Pythagorean fuzzy weak bi-ideals of Γ- near ring
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The document discusses the concept of Pythagorean fuzzy weak bi-ideals and their properties within the framework of γ-near-rings. It introduces both Pythagorean fuzzy weak bi-ideals and interval-valued Pythagorean fuzzy weak bi-ideals, and establishes their characteristics and applications in homomorphism. The research emphasizes the mathematical definitions and theorems associated with these concepts, contributing to the field of fuzzy algebra.
![Corresponding Author: marynirla@gmail.com
http://dx.doi.org/10.22105/jfea.2021.300356.1160
E-ISSN: 2717-3453 | P-ISSN: 2783-1442
|
Abstract
1 | Introduction
Zadeh [26] defined Fuzzy Set (FS) to deal with uncertainty. Atanassov [2] presented the notion of
Intuitionistic FS (IFS) and studied some of its properties. Later, Yager [24], [25] defined and studied
the properties of Pythagorean Fuzzy Set (PFS) and also used PFS to solve Multi-Criteria Decision-
Making (MCDM) problems. Booth [3] presented the properties of Γ-near-rings. Chinnadurai and
Kadalarasi [7] studied the near-ring properties of Fuzzy Weak Bi-Ideals (FWBI). Chinnadurai et al.
[4],[5] studied the Γ-near-rings characterization of fuzzy weak bi-ideal and interval-valued fuzzy weak
bi-ideal. Later, Chinnadurai et al. [6] discussed the Γ-near-rings properties of interval-valued fuzzy
ideals.
Akram [1] established the properties of fuzzy lie algebras. Kim and Kim [13] studied the near-rings
concept of fuzzy ideals. Kaviyarasu et al. [10]-[12] studied the different type of ideals in INK- algebras.
Jun et al. [9] presented the notion of fuzzy ideals and studied their properties in Γ-near-rings.
Manikantan [14] defined and studied some of the near-rings properties of fuzzy bi-ideals.
Meenakumari and Chelvam [15] presented the Γ-near-ringsproperties of fuzzy bi-ideals. Narayanan
and Manikatan [16] introduced the near-rings notions of fuzzy subnear-ring, fuzzy ideal, and fuzzy
quasi-ideal.
Journal of Fuzzy Extension and Applications
www.journal-fea.com
J. Fuzzy. Ext. Appl. Vol. 2, No. 3 (2021) 297–320.
Paper Type: Research Paper
Pythagorean Fuzzy Weak Bi-Ideals of 𝚪- Near Ring
J. Marynirmala1,* , D. Sivakumar1
1 Department of Mathematics, Annamalai University,Annamalainagar, Tamilnadu, India; marynirla@gmail.com.
Citation:
Marynirmala, J., & Sivakumar, D. (2021). Pythagorean fuzzy weak bi-ideals of Γ- near ring. Journal
of fuzzy extension and application, 2 (3), 297-320.
Accept: 20/08/2021
Revised: 23/07/2021
Reviewed: 30/05/2021
Received: 18/05/2021
We present the notion of Pythagorean Fuzzy Weak Bi-Ideals (PFWBI) and interval valued Pythagorean fuzzy weak bi-
ideals of Γ-near-rings and studies some of its properties. We present the notion of interval valued Pythagorean fuzzy
weak bi-ideal and establish some of its properties. We study interval valued Pythagorean fuzzy weak bi-ideals of Γ-near-
ring using homomorphism.
Keywords: Pythagorean fuzzy, Fuzzy ideals, Homomorphism, Near ring.
Licensee Journal
of Fuzzy Extension and
Applications. This article
is an open access article
distributed under the
terms and conditions of
the Creative Commons
Attribution (CC BY)
license
(http://creativecommons.
org/licenses/by/4.0).](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-1-320.jpg)
![298
Marynirmala
and
Sivakumar
|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
Pilz [17] introduced the concept of anti fuzzy soft gamma rings and studied their properties. Rao and
Swaminathan [20] presented the notion of anti-homomorphism between two fuzzy rings and established
its properties. Rao and Venkateswarlu [21] studied the properties of anti fuzzy ideal and pre-image of
fuzzy ideal. Satyanarayana [18] dealt with the theory of near-rings. Salah Abou-Zaid [19] studied fuzzy
ideals of a near-ring. Chelvam and Meenakumari [22] obtained the characterization for gamma near-
fields. Thillaigovindan et al. [23] introduced the notion of generalized T-fuzzy bi-ideals of a gamma-
semigroup. Cho et al. [8] presented the notion of bi-ideals in near-rings and used it in near-fields.
We introduce the notion of Pythagorean fuzzy weak bi-ideal of Γ-near-rings and interval valued
Pythagorean fuzzy weak bi-ideal of Γ-near-rings. We discuss and present some properties of
homomorphism of Pythagorean fuzzy weak bi-ideal and homomorphism of an interval valued
Pythagorean fuzzy weak bi-ideal in gamma near-ring.
2| Preliminaries
Definition 1. [16]. A fuzzy set 𝜋 of a 𝛤-near-ring 𝑀 is called a fuzzy left (resp. right) ideal of 𝑀 if
I. 𝜋(𝑘 − 𝑙) ≥ 𝑚𝑖𝑛{𝜋(𝑘),𝜋(𝑙)}, for all 𝑘, 𝑙 ∈ 𝑀,
II. 𝜋(𝑦 + 𝑥 − 𝑦) ≥ 𝜋(𝑥) , for all 𝑥, 𝑦 ∈ 𝑀,
III. 𝜋(𝑢𝛼(𝑥 + 𝑣) − 𝑢𝛼𝑣) ≥ 𝜋(𝑥) , (resp. 𝜋(𝑥𝛼𝑢) ≥ 𝜋(𝑥)) for all 𝑥, 𝑢, 𝑣 ∈ 𝑀 and 𝛼 ∈ 𝛤.
Definition 2. [15]. A fuzzy set 𝜋 of 𝑀 is called a fuzzy bi-ideal of 𝑀 if
I. 𝜋(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋(𝑥),𝜋(𝑦)} for all 𝑥, 𝑦 ∈ 𝑀,
II. 𝜋(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋(𝑥),𝜋(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤.
Definition 3. [2]. An intuitionistic fuzzy set A is a nonempty set X is an object having the form 𝐴 =
{𝑥,(𝜋𝐴(𝑥), 𝜗𝐴(𝑥)): 𝑥 ∈ 𝑋} where the functions 𝜋𝐴 : 𝑋 → [0,1] and 𝜗𝐴 : 𝑋 → [0,1] define the degree of
membership and non-membership of the element 𝑥 ∈ 𝑋 to the set 𝐴, which is a subset of X respectively
0 ≤ 𝜋𝐴(𝑥) + 𝜗𝐴(𝑥) ≤ 1 we use the simple 𝐴 = (𝜋𝐴, 𝜗𝐴) .
