) and % = (S; %p,%n), define products ζp◦%p and ζn◦%n as follows: For
u ∈ S

(ζp ◦%p)(k) =




∨
(y,z)∈Fk

{ζp(y) ∧%p(z)} if k = yz

0 if otherwise.

and

(ζn ◦%n)(k) =




∧
(y,z)∈Fk

{ζn(y) ∨%n(z)} if k = yz

0 if otherwise.

Definition 2.4. [2] Let I be a non-empty set of a semigroup S. A positive characteristic function

and a negative characteristic function are respectively defined by

λ
p
I : S → [0, 1],k 7→

p
I(u) :=

{
1 k ∈ I,
0 k /∈ I,

and



4 Int. J. Anal. Appl. (2023), 21:1

λnI : S → [−1, 0],k 7→
n
I (k) :=

{
−1 k ∈ I,
0 k /∈ I.

Remark 2.2. For the sake of simplicity we shall use the symbol λI = (S; λ
p
I,λ

n
I) for the BF set

I := {(k,λ
p
I(k),λ

n
I(k)) | k ∈ I}.

Definition 2.5. [2] A BF set ζ = (S; ζp,ζn) on a semigroup S is called a BF ideal on S if it satisfies

the following conditions: ζp(uv) ≥ ζp(u) ∨ζp(v) and ζn(uv) ≤ ζn(v) ∧ζn(u) for all u,v ∈ S.

The following theorems are true.

Theorem 2.1. [2] Let K be a nonempty subset of semigroup S. Then K is an ideal of S if and only

if characteristic function λK = (S; λ
p
K,λ

n
K) is a BF ideal of S.

Theorem 2.2. [2] Let L and J be subsets of a non-empty set S. Then the following holds.

(1) λpL∩J = λ
p
L ∧λ

p
J.

(2) λnL∪J = λ
n
L ∨λ

n
J.

(3) λpL ◦λ
p
J = λ

p
LJ.

(4) λnL ◦λ
n
J = λ

n
LJ.

Let ζ = (S; ζp,ζn) be a BF set of a non-empty of S. Then the support of ζ instead of supp(ζ) =

{u ∈ S | ζ(u) 6= 0} where ζp(u) 6= 0 and ζn(u) 6= 0 for all u ∈ S.

Theorem 2.3. Let ζ = (S; ζp,ζn) be a nonzero BF set of a semigroup S. Then ζ = (S; ζp,ζn) is a

BF ideal of S if and only if supp(ζ) is an ideal of S.

Proof. Supposet that ζ = (S; ζp,ζn) is a BF ideal of S and let u,v ∈ S, with
u,v ∈ supp(ζ) Then ζp(u) 6= 0, ζp(v) 6= 0 and ζn(u) 6= 0, ζn(v) 6= 0 .
Since ζ = (S; ζp,ζn) is a BF ideal of S we have ζp(uv) ≥ ζp(u) ∨ζp(v) and ζn(uv) ≤ ζn(u) ∧ζn(v)
Thus, ζp(uv) 6= 0 and ζn(uv) 6= 0. It implies that uv ∈ supp(ζ). Hence, supp(ζ) is an ideal of S.

Conversely, suppose that supp(ζ) is an ideal of S and let u,v,∈ S.
If u,v ∈ supp(ζ), then uv ∈ supp(ζ). Thus ζp(v) 6= 0 and ζp(uv) 6= 0.

Hence ζp(uv) ≥ ζp(u) ∨ζp(v).
If u /∈ supp(ζ) or v /∈ supp(ζ) then ζp(uv) ≥ ζp(u) ∨ζp(v).
Similarly, we can show that, ζn(uv) ≤ ζp(u) ∧ζn(v).
Thus, ζ = (S; ζp,ζn) is a BF ideal of S. �

3. Essential Bipolar Valued Fuzzy Ideals in a Semigroup.

Definition 3.1. An essential BF ideal ζ = (S; ζp,ζn) of a semigroup S if ζ = (S; ζp,ζn) is a nonzero

BF ideal of S and ζp ∧%p 6= 0 and ζn ∨%n 6= 0 for every nonzero BF ideal % = (S; %p,%n) of S.



Int. J. Anal. Appl. (2023), 21:1 5

Theorem 3.1. Let I be an ideal of a semigroup S. Then I is an essential ideal of S if and only if

λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S.

