EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 1, 2023, 121-130
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On Interval Valued Fuzzy Bi-interior Ideals in
Semigroups

Thiti Gaketem1, Tanaphong Prommai1,∗

1 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 study the concept of an interval valued fuzzy bi-interior ideal. We
investigate the properties of an interval valued fuzzy bi-interior ideal in semigroups. We characterize
a regular semigroup in terms of an interval valued fuzzy bi-interior ideal.

2020 Mathematics Subject Classifications: 03E72, 18B40

Key Words and Phrases: Bi-interior ideals, interval valued fuzzy bi-interior ideals

1. Introduction

Uncertainties cannot be handled using traditional mathematical tools but maybe deal
with using a wide range of existing theories such as probability theory, theory of fuzzy sets,
interval valued fuzzy sets. In 1975, Zadeh [11] introduced the theory of interval valued
fuzzy sets as a generalization of the notion of fuzzy sets. Interval valued fuzzy sets have
various applications in several areas like medical science [2], image processing [1], decision
making [12], etc. In 2006, Narayanan and Manikantan [9] for the first time employed
the theory of interval valued fuzzy subsemigroup and studied types of interval valued
fuzzy ideals in semigroups. In 2018, MK. Rao [5] introduced and studied the definition
and properties of the bi-interior ideal in the semigroup. In 2019, A. Mahboob et al.
[8] characterizations of regular ordered semigroups by (ε, ε

∨
(k,qk))-fuzzy quasi ideals. G

Muhiuddin et al. discussed a new type of fuzzy semiprime subsets in ordered semigroups.
Many researchers studied in interval valued fuzzy semigroup such that in 2020 Ahsan et
al. [6] extend the ideals of (m,n)-ideals in semigroups to fuzzy sets in semigroup and
they characterize the regular semigroup by using fuzzy (m,n)-ideals. I. Crista et al. [3]
studied a new type fuzzy quasi-ideal in ordered semigroups. In 2021 T. Gaketem [4],
studied interval valued fuzzy almost (m,n)-bi-ideal in semigroups. A. Mahboob and G.
Muhiuddin [7] studied a fuzzy prime subset in ordered semigroups.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4616

Email addresses: thiti.ga@up.ac.th (T. Gaketem), , tanaphong.pr@up.ac.th (T. Prommai)

https://www.ejpam.com 121 © 2023 EJPAM All rights reserved.



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 122

In this work, we establish the concept of an interval valued fuzzy bi-interior ideal.
We investigate the properties of an interval valued fuzzy bi-interior ideal in semigroups.
Finaly, we characterize a regular semigroup in terms of an interval valued fuzzy bi-interior
ideal in semigroups.

2. Preliminaries

In this section, we begin with elementary some fundamental concepts about semi-
groups, fuzzy sets, and interval valued fuzzy sets that are necessary for this paper.

By a subsemigroup of a semigroup S we mean a non-empty subset M of S such that
M2 ⊆ M , and by a left (right) ideal of S we mean a non-empty subset M of S such that
SM ⊆ M(MS ⊆ M). By a two-sided ideal or simply an ideal, we mean a non-empty
subset of a semigroup S that is both a left and a right ideal of S. A non-empty subset
M of S is called a quasi-ideal of S if MS ∩ SM ⊆ M . A subsemigroup M of S is called
a bi-ideal of S if MSM ⊆ M . A subsemigroup M of a semigroup S is called an interior
ideal of S if SMS ⊆ M . A subsemigroup M of a semigroup S is said to be a bi-interior
ideal of S if M is a subsemigroup of S and SMS ∩MSM ⊆ M .[5]. We note here that the
properties is hold:

(1) Every left ideal is a bi-interior ideal of S.

(2) Every right ideal is a bi-interior ideal of S.

(3) Every ideal is a bi-interior ideal of S.

(4) Every quasi ideal is a bi-interior ideal of S.

(5) The arbitrary intersection of bi-interior of S is also bi-interior ideal of S.