Definition 4. [25]. A Pythagorean fuzzy subset 𝑃 is a nonempty set X is an object having the form 𝑃 =
{(𝑥,𝜋𝑃(𝑥), 𝜗𝑃(𝑥))/𝑥 ∈ 𝑋}, where the functions 𝜋𝑃: 𝑋 → [0,1] and 𝜗𝑃:𝑋 → [0,1] denote the degree of
membership and non membership of each element 𝑥 ∈ 𝑋 to the set 𝑃, respectively, and 0 ≤ (𝜋𝑃(𝑥))2
+
(𝜗𝑃(𝑥))2
≤ 1 for all 𝑥 ∈ 𝑋. For the sake of simplicity, for the Pythagorean fuzzy subset 𝑃 =
{(𝑥,𝜋𝑃(𝑥), 𝜗𝑃(𝑥)) /𝑥 ∈ 𝑋}.
3| Pythagorean Fuzzy Weak Bi-Ideals of 𝚪near Ring
In this section, we initiate the notion of Pythagorean fuzzy weak bi-ideal of 𝑀 and discuss some of its
properties.
Definition 4. A subgroup 𝑊 of (𝑀, +) is said to be a weak bi-ideal of 𝑀 if 𝑊𝛤𝑊𝛤𝑊 ⊆ 𝑊.
Definition 5. A Pythagorean fuzzy set𝑃 = (𝜋𝑃, 𝜗𝑃) of 𝑀 is called a Pythagorean fuzzy weak bi-ideal of
𝑀, if
I. 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)}.
II. 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
III. 𝜋𝑃(𝑥𝛾𝑦𝛾𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}.
IV. 𝜗𝑃(𝑥𝛾𝑦𝛾𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤.](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-2-320.jpg)
![299
Pythagorean
fuzzy
weak
bi-ideals
of
Γ-
near
ring
Example 1. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation ` + ′ and 𝛤 = {𝛾} be a nonempty
set of binary operations as the following tables:
and
Let 𝜋𝑃 : 𝑀 → [0,1] be a Pythagorean fuzzy subset defined by 𝜋𝑃(𝑤) = 0.7, 𝜋𝑃(𝑥) = 0.6, 𝜋𝑃(𝑦) = 𝜋𝑃(𝑧) = 0.5.
and 𝜗𝑃(𝑤) = 0.3, 𝜗𝑃(𝑥) = 0.5, 𝜗𝑃(𝑦) = 0.8 = 𝜗𝑃(𝑧). Then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal
of 𝑀.
Theorem 1. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy subgroup of M. Then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean
fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑥, 𝑦, 𝑧, 𝑦1, 𝑦2 ∈ 𝑀 and
𝛼, 𝛽 ∈ 𝛤 such that 𝑥 = 𝑦𝛼𝑧 and 𝑦 = 𝑦1𝛽𝑦2. Then
(πP ⋆ πP ⋆ πP)(x) = sup
x=yαz
{min{(πP ⋆ πP)(y), πP(z)}}
= sup
x=yαz
{min{ sup
y=y1βy2
min{πP(y1), πP(y2)}, πP(z)}}
= sup
x=yαz
sup
y=y1βy2
{min{min{πP(y1), πP(y2)}, πP(z)}}
= sup
x=y1βy2αz
{min{πP(y1), πP(y2), πP(z)}},
since 𝜋𝑃 is a fuzzy weak bi-ideal of 𝑀,
πP(y1βy2αz) ≥ min{πP(y1), πP(y2), πP(z)}
≤ sup
x=y1βy2αz
πP(y1βy2αz)
= πP(x).
And
(ϑP ⋆ ϑP ⋆ ϑP)(x) = inf
x=yαz
{min{(ϑP ⋆ ϑP)(y), ϑP(z)}},
= inf
x=yαz
{max{ inf
y=y1βy2
min{ϑP(y1), ϑP(y2)}, ϑP(z)}}
+ 𝐰 𝐱 𝐲 𝐳
𝐰 w x y z
𝐱 x w z y
𝐲 y z w x
𝐳 z y x w
𝛄 𝐰 𝐱 𝐲 𝐳
𝐰 w x w x
𝐱 w x w x
𝐲 w x y z
𝐳 w x y z](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-3-320.jpg)
![302
Marynirmala
and
Sivakumar
|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
Theorem 2. Every Pythagorean fuzzy bi-ideal of 𝑀 is a Pythagorean fuzzy weak bi-ideal of 𝑀.
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy bi-ideal of 𝑀.
Then 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
We have 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃.
This implies that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃
and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of M.
Example 2. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation+ and 𝛤 = {𝛼} be a nonempty
set of binary operations as the following tables:
and
Let 𝜋𝑃 : 𝑀 → [0,1] be a fuzzy set defined by 𝜋𝑃(𝑤) = 0.9, 𝜋𝑃(𝑥) = 0.4 = 𝜋𝑃(𝑦) and 𝜋𝑃(𝑧) = 0.6, and
𝜗𝑃(𝑤) = 0.1,𝜗𝑃(𝑥) = 0.5 = 𝜗𝑃(𝑦),𝜗𝑃(𝑧) = 0.3. Then 𝜋𝑃 is a fuzzy weak bi-ideal of 𝑀. But 𝜋𝑃 is not a fuzzy
ideal and bi-ideal of 𝑀 and 𝜋𝑃(𝑧𝛾𝑦𝛾𝑧) = 𝜋𝑃(𝑦) = 0.4 > 0.6 = 𝑚𝑖𝑛{𝜋𝑃(𝑧), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼(𝑧 + 𝑤) −
𝑥𝛼𝑤) ≤ 𝜗𝑃(𝑧) = 0.5 ≤ 0.3 and 𝜗𝑃(𝑧𝛾𝑥𝛾𝑧) = 𝜗𝑃(𝑥) = 0.5 ≤ 0.3 = min{𝜗𝑃(𝑧),𝜗𝑃(𝑧)}.
Theorem 3. Let {(𝜋𝑃𝑖
, 𝜗𝑃𝑖
)|𝑖 ∈ 𝛺} be family of Pythagorean fuzzy weak bi-ideals of a near-ring 𝑀, then
⋂𝑖∈𝛺 𝜋𝑃𝑖
and ∪𝑖∈𝛺 𝜗𝑃𝑖
are also a Pythagorean fuzzy weak bi-idea of 𝑀, where 𝛺 is any index set.
Proof. Let {𝜋𝑖}𝑖∈𝛺 be a family of Pythagorean fuzzy weak bi-ideals of 𝑀.
Let 𝑥, 𝑦, 𝑧 ∈ 𝑀, 𝛼, 𝛽 ∈ 𝛤 and 𝜋 = ⋂𝑖∈𝛺 𝜋𝑖.
Then,⋂𝑖∈𝛺 𝜋𝑃𝑖
(𝑥) = ⋂𝑖∈𝛺 𝜋𝑃𝑖
(𝑥) = (𝑖𝑛𝑓
𝑖∈𝛺
𝜋𝑃𝑖
)(𝑥) = 𝑖𝑛𝑓
𝑖∈𝛺
𝜋𝑃𝑖
(𝑥)and
∪𝑖∈𝛺 𝜗𝑃𝑖
(𝑥) =∪𝑖∈𝛺 𝜗𝑃𝑖
(𝑥) = (𝑠𝑢𝑝
𝑖∈𝛺
𝜗𝑃𝑖
)(𝑥) = 𝑠𝑢𝑝
𝑖∈𝛺
𝜗𝑃𝑖
(𝑥).