Proof. Suppose that I is an essential ideal of S and let % = (S; %p,%n) be a nonzero BF ideal of

S. Then by Theorem 2.3 supp(%) is an ideal of S. Since I is an essential ideal of S we have I

is an ideal of S. Thus I ∩ supp(%) 6= ∅. So there exists u ∈ I ∩ supp(%). Since I is an ideal of
S we have λI = (S; λ

p
I,λ

n
I) is a BF ideal of S. Since % = (S; %

p,%n) is a nonzero BF ideal of S

we have (λpI ∧ %
p)(u) 6= 0 and (λnI ∨ %

n)(u) 6= 0 Thus, λpI ∧ %
p 6= 0 and λnI ∨ %

n 6= 0. Therefore,
λI = (S; λ

p
I,λ

n
I) is an essential BF ideal of S.

Conversely, assume that λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S and let J be an ideal of S.

Then λJ = (S; λ
p
J,λ

n
J) is a nonzero BF ideal of S.

Since λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S we have λI = (S; λ

p
I,λ

n
I) is a BF ideal of S.

Thus, λpI ∧λ
p
J 6= 0 and λ

n
I ∨λ

n
J 6= 0 So by Theorem 2.2, λ

p
I∩J 6= 0 and λ

n
I∪J 6= 0. Hence, I ∩ J 6= ∅.

Therefore, I is an essential ideal of S. �

Theorem 3.2. Let ζ = (S; ζp,ζn) be a nonzero BF ideal of a semigroup S. Then ζ = (S; ζp,ζn) is

an essential BF ideal of S if and only if supp(ζ) is an essential ideal of S.

Proof. Assume that ζ = (S; ζp,ζn) is an essential BF ideal of S and let J be an ideal of S. Then by

Theorem 2.1, λJ = (S; λ
p
J,λ

n
J) is a BF ideal of S. Since ζ = (S; ζ

p,ζn) is an essential BF ideal of

S we have ζ = (S; ζp,ζn) is a BF ideal of S. Thus, ζp ∧λpJ 6= 0 and ζ
n ∨λnJ 6= 0. So there exists

u ∈ S such that (ζp ∧λpJ)(u) 6= 0 and (ζ
n ∨λnJ)(u) 6= 0. It implies that ζ

p(u) 6= 0, λpJ(u) 6= 0 and
ζn(u) 6= 0, λnJ(u) 6= 0. Hence, u ∈supp(ζ) ∩J so supp(ζ) ∩J 6= ∅. Therefore, supp(ζ) is an essential
ideal of S.

Conversely, assume that supp(ζ) is an essential ideal of S and let % = (S; %p,%n) be a nonzero BF

ideal of S. Then by Thoerem 2.3 supp(%) is an ideal of S. Since supp(ζ) is an essential ideal of S we

have supp(ζ) is an ideal of S. Thus supp(ζ)∩supp(%) 6= ∅. So, there exists u ∈ supp(ζ)∩supp(%). It
implies that ζp(u) 6= 0, ζn(u) 6= 0 and %p(u) 6= 0, %n(u) 6= 0. for all u ∈ S. Hence, (ζp ∧%p)(u) 6= 0
and (ζn ∨ %n)(u) 6= 0 for all u ∈ S. Therefore, ζp ∧ %p 6= 0 and ζn ∨ %n 6= 0. We conclude that
ζ = (S; ζp,ζn) is an essential BF ideal of S. �

Theorem 3.3. Let ζ = (S; ζp,ζn) be an essential BF ideal of a semigroup S. If % = (S; %p,%n) is a

BF ideal of S such that ζp ≤ %p and ζn ≥ %n, then % = (S; %p,%n) is also an essential BF ideal of S.

Proof. Let % = (S; %p,%n) is a BF ideal of S such that ζp ≤ %p and ζn ≥ %n and let ξ = (S; ξp,ξn)
be any BF ideal of S. Since ζ = (S; ζp,ζn) is an essential BF ideal of S we have ζ = (S; ζp,ζn)

is a BF ideal of S. Thus ζp ∧ ξp 6= 0 and ζn ∨ ξn 6= 0. So %p ∧ ξp 6= 0 and %n ∨ ξn 6= 0. Hence
% = (S; %p,%n) is an essential BF ideal of S. �

Next, we study the intersection and union of BF sets as define.

Let ζ = (S; ζp,ζn) and % = (S; %p,%n) are BF sets of a semigroup S.



6 Int. J. Anal. Appl. (2023), 21:1

Define ζ ∩% = (ζp ∩%p,ζn ∩%n) where (ζp ∩%p)(k) = ζp(k) ∧%p(k) and
(ζn ∩%n)(k) = ζn(k) ∨%n(k) for all k ∈ S.