(6) If M is a bi-interior ideal of S then MS and SM are bi-interior ideals of S [5].

Definition 1. [10] A fuzzy subset η of a non-empty set X is a function η : X → [0, 1].

For any ηi ∈ [0, 1] where i ∈ A define

∨
i∈A

ηi := sup
i∈A

{ηi} and ∧
i∈A

ηi := inf
i∈A

{ηi}.

We see that for any η1, η2 ∈ [0, 1], we have

η1 ∨ η2 = max{η1, η2} and η1 ∧ η2 = min{η1, η2}.

Let Ω[0, 1] be the set of all closed subintervals of [0, 1], i.e.,

Ω[0, 1] = {p̃ = [p−, p+] | 0 ≤ p− ≤ p+ ≤ 1}.

Let p̃ = [p−, p+] and q̃ = [q−, q+] ∈ Ω[0, 1]. Define the operations ⪯, =, ⋏ and ⋎ as
follows:



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 123

(1) p̃ ⪯ q̃ if and only if p− ≤ q− and p+ ≤ q+

(2) p̃ = q̃ if and only if p− = q− and p+ = q+

(3) p̃⋏ q̃ = [(p− ∧ q−), (p+ ∧ q+)]

(4) p̃⋎ q̃ = [(p− ∨ q−), (p+ ∨ q+)].

If p̃ ⪰ q̃, we mean q̃ ⪯ p̃.

For each interval p̃i = [p−i , p
+
i ] ∈ µ[0, 1], i ∈ A where A is an index set, we define

⋏
i∈A

p̃i = [ ∧
i∈A

p−i , ∧
i∈A

p+i ] and ⋎
i∈A

p̃i = [ ∨
i∈A

p−i , ∨
i∈A

p+i ].

Definition 2. [9] Let T be a non-empty set. Then the function µ̃ : T → Ω[0, 1] is called
an interval valued fuzzy set (shortly, IVF set) of T .

Definition 3. [9] Let M be a subset of a non-empty set T . An interval valued character-
istic function of T is defined to be a function χ̃M : T → Ω[0, 1] by

χ̃M (e) =

{
[1, 1] if e ∈ M,

[0, 0] if e /∈ M

for all e ∈ T .

For two IVF sets µ̃ and ϖ̃ of a non-empty set T , define

(1) µ̃ ⊑ ϖ̃ ⇔ µ̃(e) ⪯ ϖ̃(e) for all e ∈ T ,

(2) µ̃ = ϖ̃ ⇔ µ̃ ⊑ ϖ̃ and ϖ̃ ⊑ µ̃,

(3) (µ̃ ⊓ ϖ̃)(e) = µ̃(e)⋏ ϖ̃(e) for all e ∈ T ,

(4) (µ̃ ⊔ ϖ̃)(e) = µ̃(e)⋎ ϖ̃(e) for all e ∈ T .

For two IVF sets µ̃ and ϖ̃ in a semigroup S, define the product µ̃ ◦ ϖ̃ as follows : for
all e ∈ S,

(µ̃ ◦ ϖ̃)(e) =


⋃

e=th

{µ̃(t)⋏ ϖ̃(h)},

[0, 0].

Definition 4. [9] An IVF subset µ̃ of a semigroup S is said to be

(1) an IVF subsemigroup of S if µ̃(uv) ⪰ µ̃(u)⋏ µ̃(v) for all u, v ∈ S,

(2) an IVF left (right) ideal of S if µ̃(uv) ⪰ µ̃(v)(µ̃(uv) ⪰ µ̃(u)) for all u, v ∈ S. An IVF
subset µ̃ of S is called an IVF ideal of S if it is both an IVF left ideal and an IVF
right ideal of S,



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 124

(3) an IVF bi-ideal of S if µ̃ is an IVF subsemigroup and µ̃(uvw) ⪰ µ̃(u) ⋏ µ̃(w) for all
u, v, w ∈ S,

(4) an IVF interior ideal of S if µ̃ is an IVF subsemigroup and µ̃(uav) ⪰ µ̃(a) for all
a, u, v ∈ S,

(5) an IVF quasi-ideal of S if (S̃ ◦ µ̃)(u)⋏ (µ̃ ◦ S̃)(u) ⪯ µ̃(u) for all u ∈ S where S̃ is an
IVF subset of S mapping every element of S on [1, 1].