⋂
i∈Ω
πPi
(x − y) = inf
i∈Ω
πPi
(x − y)
+ 𝐰 𝐱 𝐲 𝐳
𝐰 w x y z
𝐱 x w z y
𝐲 y z w x
𝐳 z y x w
𝛂 𝐰 𝐱 𝐲 𝐳
𝐰 w w w w
𝐱 w x w x
𝐲 w w y y
𝐳 w x y z](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-6-320.jpg)
![303
Pythagorean
fuzzy
weak
bi-ideals
of
Γ-
near
ring
≥ inf
i∈Ω
min{πPi
(x),πPi
(y)}
= min{inf
i∈Ω
πPi
(x), inf
i∈Ω
πPi
(y)}
= min{⋂
i∈Ω
πPi
(x), ⋂
i∈Ω
πPi
(y)}
∪i∈Ω ϑPi
(x − y) = sup
i∈Ω
πPi
(x − y)
≤ sup
i∈Ω
max{ϑPi
(x),ϑPi
(y)}
= max{sup
i∈Ω
ϑPi
(x),sup
i∈Ω
ϑPi
(y)}
= max{∪i∈Ω ϑPi
(x),∪i∈Ω ϑPi
(y)}.
And,
⋂
i∈Ω
πPi
(xαyβz) = inf
i∈Ω
πPi
(xαyβz)
≥ inf
i∈Ω
min{πPi
(x),πPi
(y), πPi
(z)}
= min{inf
i∈Ω
πPi
(x), inf
i∈Ω
πPi
(y), inf
i∈Ω
πPi
(z)}
= min{⋂
i∈Ω
πPi
(x), ⋂
i∈Ω
πPi
(y), ⋂
i∈Ω
πPi
(z)}
∪i∈Ω ϑPi
(xαyβz) = sup
i∈Ω
ϑPi
(xαyβz)
≤ sup
i∈Ω
max{ϑPi
(x), ϑPi
(y),ϑPi
(z)}
= max{sup
i∈Ω
ϑPi
(x), sup
i∈Ω
ϑPi
(y),sup
i∈Ω
ϑPi
(z)}
= max{∪i∈Ω ϑPi
(x),∪i∈Ω ϑPi
(y),∪i∈Ω ϑPi
(z)}.
Hence the set ⋂𝑖∈𝛺 𝜋𝑃𝑖
and ∪𝑖∈𝛺 𝜗𝑃𝑖
are also a family of Pythagorean fuzzy weak bi-ideals of M.
Theorem 4. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be a Pythagorean fuzzy subset of M. Then 𝑈(𝜋𝑃;𝑡) and 𝐿(𝜗𝑃;𝑠) is a
Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃𝑡 is a weak bi-ideal of 𝑀, for all 𝑡 ∈ [0,1].
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀.
Let 𝑠, 𝑡 ∈ [0,1] such that 𝑥, 𝑦 ∈ 𝑈(𝜋𝑃;𝑡).
Then 𝜋𝑃(𝑥) ≥ 𝑡 and 𝜋𝑃(𝑦) ≥ 𝑡,
Then 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)} ≥ 𝑚𝑖𝑛{𝑡,𝑡} = 𝑡 and
𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)} ≤ 𝑚𝑎𝑥{𝑠, 𝑠} = 𝑠.](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-7-320.jpg)
![304
Marynirmala
and
Sivakumar
|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
Thus 𝑥 − 𝑦 ∈ 𝑈(𝜋𝑃 𝑡).Let 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃𝑡 and 𝛼, 𝛽 ∈ 𝛤.
This implies that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)} ≥ 𝑚𝑖𝑛{𝑡,𝑡, 𝑡} = 𝑡, and
𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} ≤ 𝑚𝑎𝑥{𝑠, 𝑠, 𝑠} = 𝑠.
Therefore 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑈(𝜋𝑃; 𝑠).
Hence 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is a weak bi-ideal of 𝑀.
Conversely, assume that 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is a weak bi-ideal of 𝑀, for all 𝑠, 𝑡 ∈ [0,1].
Let 𝑥, 𝑦 ∈ 𝑀. Suppose
𝜋𝑃(𝑥 − 𝑦) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}and 𝜗𝑃(𝑥 − 𝑦) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
Choose 𝑡 such that 𝜋(𝑥 − 𝑦) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)}.
This implies that 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡 and 𝜋𝑃(𝑥 − 𝑦) < 𝑡.
Then we have 𝑥, 𝑦 ∈ 𝜋𝑃𝑡 but 𝑥 − 𝑦 ∈ 𝜋𝑃𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠 and 𝜗𝑃(𝑥 − 𝑦) > 𝑠, we have 𝑥, 𝑦 ∈ 𝜗𝑃𝑠
but 𝑥 − 𝑦 ∈ 𝜗𝑃𝑠 a contradiction.
Thus 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
If there exist 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤 such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) >
𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}.
Choose 𝑡 such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}.
Choose 𝑠 such that 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}.
Then 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡, 𝜋𝑃(𝑧) > 𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠, 𝜗𝑃(𝑧) < 𝑠 and 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡.
So, 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃 > but 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜋𝑃𝑡, and 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜗𝑃𝑠, which is a contradiction.
Hence 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}, 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦), 𝜗𝑃(𝑧)}.
Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of M.
Theorem 5. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy weak bi-ideal of 𝑀 then the set 𝑀𝜋,𝜗 = {𝑥 ∈
𝑀|𝜋𝑃(𝑥) = 𝜋𝑃(0) = 𝜗𝑃(𝑥)} is aweak bi-ideal of 𝑀.
Proof. Let 𝑥, 𝑦 ∈ 𝑀(𝜋𝑃,𝜗𝑃). Then 𝜋𝑃(𝑥) = 𝜋𝑃(0),𝜋𝑃(𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥) = 0, 𝜗𝑃(𝑦) = 0 and
𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)}
= min{𝜋𝑃(0),𝜋𝑃(0)}
= 𝜋𝑃(0), and
𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}
= max{𝜗𝑃(0), 𝜗𝑃(0)}](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-8-320.jpg)
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= 𝜗𝑃(0).
So 𝜋𝑃(𝑥 − 𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥 − 𝑦) = 𝜗𝑃(0).
Thus 𝑥 − 𝑦 ∈ 𝑀𝜋𝑃
, 𝑥 − 𝑦 ∈ 𝑀𝜗𝑃
. For every 𝑥, 𝑦, 𝑧 ∈ 𝑀𝜋𝑃
and 𝛼, 𝛽 ∈ 𝛤. We have
πP(xαyβz) ≥ min{πP(x),πP(y), πP(z)}
= min{πP(0), πP(0), πP(0)}
= πP(0).
And
ϑP(xαyβz) ≤ max{ϑP(x),ϑP(y),ϑP(z)}
= max{ϑP(0),ϑP(0), ϑP(0)}
= ϑP(0).