Define ζ ∪% = (ζp ∪%p,ζn ∪%n) where (ζp ∪%p)(k) = ζp(k) ∨%p(k) and
(ζn ∪%n)(k) = ζn(k) ∧%n(k) for all k ∈ S.

Theorem 3.4. Let ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) be essential BF ideals of a semigroup S.

Then ζ1 ∪ζ2 and ζ1 ∩ζ2 are essential BF ideals of S.

Proof. By Theorem 3.3, we have ζ1 ∪ζ2 is an essential BF ideal of S.
Since ζ1 = (S; ζ

p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) are essential BF ideals of S we have

ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) are BF ideals of S. Thus ζ1 ∩ζ2 is a BF ideal of S.

Let ξ = (S; ξp,ξn) be a BF ideal of S. Then ζp1 ∧ξ
p 6= 0 and ζn1 ∨ξ

n 6= 0. Thus there exists u ∈ S
such that (ζp1 ∧ ξ

p)(u) 6= 0 and (ζn1 ∨ ξ
n)(u) 6= 0. So ζp1(u) 6= 0, ζ

n
1(u) 6= 0 and ξ

p(u) 6= 0 and
ξn(u) 6= 0. Since ζp2 6= 0 and ζ

n
2 6= 0 and let v ∈ S such that ζ

p
2(v) 6= 0 and ζ

n
2(v) 6= 0. Since

ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) are BF ideals of S we have ζ

p
1(uv) ≥ ζ

p
1(u) ∧ ζ

p
1(v) ≥ 0,

ζn1(uv) ≤ ζ
n
1(u) ∨ ζ

n
1(v) ≤ 0 and ζ

p
2(uv) ≥ ζ

p
2(u) ∧ ζ

p
2(v) ≥ 0, ζ

n
2(uv) ≤ ζ

n
2(u) ∨ ζ

n
2(v) ≤ 0. Thus

(ζ
p
1∧ζ

p
2)(uv) = ζ

p
1(uv)∧ζ

p
2(uv) 6= 0 and (ζ

n
1∨ζ

n
2)(uv) = ζ

n
1(uv)∨ζ

n
2(uv) 6= 0. Since ξ = (S; ξ

p,ξn)

is a BF ideal of S and ξp(u) 6= 0 and ξn(u) 6= 0 we have ξp(uv) 6= 0 and ξn(uv) 6= 0 for all u,v ∈ S.
Thus ((ζp1 ∧ ζ

p
2) ∧ ξ

p)(uv) 6= 0 and ((ζn1 ∨ ζ
n
2) ∨ ξ

n)(uv) 6= 0. Hence ((ζp1 ∧ ζ
p
2) ∧ ξ

p) 6= 0 and
((ζn1 ∨ζ

n
2) ∨ξ

n) 6= 0. Therefore, ζ1 ∩ζ2 is an essential BF ideal of S. �

Definition 3.2. [1] An essential ideal I of a semigroup S is called

(1) a minimal if for every essential ideal of J of S such that J ⊆ I, we have J = I,
(2) a prime if uv ∈ I implies u ∈ I or v ∈ I,
(3) a semiprime if u2 ∈ I implies u ∈ I, for all u,v ∈ S.

Example 3.1. [1] Let S be a semigroup with zero. Then {0} is a unique minimal essential ideal of
S, since {0} is an eseential ideal of S.

Definition 3.3. An essential BF ideal ζ = (S; ζp,ζn) of a semigroup S is called

(1) a minimal if for every essential BF ideal of % = (S; %p,%n) of S such that ζp ≤ %p and ζn ≥ %n,
we have supp(ζ) = supp(%),

(2) a prime if ζp(uv) ≤ ζp(u) ∨ζp(v) and ζn(uv) ≥ ζn(u) ∧ζn(v) ,
(3) a semiprime if ζp(u2) ≤ ζp(u) and ζn(u2) ≥ ζn(u), for all u,v ∈ S.

Theorem 3.5. Let I be a non-empty subset of a semigroup S. Then the following statement holds.

(1) I is a minimal essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a minimal essential BF

ideal of S,

(2) I is a prime essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a prime essential BF ideal

of S,



Int. J. Anal. Appl. (2023), 21:1 7

(3) I is a semiprime essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a semiprime essential

BF ideal of S.

Proof. (1) Suppose that I is a minimal essential ideal of S. Then I is an essential ideal of S.