Theorem 1. [9] Let S be a semigroup and let M be non-empty subset of S. Then M is
a subsemigroup (left ideals, right ideals, interior ideals, bi-ideals, quasi-ideals) of S if and
only if the characteristic set χ̃M is an IVF subsemigroup (left ideals, right ideals, interior
ideals, bi-ideals, quasi-ideals) of S.

3. Interval valued fuzzy bi-interior ideals of Semigroups

In this section, we introduce the notion of an interval valued fuzzy bi-interior ideal and
study the properties of interval valued fuzzy bi-interior ideals of semigroups.

Definition 5. An IVF subsemigroup µ̃ of a semigroup S is called an IVF bi-interior ideal
of S if it satisfies the following condition: χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃⊆̃µ̃,

Example 1. Define µ̃ : T → Ω[0, 1] by

µ̃(e) =

{
[1, 1] if e ∈ T,

[0, 0] if e /∈ T

Then µ̃ is an IVF bi-interior ideal of T .

The next Theorems are studies IVF ideals in semigroup are IVF bi-interior ideals of
semigroups

Theorem 2. Every IVF left ideal of a semigroup S is an IVF bi-interior ideal of S.

Proof. Let µ̃ be an IVF left ideal of S. Let x ∈ S. Then

(χ̃S ◦ µ̃)(x) =
⋃

x=yz

{χ̃S(y)⋏ µ̃(z)} =
⋃

x=yz

{µ̃(z)} ⊆
⋃

x=yz

{µ̃(yz)} =
⋃

x=yz

{µ̃(x)} = µ̃(x).

We have, (µ̃ ◦ χ̃S ◦ µ̃)(x) =
⋃

x=abc{µ̃(a) ⋏ (χ̃S ◦ µ̃)(bc)} ⊆
⋃

x=abc{µ̃(a) ⋏ µ̃(bc)} = µ̃(x).
Now

(χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃)(x) = (χ̃S ◦ µ̃ ◦ χ̃S)(x) ⊓ (µ̃ ◦ χ̃S ◦ µ̃)(x)
⪯ (χ̃S ◦ µ̃ ◦ χ̃S)(x) ⊓ µ̃(x) ⪯ µ̃(x).

Therefor χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃ ⊑ µ̃. Hence µ̃ is an IVF bi-interior ideal of S.



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 125

Theorem 3. Every IVF right ideal of a semigroup S is an IVF bi-interior ideal of S.

Proof. It follows Theorem 2.

Corollary 1. Every IVF ideal of a semigroup S is an IVF bi-interior ideal of S.

Definition 6. Let µ̃ be an IVF set in a non-empty set X. Define U(µ̃; t̃) = {x ∈ X|t̃ ⊆
µ̃(x)} where t̄ ∈ Ω[0, 1] is called the IVF level set of µ̃.

Theorem 4. Let S be a semigroup and µ̃ be a non-empty IVF set of S. An IVF set µ̃ is
an IVF bi-interior ideal of a semigroup S if and only if the IVF level set U(µ̃; t̃) of S is a
bi-interior ideal of a semigroup S for every t̃ ∈ Ω[0, 1], where U(µ̃; t̃) ̸= ∅.

Proof. Assume that µ̃ is an IVF bi-interior ideal of S and let x ∈ SU(µ̃; t̃)S ∩
U(µ̃; t̃)SU(µ̃; t̃). Then x = bau = cde where b, u, d ∈ S and a, c, e ∈ U(µ̃; t̃). Then
t̃ ⪯ (χ̃S ◦ µ̃ ◦ χ̃S)(x) and t̃ ⪯ (µ̃ ◦ χ̃S ◦ µ̃)(x) implies that t̃ ⪯ µ̃(x) Then x ∈ U(µ̃; t̃).
Therefore U(µ̃; t̃) is a bi-interior ideal of S.