Thus 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜋𝑃
, 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜗𝑃
. Hence 𝑀(𝜋𝑃,𝜗𝑃) is a weak bi-ideal of M.
4| Homomorphism of Pythagorean Fuzzy Weak Bi-Ideals of 𝚪-Near-
Rings
In this section, we characterize Pythagorean fuzzy weak bi-ideals of 𝛤-near-rings using homomorphism.
Definition 6. Let 𝑓 be a mapping from a set 𝑀 to a set 𝑆. Let 𝑓 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy subsets
of 𝑀 and 𝑆, resp. then 𝑓 is image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑆 defined by
f(πP)(y) =
{
sup
x∈f−1(y)
πP(x) if f −1
(y) ≠ ∅
0 otherwise.
f(ϑP)(y) =
{
inf
x∈f−1(y)
ϑP(x) if f −1
(y) ≠ ∅
1 otherwise
And the pre-image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑀 defined by
𝑓−1
(𝜋𝑃(𝑥)) = 𝜋𝑃(𝑓(𝑥)), 𝑓−1
(𝜗𝑃(𝑥)) = 𝜗𝑃(𝑓(𝑥)) for all 𝑥 ∈ 𝑀 and 𝑓−1
(𝑦) = {𝑥 ∈ 𝑀|𝑓(𝑥) = 𝑦}.
Theorem 6. Let 𝑓: 𝑀 → 𝑆 be a homomorphism between 𝛤-near-rings 𝑀 and S. If 𝑃 = (𝜋𝑃,𝜗𝑃) is a
Pythagorean fuzzy weak bi-ideal of 𝑆, then 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃,𝜗𝑃)] is a Pythagorean fuzzy weak bi-ideal of
𝑀.
Proof. Let 𝑓 be a Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. Then
f−1
(πP)(x − y) = πP(f(x − y))
= πP(f(x) − f(y))](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-9-320.jpg)
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≥ min{πP(f(x)), πP(f(y))}
= min{f−1
(πP(x)), f −1
(πP(y))}.
f −1
(ϑP)(x − y) = ϑP(f(x − y))
= ϑP(f(x) − f(y))
≤ max{ϑP(f(x)), ϑP(f(y))}
= max{f−1
(ϑP(x)), f−1
(ϑP(y))}.
f −1
(πP)(xαyβz) = πP(f(xαyβz))
= πP(f(x)αf(y)βf(z))
≥ min{πP(f(x)), πP(f(y)), πP(f(z))}
= min{f−1
(πP(x)), f −1
(πP(y)), f−1
(πP(z))}.
f −1
(ϑP)(xαyβz) = ϑP(f(xαyβz))
= ϑP(f(x)αf(y)βf(z))
≤ max{ϑP(f(x)), ϑP(f(y)),ϑP(f(z))}
= max{f−1
(ϑP(x)), f−1
(ϑP(y)),f −1
(ϑP(z))}.
Therefore 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃, 𝜗𝑃)] is a Pythagorean fuzzy weak bi-ideal of M.
We can also state the converse of the Theorem 7 by strengthening the condition on 𝑓 as follows.
Theorem 7. Let 𝑓: 𝑀 → 𝑆 be an onto homomorphism of 𝛤-near-rings 𝑀 and S. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a
Pythagorean fuzzy subset of S. If 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃),𝑓−1
(𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀,
then 𝑃 = (𝜋𝑃,𝜗𝑃) is a Pythagorean fuuzy weak bi-ideal of 𝑆.
Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑆. Then 𝑓(𝑗) = 𝑥, 𝑓(𝑘) = 𝑦 and 𝑓(𝑙) = 𝑧 for some 𝑗, 𝑘, 𝑙 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. It follows that
πP(x − y) = πP(f(j) − f(k))
= πP(f(j − k))
= f−1
(πP)(j − k)
≥ min{f−1
(πP)(j), f−1
(πP)(k)}
= min{πP(f(j)), πP(f(k))}
= min{πP(x),πP(y)}.
ϑP(x − y) = ϑP(f(j) − f(k))](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-10-320.jpg)
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5| Interval Valued Pythagorean Fuzzy Weak Bi-Ideals of 𝚪-Near-
Rings
In this section, we initiate the notion of interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 and discuss
some of its properties.
Definition 7. An interval valued Pythagorean fuzzy set 𝑃 = (𝜋𝑃,𝜗𝑃) of 𝑀 is called an interval valued
Pythagorean fuzzy weak bi-ideal of 𝑀, if
I. 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}.
II. 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
III. 𝜋𝑃(𝑥𝛾𝑦𝛾𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}.
IV. 𝜗𝑃(𝑥𝛾𝑦𝛾𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤.
Example 3. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation ` + ′ and 𝛤 = {𝛾} be a nonempty
set of binary operations as the following tables:
and
Let 𝜋𝑃 : 𝑀 → 𝐷[0,1] and 𝜗𝑃 : 𝑀 → 𝐷[0,1] be aninterval valued fuzzy subsets defined by 𝜋𝑃(𝑤) =
[0.6,0.7], 𝜋𝑃
(𝑥) = [0.5,0.6], 𝜋𝑃(𝑦) = 𝜋𝑃(𝑧) = [0.4,0.5]. And 𝜗𝑃(𝑤) = [0.2,0.3], 𝜗𝑃(𝑥) = [0.4,0.5], 𝜗𝑃(𝑦) =
[0.7,0.8] = 𝜗𝑃(𝑧). Then 𝑃 = (𝜋𝑃,𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀.
Theorem 9. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be an interval valued Pythagorean fuzzy subgroup of M. Then 𝑃 = (𝜋𝑃, 𝜗𝑃)
is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆
𝜗𝑃 ⊇ 𝜗𝑃.
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Let
𝑥, 𝑦, 𝑧, 𝑦1, 𝑦2 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤 such that 𝑥 = 𝑦𝛼𝑧 and 𝑦 = 𝑦1𝛽𝑦2. Then
(πP ⋆ πP ⋆ πP)(x) = sup
x=yαz
{min{(πP ⋆ πP)(y),πP(z)}}
= sup
x=yαz
{min{ sup
y=y1βy2
min{πP(y1), πP(y2)}, πP(z)}}
= sup
x=yαz
sup
y=y1βy2
{min{min{πP(y1), πP(y2)}, πP(z)}}
+ 𝐰 𝐱 𝐲 𝐳
𝐰 w x y z
𝐱 x w z y
𝐲 y z w x
𝐳 z y x w
𝛄 𝐰 𝐱 𝐲 𝐳
𝐰 w x w x
𝐱 w x w x
𝐲 w x y z
𝐳 w x y z](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-13-320.jpg)
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(2021)
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Therefore 𝜋𝑃 ⋆ 𝜗𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Similarly 𝜗𝑃 ⋆ 𝜋𝑃 is an
interval valued Pythagorean fuzzy weak bi-ideal of M.
Lemma 4. Every interval valued Pythagorean fuzzy ideal of 𝑀 is an interval valued Pythagorean fuzzy
bi-ideal of 𝑀.