By Theorem 3.1, λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S. Let ζ = (S; ζ

p,ζn) be an

essential BF ideal of S such that ζp ≤ λpI and ζ
n ≥ λnI. Then supp(ζ) ⊆ supp(λI). Thus,

supp(ζ) ⊆ supp(λI) = I. Hence, supp(ζ) ⊆ I. Since ζ = (S; ζp,ζn) is an essential BF ideal
of S we have supp(ζ) is an essential ideal of S. By assumption, supp(ζ) = I = supp(λI).

Hence, λI = (S; λ
p
I,λ

n
I) is a minimal essential BF ideal of S.

Conversely, λI = (S; λ
p
I,λ

n
I) is a minimal essential BF ideal of S and let B be an essential

ideal of S such that B ⊆ I. Then B is an ideal of S. Thus by Theorem 3.1, λB = (S; λ
p
B,λ

n
B)

is an essential BF ideal of S such that λpB ≥ λ
p
I and λ

n
B ≤ λ

n
I. So λB = λI. Hence

B = supp(λB) = supp(λI) = I. Therefore I is a minimal essential ideal of S.

(2) Suppose that I is a prime essential ideal of S. Then I is an essential ideal of S. Thus by

Theorem 3.1 λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S. Let u,v ∈ S.

If uv ∈ I, then u ∈ I or v ∈ I. Thus λpI(u) ∨λ
p
I(v) = 1 ≥ λ

p
I(uv) and λ

n
I(u) ∧λ

n
I(v) =

−1 ≤ λnI(uv).
If uv /∈ I, then λpI(u) ∨λ

p
I(v) ≥ λ

p
I(uv) and λ

n
I(u) ∧λ

n
I(v) ≤ λ

n
I(uv).

Thus λI = (S; λ
p
I,λ

n
I) is a prime essential BF ideal of S.

Conversely, suppose that λI = (S; λ
p
I,λ

n
I) is a prime essential BF ideal of S. Then

λI = (S; λ
p
I,λ

n
I) is an essential BF ideal. Thus by Theorem 3.1, I is an essential ideal

of S. Let u,v ∈ S. If uv ∈ I, then λpI(uv) = 1 and λ
n
I(uv) = −1. By assumption,

λ
p
I(uv) ≤ λ

p
I(u) ∨ λ

p
I(v) and λ

n
I(uv) ≥ λ

n
I(u) ∧ λ

n
I(v). Thus λ

p
I(u) ∨ λ

p
I(v) = 1 and

λnI(u) ∧λ
n
I(v) = −1 so u ∈ I or v ∈ I. Hence I is a prime essential ideal of S.

(3) Suppose that I is a semiprime essential ideal of S. Then I is an essential ideal of S. Thus

by Theorem 4.1, λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S. Let u ∈ S.

If u2 ∈ I, then u ∈ I. Thus, λpI(u) = λ
p
I(u
2) = 1 and λnI(u) = λ

n
I(u
2) = −1. Hence,

λ
p
I(u
2) ≤ λpI(u) and λ

n
I(u
2) ≥ λnI(u).

If u2 /∈ I, then λpI(u
2) = 0 ≤ λpI(u) and λ

n
I(u
2) = 0 ≥ λnI(u).

Thus λI = (S; λ
p
I,λ

n
I) is a semiprime essential BF ideal of S.

Conversely, suppose that λI = (S; λ
p
I,λ

n
I) is a semiprime essential BF ideal of S. Then

λI = (S; λ
p
I,λ

n
I) is an essential BF ideal of S. Thus by Theorem 4.1, I is an essential

ideal of S. Let u ∈ S with u2 ∈ I. Then λpI(u
2) = 1 and λnI(u

2) = −1. By assumption,
λ
p
I(u
2) ≤ λpI(u) and λ

n
I(u
2) ≥ λnI(u). Thus λ

p
I(u) = 1 and λ

n
I(u) = −1 so u ∈ I. Hence, I

is a semiprime essential ideal of S.

�



8 Int. J. Anal. Appl. (2023), 21:1

Theorem 3.6. Let ζ = (S; ζp,ζn) be a minimal essential BF ideal of a semigroup S.

If % = (S; %p,%n) is a BF ideal of S such that ζp ≤ %p and ζn ≥ %n, then % = (S; %p,%n) is also a
minimal essential BF ideal of S.