Conversely suppose that U(µ̃; t̃) is a bi-interior ideal of S, for all t̃ ∈ Im(µ̃).
Let x, y ∈ S. Then µ̃(x) = t̃1, µ̃(y) = t̃2, t̃1 ⪰ t̃2. Then x, y ∈ U(µ̃; t̃).
Thus SU(µ̃; l̃)S ⊓ U(µ̃; l̃)SU(µ̃; l̃) ⪯ U(µ̃; l̃), for all l̃ ∈ Im(µ̃).
Suppose t̃ = min{Im(µ̃)}. Then SU(µ̃; t̃)S ⊓ U(µ̃; t̃)SU(µ̃; t̃) ⪯ U(µ̃; t̃).
Therefor χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃ ⊑ µ̃. Hence µ̃ is an IVF bi-interior ideal of S.

Theorem 5. Let M be a non-empty subset of a semigroup S and χ̃M be the characteristic
IVF set of M . Then M is a bi-interior ideal of a semigroup S if and only if χ̃M is an
IVF bi-interior ideal of a semigroup S.

Proof. Suppose M is a bi-interior ideal of S. Then M is a subsemigroup of S. Thus
by Theorem 1, χ̃M is an IVF subsemigroup of S. Let x ∈ S. Since M is a bi-interior ideal
of S, we have SMS ∩MSM ⊆ M . Thus

(χ̃S ◦ χ̃M ◦ χ̃S ⊓ χ̃M ◦ χ̃S ◦ χ̃M )(x) = (χ̃S ◦ χ̃M ◦ χ̃S)(x)⋏ (χ̃M ◦ χ̃S ◦ χ̃M )(x)

= χ̃SMS(x)⋏ ˜χMSM (x)

= χ̃SIS∩MSM (x) ⪯ χ̃M (x).

Therefore χ̃S ◦ χ̃M ◦ χ̃S ⊓ χ̃M ◦ χ̃S ◦ χ̃M ⊑ χ̃M .
Hence χ̃M is an IVF bi-interior ideal of S.

Conversely, suppose that χ̃M is an IVF bi-interior ideal of S.
Then χ̃M is an IVF subsemigroup of S. Thus by Theorem 1,
M is a subsemigroup of S. Let x ∈ M . We have

(χ̃S ◦ χ̃M ◦ χ̃S)(x)⋏ (χ̃M ◦ χ̃S ◦ χ̃M )(x) ⪯ χ̃M (x)

⇒ χ̃SMS(x)⋏ χ̃MSM (x) ⪯ χ̃M (x)

⇒ χ̃SMS∩MSM (x) ⪯ χ̃M (x).

Therefore SMS ∩MSM ⊆ M . Hence M is a bi-interior ideal of S.



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 126

Theorem 6. If µ̃ and λ̃ are IVF bi-interior ideals of a semigroup S, then µ̃⊓ λ̃ is an IVF
bi-interior ideal of S.

Proof. Let µ̃ and λ̃ be IVF bi-interior ideals of S. Then

(χ̃S ◦ µ̃∩̃λ̃)(x) =
⋃
x=ab

{χ̃S(a)⋏ (µ̃∩̃λ̃)(b)}

=
⋃
x=ab

{χ̃S(a)⋏ µ̃(b)⋏ λ̃(b)}

=
⋃
x=ab

{{χ̃S(a)⋏ µ̃(b)}⋏ {χ̃S(a) ∩ λ̃(b)}}

=
⋃
x=ab

{χ̃S(a)⋏ µ̃(b)} ∩
⋃
x=ab

{χ̃S(a) ∩ λ̃(b)}

= (χ̃S ◦ µ̃)(x)⋏ (χ̃S ◦ λ̃)(x)
= (χ̃S ◦ µ̃∩̃χ̃S ◦ λ̃)(x).