Proof. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy ideal of 𝑀. Then
𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ M ⊆ 𝜋𝑃 ⋆ M ⊆ 𝜋𝑃,
𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ M ⊇ 𝜗𝑃 ⋆ M ⊇ 𝜗𝑃,
since 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy ideal of 𝑀.
This implies that 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy bi-ideal of M.
Theorem 10. Every interval valued Pythagorean fuzzy bi-ideal of 𝑀 is an interval valued Pythagorean
fuzzy weak bi-ideal of 𝑀.
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy bi-ideal of 𝑀.
Then 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
We have 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃.
This implies that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃.
Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of M.
Example 4. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation+ and 𝛤 = {𝛼} be a nonempty
set of binary operations as the following tables:
and
Let 𝑃 : 𝑀 → 𝐷[0,1] be aninterval valued Pythagorean fuzzy set defined by 𝜋𝑃(𝑤) = [0.8,0.9], 𝜋𝑃(𝑥) =
[0.3,0.4] = 𝜋𝑃(𝑦) and 𝜋𝑃(𝑧) = [0.5,0.6], and 𝜗𝑃(𝑤) = [0,0.1],𝜗𝑃(𝑥) = [0.4,0.5] = 𝜗𝑃(𝑦), 𝜗𝑃(𝑧) = [0.2,0.3].
Then 𝜋𝑃 is an interval valued fuzzy weak bi-ideal of 𝑀. But 𝜋𝑃 is not a fuzzy ideal and bi-ideal of 𝑀 and
+ 𝐰 𝐱 𝐲 𝐳
𝐰 w x y z
𝐱 x w z y
𝐲 y z w x
𝐳 z y x w
𝛂 𝐰 𝐱 𝐲 𝐳
w w w w w
x w x w x
y w w y y
z w x y z](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-16-320.jpg)
![313
Pythagorean
fuzzy
weak
bi-ideals
of
Γ-
near
ring
𝜋𝑃(𝑧𝛾𝑦𝛾𝑧) = 𝜋𝑃(𝑦) = [0.3,0.4] > [0.5,0.6] = 𝑚𝑖𝑛{𝜋𝑃(𝑧), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼(𝑧 + 𝑤) − 𝑥𝛼𝑤) ≤ 𝜗𝑃(𝑧) =
[0.4,0.5] ≤ [0.2,0.3] and 𝜗𝑃(𝑧𝛾𝑥𝛾𝑧) = 𝜗𝑃(𝑥) = [0.4,0.5] ≤ [0.2,0.3] = min{𝜗𝑃(𝑧),𝜗𝑃(𝑧)}.
Theorem 11. Let {(𝜋𝑃𝑖
, 𝜗𝑃𝑖
)|𝑖 ∈ 𝛺} be family of interval valued Pythagorean fuzzy weak bi-ideals of a near-
ring 𝑀, then ⋂𝑖∈𝛺 𝜋𝑃𝑖
and ∪𝑖∈𝛺 𝜗𝑃𝑖
are also an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, where
𝛺 is any index set.
Proof. Let {𝜋𝑃𝑖
, 𝜗𝑃𝑖
}𝑖∈Ω be a family of interval valued Pythagorean fuzzy weak bi-ideals of 𝑀.
Let 𝑥, 𝑦, 𝑧 ∈ 𝑀, 𝛼, 𝛽 ∈ Γ and 𝜋𝑃𝑖
= ⋂𝑖∈Ω 𝜋𝑖, 𝜗𝑃𝑖
=∪𝑖∈𝛺 𝜗𝑃𝑖
Then, 𝜋𝑃𝑖
(𝑥) = ⋂𝑖∈Ω 𝜋𝑃𝑖
(𝑥) = (inf
𝑖∈Ω
𝜋𝑃𝑖
)(𝑥) = inf
𝑖∈Ω
𝜋𝑃𝑖
(𝑥)
and 𝜗𝑃𝑖
(𝑥) =∪𝑖∈𝛺 𝜗𝑃𝑖
(𝑥) = (sup
𝑖∈Ω
𝜗𝑃𝑖
)(𝑥) = sup
𝑖∈Ω
𝜗𝑃𝑖
(𝑥) .
πPi
(x − y) = inf
i∈Ω
πPi
(x − y)
≥ inf
i∈Ω
min{πPi
(x),πPi
(y)}
= min{inf
i∈Ω
πPi
(x), inf
i∈Ω
πPi
(y)}
= min{⋂
i∈Ω
πPi
(x), ⋂
i∈Ω
πPi
(y)}
= min{πPi
(x), πPi
(y)}.
ϑPi
(x − y) = sup
i∈Ω
ϑPi
(x − y)
≤ sup
i∈Ω
max{ϑPi
(x),ϑPi
(y)}
= max{sup
i∈Ω
ϑPi
(x), sup
i∈Ω
ϑPi
(y)}
= max{∪i∈Ω ϑPi
(x),∪i∈Ω ϑPi
(y)}
= max{ϑPi
(x),ϑPi
(y)}.
And,
πPi
(xαyβz) = inf
i∈Ω
πPi
(xαyβz)
≥ inf
i∈Ω
min{πPi
(x),πPi
(y), πPi
(z)}
= min{inf
i∈Ω
πPi
(x), inf
i∈Ω
πPi
(y), inf
i∈Ω
πPi
(z)}
= min{⋂
i∈Ω
πPi
(x), ⋂
i∈Ω
πPi
(y), ⋂
i∈Ω
πPi
(z)}](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-17-320.jpg)
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Marynirmala
and
Sivakumar
|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
= min{πPi
(x),πPi
(y), πPi
(z)}.
ϑPi
(xαyβz) = sup
i∈Ω
ϑPi
(xαyβz)
≤ sup
i∈Ω
max{ϑPi
(x), ϑPi
(y),ϑPi
(z)}
= max{sup
i∈Ω
ϑPi
(x), sup
i∈Ω
ϑPi
(y), sup
i∈Ω
ϑPi
(z)}
= max{∪i∈Ω ϑPi
(x),∪i∈Ω ϑPi
(y),∪i∈Ω ϑPi
(z)}
= max{ϑPi
(x),ϑPi
(y), ϑPi
(z)}.
Theorem 12. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be an interval valued Pythagorean fuzzy subset of M. Then 𝑈(𝜋𝑃; 𝑡) and
𝐿(𝜗𝑃;𝑠) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃𝑡 is a weak bi-ideal
of 𝑀, for all 𝑡 ∈ [0,1].
Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑅.
Let 𝑠, 𝑡 ∈ [0,1] such that 𝑥, 𝑦 ∈ 𝑈(𝜋𝑃; 𝑡).
Then 𝜋𝑃(𝑥) ≥ 𝑡 and 𝜋𝑃(𝑦) ≥ 𝑡,then 𝜋𝑃(𝑥 − 𝑦) ≥ min{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} ≥ min{𝑡,𝑡} = 𝑡 and
𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)} ≤ 𝑚𝑎𝑥{𝑠,𝑠} = 𝑠.
Thus 𝑥 − 𝑦 ∈ 𝑈(𝜋𝑃 𝑡).Let 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃𝑡 and 𝛼, 𝛽 ∈ Γ.