Proof. Let % = (S; %p,%n) is a BF ideal of S such that ζp ≤ %p and ζn ≥ %n and
let ξ = (S; ξp,ξn) be any BF ideal of S. Since ζ = (S; ζp,ζn) is a minimal essential BF ideal of S

we have ζ = (S; ζp,ζn) is a BF ideal of S. Thus, ζp ∧ξp 6= 0 and ζn ∨ξn 6= 0. So %p ∧ξp 6= 0 and
%n ∨ξn 6= 0. Hence, % = (S; %p,%n) is a minimal essential BF ideal of S. �

Corollary 3.1. Let ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) be minimal essential BF ideals of a semigroup

S. Then ζ1 ∪ζ2 is a minimal essential BF ideals of S.

4. 0-Essential BF ideal.

In this section, we let S be a semigroup with zero. begin we review the definition 0-essential ideal

of S as follows:

Definition 4.1. [1] A nonzero ideal I of a semigroup with zero S is called a 0-essential ideal of S if

I∩J 6= {0} for every nonzero ideal of J of S.

Example 4.1. [1] Let (Z12, +) be semigroup. Then {0, 2, 4, 6, 8, 10} and Z12 are 0-essential ideal of
Z12.

Definition 4.2. A BF ideal ζ = (S; ζp,ζn) of a semigroup with zero S is called a nontrivial BF ideal

of S if there exists a nonzero element u ∈ S such that ζp(u) 6= 0 and ζn(u) 6= 0.

We define the definition of 0-essential BF ideals of a semigroup with zero as follows:

Definition 4.3. A 0-essential BF ideal ζ = (S; ζp,ζn) of a semigroup with zero S if ζ = (S; ζp,ζn)

is a nonzero BF ideal of S and supp(ζ ∧%) 6= {0} for every nonzero BF ideal % = (S; %p,%n) of S.

Theorem 4.1. Let I be a nonzero ideal of a semigroup with zero S. Then I is a 0-essential ideal of

S if and only if λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S.

Proof. Suppose that I is a 0-essential ideal of S and let % = (S; %p,%n) be a nontrivial BF ideal of

S. Then by Theorem 2.3, supp(%) is a nonzero ideal of S. Since I is a 0-essential ideal of S we

have I is a nonzero ideal of S. Thus I∩supp(%) 6= {0}. So there exists u ∈ I∩ supp(%). Since I is a
nonzero ideal of S we have λI = (S; λ

p
I,λ

n
I) is a nonzero BF ideal of S. Since % = (S; %

p,%n) is a

nonzero BF ideal of S we have supp(λI ∧%)(u) 6= 0. Thus, λ
p
I ∧%

p 6= 0 and λnI ∨%
n 6= 0. Therefore,

λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S.

Conversely, assume that λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S and let J be a nonzero

ideal of S. Then λJ = (S; λ
p
J,λ

n
J) is a nonzero BF ideal of S. Sicne λI = (S; λ

p
I,λ

n
I) is a 0-essential

BF ideal of S we have λI = (S; λ
p
I,λ

n
I) is a nontrivial BF ideal of S. Thus, supp(λI ∧λJ) 6= {0}.



Int. J. Anal. Appl. (2023), 21:1 9

So by Theorem 2.2, λpL∩J 6= 0 and λ
n
L∪J 6= 0. Hence, I ∩ J 6= {0}. Therefore I is a 0-essential ideal

of S. �

Theorem 4.2. Let ζ = (S; ζp,ζn) be a nonzero BF ideal of a semigroup with zero S. Then ζ is a

0-essential BF ideal of S if and only if supp(ζ) is a 0-essential ideal of S.

Proof. Assume that ζ = (S; ζp,ζn) is a 0-essential BF ideal of S and let J be a nontrivial ideal of

S. Then by Theorem 2.1, λJ = (S; λ
p
J,λ

n
J) is a nonzero BF ideal of S. Since ζ = (S; ζ

p,ζn) is a

0-essential BF ideal of S we have ζ = (S; ζp,ζn) is a nonzero BF ideal of S. Thus ζp ∧ λpJ 6= 0
and ζn ∨ λnJ 6= 0. So there exists a nonzero element u ∈ S such that (ζ

p ∧ λpJ)(u) 6= 0 and
(ζn ∨ λnJ)(u) 6= 0. It implies that ζ

p(u) 6= 0, ζn(u) 6= 0 and λpJ(u) 6= 0 , λ
n
J(u) 6= 0. Hence,

u ∈ supp(ζ) ∩J so supp(ζ) ∩J 6= {0}. Therefore supp(ζ) is a 0-essential ideal of S.
Conversely, assume that supp(ζ) is a 0-essential ideal of S and let % = (S; %p,%n) be a nonzero BF