Therefore χ̃S ◦ µ̃ ⊓ λ̃ = χ̃S ◦ µ̃ ⊓ χ̃S ◦ λ̃.

(µ̃∩̃λ̃ ◦ χ̃S ◦ µ̃ ⊓ λ̃)(x) =
⋃

x=abc

{(µ̃∩̃λ̃)(a)⋏ (χ̃S ◦ µ̃∩̃λ̃)(bc)}

=
⋃

x=abc

{(µ̃ ⊓ λ̃)(a)⋏ {(χ̃S ◦ µ̃∩̃χ̃S ◦ λ̃)(bc)}}

=
⋃

x=abc

{(µ̃ ⊓ λ̃)(a)⋏ {(χ̃S ◦ µ̃)(bc)⋏ (χ̃S ◦ λ̃)(bc)}}

=
⋃

x=abc

{{µ̃(a)⋏ (χ̃S ◦ µ̃)(bc)} ∩ {λ̃(a)⋏ (χ̃S ◦ λ̃)(bc)}}

= (µ̃ ◦ χ̃S ◦ µ̃)(x)⋏ (λ̃ ◦ χ̃S ◦ λ̃)(x)
= (µ̃ ◦ χ̃S ◦ µ̃∩̃λ̃ ◦ χ̃S ◦ λ̃)(x).

Therefore µ̃ ⊓ λ̃ ◦ χ̃S ◦ µ̃ ⊓ λ̃ = µ̃ ◦ χ̃S ◦ µ̃ ⊓ λ̃ ◦ χ̃S ◦ λ̃.
Then

(χ̃S ◦ µ̃ ⊓ λ̃ ◦ χ̃S)(x)⋏ (µ̃ ⊓ λ̃ ◦ χ̃S ◦ µ̃ ⊓ λ̃)(x)
= (χ̃S ◦ µ̃ ◦ χ̃S)(x)⋏ (µ̃ ◦ χ̃S ◦ µ̃)(x)⋏ (χ̃S ◦ λ̃ ◦ µ̃χS )(x)⋏ (λ̃ ◦ χ̃S ◦ λ̃)(x)
⪯ (µ̃ ⊓ λ̃)(x).

Therefore (χ̃S ◦ µ̃ ⊓ λ̃ ◦ χ̃S) ⊓ (µ̃ ⊓ λ̃ ◦ χ̃S ◦ µ̃ ⊓ λ̃) ⊑ µ̃ ⊓ λ̃.
Hence µ̃∩̃λ̃ is an IVF bi-interior ideal of a semigroup S.

We know that every IVF ideal is an IVF bi-interior ideal then the following theorem
holds.

Theorem 7. If µ̃ and λ̃ are IVF right ideals and an IVF left ideal of a semigroup S
respectively. Then µ̃ ⊓ λ̃ is an IVF bi-interior ideal of S.



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 127

Proof. Assume that µ̃ and λ̃ are IVF right ideals and an IVF left ideal of S respectively.
Then by Theorems 2 and 3, we have µ̃ and λ̃ are IVF bi-interior ideals of S. By Theorem
6 we have µ̃ ⊓ λ̃ is an IVF bi-interior ideal of S.

The following are tools the converse of an IVF bi-interior ideals is IVF ideals on
semigroups.

Definition 7. A semigroup S is called regular if for all a ∈ S there exists x ∈ S such that
a = axa.

Theorem 8. If µ̃ be an IVF quasi-ideal of a regular semigroup S.Then µ̃ is an IVF ideal
of a semigroup S.

Proof. Assume that µ̃ is an IVF quasi-ideal of S and let x, y ∈ S. Then

µ̃(xy) ⪰ (µ̃ ◦ χ̃S)(xy)⋏ (χ̃S ◦ µ̃)(xy)
=

⋃
xy=ab

{µ̃(a)⋏ χ̃S(b)}⋏
⋃

xy=ij
{χ̃S(i)⋏ µ̃(j)}

⪰ µ̃(x) ∩ χ̃S(y)⋏ χ̃S(x)⋏ µ̃(y)
= (µ̃(x)⋏ [1, 1])⋏ ([1, 1]⋏ µ̃(y)) = µ̃(x) ∩ µ̃(y).