This implies that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ min{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)} ≥ min{𝑡, 𝑡, 𝑡} = 𝑡, and
𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} ≤ max{𝑠, 𝑠, 𝑠} = 𝑠.
Therefore 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑈(𝜋𝑃; 𝑠).
Hence 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀.
Conversely, assume that 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is an interval valuedPythagorean fuzzy weak bi-ideal of 𝑀,
for all 𝑠, 𝑡 ∈ [0,1].
Let 𝑥, 𝑦 ∈ 𝑀. Suppose 𝜋𝑃(𝑥 − 𝑦) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}and 𝜗𝑃(𝑥 − 𝑦) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
Choose 𝑡 such that 𝜋𝑃(𝑥 − 𝑦) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
This implies that 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡 and 𝜋𝑃(𝑥 − 𝑦) < 𝑡.
Then we have 𝑥, 𝑦 ∈ 𝜋𝑃𝑡 but 𝑥 − 𝑦 ∈ 𝜋𝑃𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠 and 𝜗𝑃(𝑥 − 𝑦) > 𝑠, we have 𝑥, 𝑦 ∈ 𝜗𝑃𝑠 but
𝑥 − 𝑦 ∈ 𝜗𝑃𝑠 a contradiction.
Thus 𝜋𝑃(𝑥 − 𝑦) ≥ min{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-18-320.jpg)
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|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
Definition 8. Let 𝑓 be a mapping from a set 𝑀 to a set 𝑆. Let 𝑓 = (𝜋𝑃, 𝜗𝑃) be an interval valued
Pythagorean fuzzy subsets of 𝑀 and 𝑆, resp. then 𝑓 is image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of
𝑆 defined by
f(πP)(y) =
{
sup
x∈f−1(y)
πP(x) if f−1
(y) ≠ ∅
0 otherwise.
f(ϑP)(y) =
{
inf
x∈f−1(y)
ϑP(x) if f−1
(y) ≠ ∅
1 otherwise
And the pre-image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑀 defined by
𝑓−1
(𝜋𝑃(𝑥)) = 𝜋𝑃(𝑓(𝑥)), 𝑓−1
(𝜗𝑃(𝑥)) = 𝜗𝑃(𝑓(𝑥)) for all 𝑥 ∈ 𝑀 and 𝑓−1
(𝑦) = {𝑥 ∈ 𝑀|𝑓(𝑥) = 𝑦}.
Theorem 14. Let 𝑓: 𝑀 → 𝑆 be a homomorphism between 𝛤-near-rings 𝑀 and S. If 𝑃 = (𝜋𝑃,𝜗𝑃) is an
interval valued Pythagorean fuzzy weak bi-ideal of 𝑆, then 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃,𝜗𝑃)] is an interval valued
fuzzy weakbi-ideal of 𝑀.
Proof. Let 𝑓 be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑆. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤.
Then
f −1
(πP)(x − y) = πP(f(x − y))
= πP(f(x) − f(y))
≥ min{πP(f(x)), πP(f(y))}
= min{f−1
(πP(x)), f −1
(πP(y))}.
f −1
(ϑP)(x − y) = ϑP(f(x − y))
= ϑP(f(x) − f(y))
≤ max{ϑP(f(x)), ϑP(f(y))}
= max{f−1
(ϑP(x)), f−1
(ϑP(y))}.
f −1
(πP)(xαyβz) = πP(f(xαyβz))
= πP(f(x)αf(y)βf(z))
≥ min{πP(f(x)), πP(f(y)), πP(f(z))}
= min{f−1
(πP(x)), f −1
(πP(y)), f−1
(πP(z))}.
f −1
(ϑP)(xαyβz) = ϑP(f(xαyβz))
= ϑP(f(x)αf(y)βf(z))](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-20-320.jpg)
![317
Pythagorean
fuzzy
weak
bi-ideals
of
Γ-
near
ring
≤ max{ϑP(f(x)), ϑP(f(y)),ϑP(f(z))}
= max{f−1
(ϑP(x)),f −1
(ϑP(y)),f −1
(ϑP(z))}.
Therefore 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean fuzzy weak bi-ideal of M.
We can also state the converse of the Theorem 7 by strengthening the condition on 𝑓 as follows.
Theorem 15. Let 𝑓: 𝑀 → 𝑆 be an onto homomorphism of 𝛤-near-rings 𝑀 and S. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be an
interval valued Pythagorean fuzzy subset of S. If 𝑓−1
(𝑃) = [𝑓−1
(𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean
fuzzy weak bi-ideal of 𝑀, then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑆.
Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑆. Then 𝑓(𝑗) = 𝑥, 𝑓(𝑘) = 𝑦 and 𝑓(𝑙) = 𝑧 for some 𝑗, 𝑘, 𝑙 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. It follows that
πP(x − y) = πP(f(j) − f(k))
= πP(f(j − k))
= f −1
(πP)(j − k)
≥ min{f−1
(πP)(j),f −1
(πP)(k)}
= min{πP(f(j)), πP(f(k))}
= min{πP(x), πP(y)}.
ϑ(x − y) = ϑ(f(j) − f(k))
= ϑ(f(j − k))
= f −1
(ϑ)(j − k)
≤ max{f−1
(ϑ)(j), f−1
(ϑ)(k)}
= max{ϑ(f(j)), ϑ(f(k))}
= max{ϑ(x),ϑ(y)}.
And
πP(xαyβz) = πP(f(j)αf(k)βf(l))
= πP(f(jkl))
= f −1
(πP)(jkl)
≥ min{f−1
(πP)(j),f −1
(πP)(k), f−1
(πP)(l)}
= min{πP(f(j)), πP(f(k)), πP(f(l))}](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-21-320.jpg)
![318
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and
Sivakumar
|J.
Fuzzy.
Ext.
Appl.
2(3)
(2021)
297-320
= min{πP(x),πP(y), πP(z)}.
ϑ(xαyβz) = ϑ(f(j)αf(k)βf(l))
= ϑ(f(jkl))
= f−1(ϑ)(jkl)
≤ max{f−1
(ϑ)(j), f−1
(ϑ)(k),f −1
(ϑ)(l)}
= max{ϑ(f(j)), ϑ(f(k)), ϑ(f(l))}
= max{ϑ(x),ϑ(y),ϑ(z)}.
Hence 𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of S.
Theorem 16. Let 𝑓 : 𝑀 → 𝑆 be an onto 𝛤-near-ring homomorphism. If 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued
Pythagorean fuzzy weak bi-ideal of 𝑀, then 𝑓(𝑃) = [𝑓(𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean fuzzy
weak bi-ideal of 𝑆.