ideal of S. Then by Theorem 2.3 supp(%) is a nontrivial zero ideal of S. Since supp(ζ) is a 0-essential

ideal of S we have supp(ζ) is a nonzero ideal of S. Thus supp(ζ) ∩ supp(%) 6= {0}. So there exists
u ∈ supp(ζ)∩supp(%), this implies that ζp(u) 6= 0, ζn(u) 6= 0 and %p(u) 6= 0, %n(u) 6= 0 for all u ∈ S.
Hence, (ζp ∧%p)(u) 6= 0 and (ζn ∨%n)(u) 6= 0 for all u ∈ S. Therefore, ζp ∧%p 6= 0 and ζn ∨%n 6= 0.
We conclude that ζ = (S; ζp,ζn) is a 0-essential BF ideal of S. �

Theorem 4.3. Let ζ = (S; ζp,ζn) be a 0-essential BF ideal of a semigroup S. If % = (S; %p,%n) is a

BF ideal of S such that ζp ≤ %p and ζn ≥ %n, then % = (S; %p,%n) is also a 0-essential BF ideal of S.

Proof. Let % = (S; %p,%n) is a BF ideal of S such that ζp ≤ %p and ζn ≥ %n and
let ξ = (S; ξp,ξn) be any nonzero BF ideal of S. Since ζ = (S; ζp,ζn) is a 0-essential BF ideal of S

we have ζ = (S; ζp,ζn) is a BF ideal of S. Thus supp(ζ ∧ξ) 6= 0. So %p ∧ξp 6= 0 and %n ∨ξn 6= 0.
Hence % = (S; %p,%n) is a 0-essential BF ideal of S. �

Theorem 4.4. Let ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) be 0-essential BF ideals of a semigroup S.

Then ζ1 ∪ζ2 and ζ1 ∩ζ2 are 0-essential BF ideals of S.

Proof. By Theorem 4.3, we have ζ1 ∪ζ2 is a 0-essential BF ideal of S.
Since ζ1 = (S; ζ

p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) are 0-essential BF ideals of S we have

ζ1 = (S; ζ
p
1,ζ

n
1) and ζ2 = (S; ζ

p
2,ζ

n
2) are BF ideals of S. Thus ζ1 ∩ ζ2 is a BF ideal of S. Let

ξ = (S; ξp,ξn) be a nontrival BF ideal of S. Since ζ1 = (S; ζ
p
1,ζ

n
1) is a BF ideal of S we havesupp(ζ1)

is an ideal of S. Thus supp(ζ1 ∧ξ) 6= {0}. Thus there exists u ∈ S such that (ζ
p
1 ∧ξ

p)(u) 6= 0 and
(ζn1∨ξ

n)(u) 6= 0. Since ζ2 = (S; ζ
p
2,ζ

n
2) is a 0-essential BF ideal of S we have supp(ζ2) is a 0-essential

BF ideal of S. Thus, supp(ζ2 ∧ ξ) 6= {0}. So, there exists a nonzero element v ∈ supp(ζ2 ∧ ξ)(u)
implies ζp2(v) 6= 0 and ζ

n
2(v) 6= 0. Since ζ1 = (S; ζ

p
1,ζ

n
1) and ξ = (S; ξ

p,ξn) are BF ideals of S we

have ζp1(v) ≥ ζ
p
1(u), ξ

p(v) ≥ ξp(u) and ζn1(v) ≤ ζ
n
1(u), ξ

n(v) ≤ ξn(u). So ((ζp1 ∧ ζ
p
2) ∧ ξ

p)(v) 6= 0
and ((ζn1 ∨ζ

n
2) ∨ξ

n)(v) 6= 0. Thus, supp((ζ1 ∧ζ2) ∧ξ) 6= {0}. Therefore, ζ1 ∩ζ2 is a 0-essential BF
ideals of S. �



10 Int. J. Anal. Appl. (2023), 21:1

Definition 4.4. [1] A 0-essential ideal I of a semigroup with zero S is called

(1) a minimal if for every 0-essential ideal of J of S such that J ⊆ I, we have J = I,
(2) a prime if uv ∈ I implies u ∈ I or v ∈ I,
(3) a semiprime if u2 ∈ I implies u ∈ I, for all u,v ∈ S.

Example 4.2. Let (Z12, +) be a semigroup with zero. Then {0, 2, 4, 6, 8, 10} is a minimal 0-essential
ideal of S.