Thus µ̃(xy) ⪰ µ̃(x) ∩ µ̃(y). Hence µ̃ is an IVF subsemigroup of S. Let x, y, z ∈ S. Then

µ̃(xyz) ⪰ (µ̃ ◦ χ̃S)(xyz)⋏ (χ̃S ◦ µ̃)(xyz)
=

⋃
xyz=ab

{µ̃(a)⋏ χ̃S(b)}⋏
⋃

xyz=ij
{χ̃S(i)⋏ µ̃(j)}

⪰ µ̃(x)⋏ χ̃S(yz)⋏ χ̃S(xy)⋏ µ̃(z)
= (µ̃(x)⋏ [1, 1])⋏ ([1, 1]⋏ µ̃(z)) = µ̃(x)⋏ µ̃(z).

Thus µ̃(xyz) ⪰ µ̃(x) ⋏ µ̃(z). Hence µ̃ is an IVF bi-ideal of S. Since S is regular, µ̃ is an
IVF bi-ideal of S and x, y ∈ S we have xy ∈ (xSx)S ⊆ xSx. Thus there exists k ∈ S such
that xy = xkx. So µ̃(xy) = µ̃(xkx) ⪰ µ̃(x) ⋏ µ̃(x) = µ̃(x). Similarly, we can show that
µ̃(xy) ⪰ µ̃(y). Thus µ̃ is an IVF left ideal of S. Hence µ̃ is an IVF ideal of S.

Theorem 9. Let S be a regular semigroup. Then µ̃ is an IVF bi-interior ideal of S if and
only if µ̃ is an IVF quasi-ideal of S.

Proof. Let µ̃ be an IVF bi-interior ideal of S and x ∈ S. Then
(χ̃S ◦ µ̃ ◦ χ̃S)(x)⋏ (µ̃ ◦ µ̃χS ◦ µ̃)(x) ⪰ µ̃(x). Suppose (χ̃S ◦ µ̃)(x) ⪰ µ̃(x). Since S is regular,
there exists y ∈ S such that x = xyx. Then

(µ̃ ◦ χ̃S ◦ µ̃)(x) =
⋃

x=xyx

{µ̃(xy)⋏ (χ̃S ◦ µ̃)(x)} ⊇
⋃

x=xyx

{µ̃(x)⋏ µ̃(x)} = µ̃(x).

Which is a contradiction. Therefore µ̃ is an IVF quasi-ideal of S. By Theorem 8, converse
is true.



T. Gaketem, T. Prommai / Eur. J. Pure Appl. Math, 16 (1) (2023), 121-130 128

Theorem 10. If µ̃ be an IVF bi-interior ideal of a regular semigroup S. Then µ̃ is an
IVF ideal of a semigroup S.

Proof. Suppose that µ̃ is an IVF bi-interior ideal of S. Then by Theorem 9, µ̃ is an
IVF quasi-ideal of S. Thus by Theorem 8, µ̃ is an IVF ideal of S. Hence the theorem is
complete.

The following theorems are a tool in characterization regular semigroup in terms of
IVF bi-interior ideals on semigroups.

Theorem 11. [4] For non-empty subsets G and H of a semigroup S, we have

(1) χ̃G◦χ̃H = ˜χGH ,

(2) χ̃G ⊓ χ̃H = ˜χG∩H .

Proof. It is straightforward.

Theorem 12. Let S be a semigroup. Then S is a regular semigroup if and only if
B = SBS ∩ SBS, for every bi-interior ideal of S.

The following theorems are characterization regular semigroup in terms of IVF bi-
interior ideals in semigroups.