Proof. Let 𝑃 be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Since 𝑓(𝜋𝑃)(𝑥′) = 𝑠𝑢𝑝
𝑓(𝑥)=𝑥′
(𝜋𝑃(𝑥))
and 𝑓(𝜗𝑃)(𝑥′) = 𝑖𝑛𝑓
𝑓(𝑥)=𝑥′
(𝜗𝑃(𝑥)) , for 𝑥′ ∈ 𝑆 and hence 𝑓(𝑃) is nonempty. Let 𝑥′, 𝑦′ ∈ 𝑆 and 𝛼, 𝛽 ∈ 𝛤. Then
we have {𝑥|𝑥 ∈ 𝑓−1
(𝑥′ − 𝑦) ⊇ {𝑥 − 𝑦|𝑥 ∈ 𝑓−1
(𝑥′) and 𝑦 ∈ 𝑓−1
(𝑦′)} and {𝑥|𝑥 ∈ 𝑓−1
(𝑥′𝑦′)} ⊇ {𝑥𝛼𝑦|𝑥 ∈ 𝑓−1
(𝑥′)
and 𝑦 ∈ 𝑓−1
(𝑦′)}.
f(πP)(x′ − y′) = sup
f(z)=(x′−y′)
{πP(z)}
≥ sup
f(x)=x′,f(y)=y′
{πP(x − y)}
≥ sup
f(x)=x′,f(y)=y′
{min{πP(x), πP(y)}}
= min{ sup
f(x)=x′
{πP(x)}, sup
f(y)=y′
{πP(y)}}
= min{f(πP)(x′), f(πP)(y′)}.
And
f(ϑP)(x′ − y′) = inf
f(z)=x′−y′
{ϑP(z)}
≤ inf
f(x)=x′,f(y)=y′
{ϑP(x − y)}
≤ inf
f(x)=x′,f(y)=y′
{max{ϑP(x),ϑP(y)}}
= max{ inf
f(x)=x′
{ϑP(x)}, inf
f(y)=y′
{ϑP(y)}}](https://image.slidesharecdn.com/jfeavolume2issue3pages297-320-220419053846/85/Pythagorean-fuzzy-weak-bi-ideals-of-near-ring-22-320.jpg)
Pythagorean fuzzy weak bi-ideals of Γ- near ring
- 1. Corresponding Author: marynirla@gmail.com http://dx.doi.org/10.22105/jfea.2021.300356.1160 E-ISSN: 2717-3453 | P-ISSN: 2783-1442 | Abstract 1 | Introduction Zadeh [26] defined Fuzzy Set (FS) to deal with uncertainty. Atanassov [2] presented the notion of Intuitionistic FS (IFS) and studied some of its properties. Later, Yager [24], [25] defined and studied the properties of Pythagorean Fuzzy Set (PFS) and also used PFS to solve Multi-Criteria Decision- Making (MCDM) problems. Booth [3] presented the properties of Γ-near-rings. Chinnadurai and Kadalarasi [7] studied the near-ring properties of Fuzzy Weak Bi-Ideals (FWBI). Chinnadurai et al. [4],[5] studied the Γ-near-rings characterization of fuzzy weak bi-ideal and interval-valued fuzzy weak bi-ideal. Later, Chinnadurai et al. [6] discussed the Γ-near-rings properties of interval-valued fuzzy ideals. Akram [1] established the properties of fuzzy lie algebras. Kim and Kim [13] studied the near-rings concept of fuzzy ideals. Kaviyarasu et al. [10]-[12] studied the different type of ideals in INK- algebras. Jun et al. [9] presented the notion of fuzzy ideals and studied their properties in Γ-near-rings. Manikantan [14] defined and studied some of the near-rings properties of fuzzy bi-ideals. Meenakumari and Chelvam [15] presented the Γ-near-ringsproperties of fuzzy bi-ideals. Narayanan and Manikatan [16] introduced the near-rings notions of fuzzy subnear-ring, fuzzy ideal, and fuzzy quasi-ideal. Journal of Fuzzy Extension and Applications www.journal-fea.com J. Fuzzy. Ext. Appl. Vol. 2, No. 3 (2021) 297–320. Paper Type: Research Paper Pythagorean Fuzzy Weak Bi-Ideals of 𝚪- Near Ring J. Marynirmala1,* , D. Sivakumar1 1 Department of Mathematics, Annamalai University,Annamalainagar, Tamilnadu, India; marynirla@gmail.com. Citation: Marynirmala, J., & Sivakumar, D. (2021). Pythagorean fuzzy weak bi-ideals of Γ- near ring. Journal of fuzzy extension and application, 2 (3), 297-320. Accept: 20/08/2021 Revised: 23/07/2021 Reviewed: 30/05/2021 Received: 18/05/2021 We present the notion of Pythagorean Fuzzy Weak Bi-Ideals (PFWBI) and interval valued Pythagorean fuzzy weak bi- ideals of Γ-near-rings and studies some of its properties. We present the notion of interval valued Pythagorean fuzzy weak bi-ideal and establish some of its properties. We study interval valued Pythagorean fuzzy weak bi-ideals of Γ-near- ring using homomorphism. Keywords: Pythagorean fuzzy, Fuzzy ideals, Homomorphism, Near ring. Licensee Journal of Fuzzy Extension and Applications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons. org/licenses/by/4.0).
- 2. 298 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 Pilz [17] introduced the concept of anti fuzzy soft gamma rings and studied their properties. Rao and Swaminathan [20] presented the notion of anti-homomorphism between two fuzzy rings and established its properties. Rao and Venkateswarlu [21] studied the properties of anti fuzzy ideal and pre-image of fuzzy ideal. Satyanarayana [18] dealt with the theory of near-rings. Salah Abou-Zaid [19] studied fuzzy ideals of a near-ring. Chelvam and Meenakumari [22] obtained the characterization for gamma near- fields. Thillaigovindan et al. [23] introduced the notion of generalized T-fuzzy bi-ideals of a gamma- semigroup. Cho et al. [8] presented the notion of bi-ideals in near-rings and used it in near-fields. We introduce the notion of Pythagorean fuzzy weak bi-ideal of Γ-near-rings and interval valued Pythagorean fuzzy weak bi-ideal of Γ-near-rings. We discuss and present some properties of homomorphism of Pythagorean fuzzy weak bi-ideal and homomorphism of an interval valued Pythagorean fuzzy weak bi-ideal in gamma near-ring. 2| Preliminaries Definition 1. [16]. A fuzzy set 𝜋 of a 𝛤-near-ring 𝑀 is called a fuzzy left (resp. right) ideal of 𝑀 if I. 