Definition 4.5. A 0-essential BF ideal ζ = (S; ζp,ζn) of a semigroup S is called

(1) a minimal if for every 0-essential BF ideal of % = (S; %p,%n) of S such that ζp ≤ %p and
ζn ≥ %n, we have supp(ζ) = supp(%),

(2) a prime if ζp(uv) ≤ ζp(u) ∨ζp(v) and ζn(uv) ≥ ζn(u) ∧ζn(v) ,
(3) a semiprime if ζp(u2) ≤ ζp(u) and ζn(u2) ≥ ζn(u), for all u,v ∈ S.

Theorem 4.5. Let I be a non-empty subset of a semigroup S. Then the following statement holds.

(1) I is a minimal 0-essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a minimal 0-essential

BF ideal of S,

(2) I is a prime 0-essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a prime 0-essential BF

ideal of S,

(3) I is a semiprime 0-essential ideal of S if and only if λI = (S; λ
p
I,λ

n
I) is a semiprime 0-essential

BF ideal of S.

Proof. (1) Suppose that I is a minimal 0-essential ideal of S. Then I is a 0-essential ideal of S.

By Theorem 4.1, λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S. Let ζ = (S; ζ

p,ζn) be a

0-essential BF ideal of S such that ζp ≤ λpI and ζ
n ≥ λnI. Then supp(ζ) ⊆ supp(λI). Thus

supp(ζ) ⊆ supp(λI) = I. Thus supp(ζ) ⊆ I. Since ζ = (S; ζp,ζn) is a 0-essential BF ideal
of S we have supp(ζ) is a 0-essential ideal of S. By assumption, supp(ζ) = I = supp(λI).

Hence, λI = (S; λ
p
I,λ

n
I) is a minimal 0-essential BF ideal of S.

Conversely, λI = (S; λ
p
I,λ

n
I) is a minimal 0-essential BF ideal of S and let B be a 0-

essential ideal of S such that B ⊆ I. Then B is an ideal of S. Thus by Theorem 4.1,
λB = (S; λ

p
B,λ

n
B) is an essential BF ideal of S such that λ

p
B ≥ λ

p
I and λ

n
B ≤ λ

n
I. So

λB = λI. Hence B = supp(λB) = supp(λI) = I. Therefore I is a minimal 0-essential ideal

of S.

(2) Suppose that I is a prime 0-essential ideal of S. Then I is a 0-essential ideal of S. Thus by

Theorem 4.1 λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S. Let u,v ∈ S.

If uv ∈ I, then u ∈ I or v ∈ I. Thus λpI(u) ∨λ
p
I(v) = 1 ≥ λ

p
I(uv) and λ

n
I(u) ∧λ

n
I(v) =

−1 ≤ λnI(uv).
If uv /∈ I, then λpI(u) ∨λ

p
I(v) ≥ λ

p
I(uv) and λ

n
I(u) ∧λ

n
I(v) ≤ λ

n
I(uv).



Int. J. Anal. Appl. (2023), 21:1 11

Thus λI = (S; λ
p
I,λ

n
I) is a prime 0-essential BF ideal of S.

Conversely, suppose that λI = (S; λ
p
I,λ

n
I) is a prime 0-essential BF ideal of S. Then

λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal. Thus by Theorem 4.1, I is a 0-essential ideal

of S. Let u,v ∈ S. If uv ∈ I, then λpI(uv) = 1 and λ
n
I(uv) = −1. By assumption

λ
p
I(uv) ≤ λ

p
I(u) ∨ λ

p
I(v) and λ

n
I(uv) ≥ λ

n
I(u) ∧ λ

n
I(v). Thus λ

p
I(u) ∨ λ

p
I(v) = 1 and

λnI(u) ∧λ
n
I(v) = −1 so u ∈ I or v ∈ I. Hence I is a prime 0-essential ideal of I.

(3) Suppose that I is a semiprime 0-essential ideal of S. Then I is a 0-essential ideal of S. Thus

by Theorem 4.1, λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal of S. Let u ∈ S.

If u2 ∈ I, then u ∈ I Thus λpI(u) = λ
p
I(u
2) = 1 and λnI(u) = λ

n
I(u
2) = −1. Hence,

λ
p
I(u
2) ≤ λpI(u) and λ

n
I(u
2) ≥ λnI(u).

If u2 /∈ I, then λpI(u
2) = 0 ≤ λpI(u) and λ

n
I(u
2) = 0 ≥ λnI(u).

Thus, λI = (S; λ
p
I,λ

n
I) is a semiprime 0-essential BF ideal of S.