Theorem 13. Let S be a semigroup. Then S is a regular if and only if
µ̃ = χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃ for every IVF bi-interior ideal of a semigroup S.

Proof. Let µ̃ be an IVF bi-interior ideal of the regular semigroup S and let x ∈ S.
Since S is regular, there exists a ∈ S such that x = xax. Thus

(µ̃ ◦ χ̃S ◦ µ̃)(x) =
⋃

x=xax

{µ̃(x)⋏ (χ̃S ◦ µ̃)(ax)}

=
⋃

x=xax

{µ̃(x)⋏
⋃

ax=yz

{χ̃S(y)⋏ µ̃(z)}}

⊇
⋃

x=xax

{µ̃(x)⋏ µ̃(x)} = µ̃(x).

Similarly, (χ̃S ◦ µ̃ ◦ χ̃S)(x) ⪯ µ̃(x). Therefore µ̃ = χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃.
Conversely, suppose that B is a bi-interior ideal of a semigroup S. Then by Theorem

5, χ̃B is an IVF bi-interior ideal of the semigroup S. Thus by Theorem 11,

µ̃χB (x) = χ̃S ◦ µ̃χB ◦ χ̃S(x)⋏ µ̃χB ◦ χ̃S ◦ µ̃χB (x)

= µ̃χSBS (x)⋏ µ̃χBSB (x) = µ̃χSBS∩BSB (x).

Therefore B = SBS ∩BSB. By Theorem 12, S is a regular semigroup.

Theorem 14. Let S be a semigroup. Then S is regular if and only if
µ̃ ⊓ λ̃⊆̃λ̃ ◦ µ̃ ◦ λ̃ ⊓ µ̃ ◦ λ̃ ◦ µ̃ for every IVF bi-interior ideal µ̃ and every IVF ideal λ̃ of S.



REFERENCES 129

Proof. Let µ̃ be an IVF bi-interior ideal and λ̃ be an IVF ideal of a regular semigroup
S and let x ∈ S. Then there exists y ∈ S such that x = xyx.

(µ̃ ◦ λ̃ ◦ µ̃)(x) =
⋃

x=xyx

{(µ̃ ◦ λ̃)(xy)⋏ µ̃(x)}

=
⋃

x=xyx

{
⋃

xy=xyxy

{µ̃(x) ∩ λ̃(yxy)}⋏ µ̃(x)}

⪰ {µ̃(x)⋏ λ̃(x)}⋏ µ̃(x) = µ̃(x)⋏ λ̃(x) = (µ̃ ⊓ λ̃)(x).

(λ̃ ◦ µ̃)(x) =
⋃

x=xyx

{λ̃(xy)⋏ µ̃(x)} ⪰ {λ̃(x)⋏ µ̃(x)} = (µ̃ ⊓ λ̃)(x).

Therefore µ̃ ◦ λ̃ ◦ µ̃ ⊑ µ̃ ⊓ λ̃. Similary, we can prove λ̃ ◦ µ̃ ◦ λ̃ ⊒ µ̃ ⊓ λ̃.
Hence µ̃ ⊓ λ̃ ⊑ λ̃ ◦ µ̃ ◦ λ̃ ⊓ µ̃ ◦ λ̃ ◦ µ̃.

Conversely, suppose that the condition holds. Let µ̃ be an IVF bi-interior ideal.
We have µ̃ ⊓ χ̃S ⊑ χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃ and implies that µ̃ ⊑ χ̃S ◦ µ̃ ◦ χ̃S ⊓ µ̃ ◦ χ̃S ◦ µ̃.
By Theorem 13, S is a regular semigroup.

4. Conclusion

In this paper, we give the concept of IVF bi-interior ideals in semigroups and we
study properties of IVF bi-interior ideals in semigroups. Moreover, we prove relationship
between IVF bi-interior ideals and bi-interior ideals. In the future we study other kinds
of IVF bi-quasi interior ideals in semigroup or algebric system.

Acknowledgements

The authors are grateful to the School of Science, University of Phayao for grant
support.

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