Conversely, suppose that λI = (S; λ
p
I,λ

n
I) is a semiprime 0-essential BF ideal of S. Then

λI = (S; λ
p
I,λ

n
I) is a 0-essential BF ideal. Thus by Theorem 4.1, I is a 0-essential ideal of S.

Let u ∈ S with u2 ∈ I Then λpI(u
2) = 1 and λnI(u

2) = −1. By assumption, λpI(u
2) ≤ λpI(u)

and λnI(u
2) ≥ λnI(u). Thus λ

p
I(u) = 1 and λ

n
I(u) = −1 so u ∈ I. Hence I is a semiprime

0-essential ideal of S.

�

Acknowledgment: This research project was supported by the thailand science research and inno-

vation fund and the University of Phayao (Grant No. FF66-RIM024) The authors also gratefully

acknowledge the helpful comments and suggestions of the reviewers, which have improved the pre-

sentation.

Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi-

cation of this paper.

References

[1] S. Baupradist, B. Chemat, K. Palanivel, R. Chinram, Essential Ideals and Essential Fuzzy Ideals in Semigroups, J.

Discrete Math. Sci. Cryptography. 24 (2020), 223–233. https://doi.org/10.1080/09720529.2020.1816643.

[2] C.S. Kim, J.G. Kang, J.M. Kang, Ideal Theory of Semigroups Based on the Bipolar Valued Fuzzy Set Theory, Ann.

Fuzzy Math. Inform. 2 (2012), 193-206.

[3] T. Gaketem, A. Iampan, Essential UP-Ideals and t-Essential Fuzzy UP-Ideals of UP-Algebras, ICIC Express Lett.

15 (2021), 1283-1289. https://doi.org/10.24507/icicel.15.12.1283.

[4] T. Gaketem, P. Khamrot, A. Iampan, Essential UP-Filters and t-Essential Fuzzy UP-Filters of UP-Algebras, ICIC

Express Lett. 16 (2021), 1057-1062. https://doi.org/10.24507/icicel.16.10.1057.

[5] K. Nobuaki, Fuzzy Bi-Ideals in Semigroups, Comment. Math. Univ. St. Paul 5 (1979), 128-132.

[6] K. Lee, Bipolar-Valued Fuzzy Sets and Their Operations, In: Proceeding International Conference on Intelligent

Technologies Bangkok, Thailand (2000), 307-312.

[7] U. Medhi, K. Rajkhowa, L.K. Barthakur, H.K. Saikia, On Fuzzy Essential Ideals of Rings, Adv. Fuzzy Sets Syst. 5

(2008), 287-299.

https://doi.org/10.1080/09720529.2020.1816643
https://doi.org/10.24507/icicel.15.12.1283
https://doi.org/10.24507/icicel.16.10.1057


12 Int. J. Anal. Appl. (2023), 21:1

[8] U. Medhi, H.K. Saikia, On T-Fuzzy Essential Ideals of Rings, J. Pure Appl. Math. 89 (2013), 343-353. https:

//doi.org/10.12732/ijpam.v89i3.5.

[9] J.N. Mordeson, D.S. Malik, N. Kuroki, Fuzzy Semigroup, Springer, Berlin, (2003).

[10] N. Panpetch, T. Muangngao, T. Gaketem, Some Essential Fuzzy Bi-Ideals and Essential Fuzzy Bi-Ideals in a

Semigroup, J. Math. Computer Sci. 28 (2023), 326-334. http://dx.doi.org/10.22436/jmcs.028.04.02.

[11] S. Wani, K. Pawar, On Essential Ideals of a Ternary Semiring, Sohag J. Math. 4 (2017), 65-69. https://doi.org/

10.18576/sjm/040301.

[12] L.A. Zadeh, Fuzzy Sets, Inform. Control. 8 (1965), 338–353. https://doi.org/10.1016/s0019-9958(65)

90241-x.

https://doi.org/10.12732/ijpam.v89i3.5
https://doi.org/10.12732/ijpam.v89i3.5
http://dx.doi.org/10.22436/jmcs.028.04.02
https://doi.org/10.18576/sjm/040301
https://doi.org/10.18576/sjm/040301
https://doi.org/10.1016/s0019-9958(65)90241-x
https://doi.org/10.1016/s0019-9958(65)90241-x

	1. Introduction
	2. Preliminaries
	3. Essential Bipolar Valued Fuzzy Ideals in a Semigroup.
	4. 0-Essential BF ideal.
	References