𝜋(𝑘 − 𝑙) ≥ 𝑚𝑖𝑛{𝜋(𝑘),𝜋(𝑙)}, for all 𝑘, 𝑙 ∈ 𝑀, II. 𝜋(𝑦 + 𝑥 − 𝑦) ≥ 𝜋(𝑥) , for all 𝑥, 𝑦 ∈ 𝑀, III. 𝜋(𝑢𝛼(𝑥 + 𝑣) − 𝑢𝛼𝑣) ≥ 𝜋(𝑥) , (resp. 𝜋(𝑥𝛼𝑢) ≥ 𝜋(𝑥)) for all 𝑥, 𝑢, 𝑣 ∈ 𝑀 and 𝛼 ∈ 𝛤. Definition 2. [15]. A fuzzy set 𝜋 of 𝑀 is called a fuzzy bi-ideal of 𝑀 if I. 𝜋(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋(𝑥),𝜋(𝑦)} for all 𝑥, 𝑦 ∈ 𝑀, II. 𝜋(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋(𝑥),𝜋(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. Definition 3. [2]. An intuitionistic fuzzy set A is a nonempty set X is an object having the form 𝐴 = {𝑥,(𝜋𝐴(𝑥), 𝜗𝐴(𝑥)): 𝑥 ∈ 𝑋} where the functions 𝜋𝐴 : 𝑋 → [0,1] and 𝜗𝐴 : 𝑋 → [0,1] define the degree of membership and non-membership of the element 𝑥 ∈ 𝑋 to the set 𝐴, which is a subset of X respectively 0 ≤ 𝜋𝐴(𝑥) + 𝜗𝐴(𝑥) ≤ 1 we use the simple 𝐴 = (𝜋𝐴, 𝜗𝐴) . Definition 4. [25]. A Pythagorean fuzzy subset 𝑃 is a nonempty set X is an object having the form 𝑃 = {(𝑥,𝜋𝑃(𝑥), 𝜗𝑃(𝑥))/𝑥 ∈ 𝑋}, where the functions 𝜋𝑃: 𝑋 → [0,1] and 𝜗𝑃:𝑋 → [0,1] denote the degree of membership and non membership of each element 𝑥 ∈ 𝑋 to the set 𝑃, respectively, and 0 ≤ (𝜋𝑃(𝑥))2 + (𝜗𝑃(𝑥))2 ≤ 1 for all 𝑥 ∈ 𝑋. For the sake of simplicity, for the Pythagorean fuzzy subset 𝑃 = {(𝑥,𝜋𝑃(𝑥), 𝜗𝑃(𝑥)) /𝑥 ∈ 𝑋}. 3| Pythagorean Fuzzy Weak Bi-Ideals of 𝚪near Ring In this section, we initiate the notion of Pythagorean fuzzy weak bi-ideal of 𝑀 and discuss some of its properties. Definition 4. A subgroup 𝑊 of (𝑀, +) is said to be a weak bi-ideal of 𝑀 if 𝑊𝛤𝑊𝛤𝑊 ⊆ 𝑊. Definition 5. A Pythagorean fuzzy set𝑃 = (𝜋𝑃, 𝜗𝑃) of 𝑀 is called a Pythagorean fuzzy weak bi-ideal of 𝑀, if I. 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)}. II. 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. III. 𝜋𝑃(𝑥𝛾𝑦𝛾𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}. IV. 𝜗𝑃(𝑥𝛾𝑦𝛾𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤.
- 3. 299 Pythagorean fuzzy weak bi-ideals of Γ- near ring Example 1. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation ` + ′ and 𝛤 = {𝛾} be a nonempty set of binary operations as the following tables: and Let 𝜋𝑃 : 𝑀 → [0,1] be a Pythagorean fuzzy subset defined by 𝜋𝑃(𝑤) = 0.7, 𝜋𝑃(𝑥) = 0.6, 𝜋𝑃(𝑦) = 𝜋𝑃(𝑧) = 0.5. and 𝜗𝑃(𝑤) = 0.3, 𝜗𝑃(𝑥) = 0.5, 𝜗𝑃(𝑦) = 0.8 = 𝜗𝑃(𝑧). Then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀. Theorem 1. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy subgroup of M. Then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑥, 𝑦, 𝑧, 𝑦1, 𝑦2 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤 such that 𝑥 = 𝑦𝛼𝑧 and 𝑦 = 𝑦1𝛽𝑦2. Then (πP ⋆ πP ⋆ πP)(x) = sup x=yαz {min{(πP ⋆ πP)(y), πP(z)}} = sup x=yαz {min{ sup y=y1βy2 min{πP(y1), πP(y2)}, πP(z)}} = sup x=yαz sup y=y1βy2 {min{min{πP(y1), πP(y2)}, πP(z)}} = sup x=y1βy2αz {min{πP(y1), πP(y2), πP(z)}}, since 𝜋𝑃 is a fuzzy weak bi-ideal of 𝑀, πP(y1βy2αz) ≥ min{πP(y1), πP(y2), πP(z)} ≤ sup x=y1βy2αz πP(y1βy2αz) = πP(x). And (ϑP ⋆ ϑP ⋆ ϑP)(x) = inf x=yαz {min{(ϑP ⋆ ϑP)(y), ϑP(z)}}, = inf x=yαz {max{ inf y=y1βy2 min{ϑP(y1), ϑP(y2)}, ϑP(z)}} + 𝐰 𝐱 𝐲 𝐳 𝐰 w x y z 𝐱 x w z y 𝐲 y z w x 𝐳 z y x w 𝛄 𝐰 𝐱 𝐲 𝐳 𝐰 w x w x 𝐱 w x w x 𝐲 w x y z 𝐳 w x y z
- 4. 300 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 = inf x=yαz sup y=y1βy2 {max{max{ϑP(y1), ϑP(y2)}, ϑP(z)}} = inf x=y1βy2αz {max{ϑP(y1), ϑP(y2), ϑP(z)}}, since 𝜗𝑃 is a fuzzy weak bi-ideal of 𝑀, ϑP(y1βy2αz) ≤ max{ϑP(y1), ϑP(y2), ϑP(z)} ≥ inf x=y1βy2αz ϑP(y1βy2αz) = ϑP(x). If 𝑥 can not be expressed as 𝑥 = 𝑦𝛼𝑧, then (𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃)(𝑥) = 0 ≤ 𝜋𝑃(𝑥) and (𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃)(𝑥) = 0 ≥ 𝜗𝑃(𝑥). In both cases 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃, and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Conversely, assume that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃. For 𝑥′, 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽, 𝛼1, 𝛽1 ∈ 𝛤. Let 𝑥′ be such that 𝑥′ = 𝑥𝛼𝑦𝛽𝑧. Then 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) = 𝜋𝑃(𝑥′) ≥ (𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃)(𝑥′) = sup x′=pα1q {min{(πP ⋆ πP)(p),πP(q)}} = sup x′=pα1q {min{ sup p=p1β1p2 min{πP(p1), πP(p2)}, πP(q)}} = sup x′=p1β1p2α1q {min{πP(p1), πP(p2), πP(q)}} ≥ min{πP(x),πP(y), πP(z)}. ϑP(xαyβz) = ϑP(x′) ≤ (ϑP ∗ ϑP ∗ ϑP)(x′) = inf x′=pα1q {max{(ϑP ∗ ϑP)(p),ϑP(q)}} = inf x′=pα1q {max{ inf p=p1β1p2 min{ϑP(p1), ϑP(p2)}, ϑP(q)}} = inf x′=p1β1p2α1q {max{ϑP(p1), ϑP(p2), ϑP(q)}} ≤ max{ϑP(x),ϑP(y),ϑP(z)}. Hence 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}, and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Lemma 1. Let 𝜋𝑃 = (𝜋𝑃1, 𝜋𝑃2) and 𝜗𝑃 = (𝜗𝑃1, 𝜗𝑃2) be Pythagorean fuzzy weak bi-ideals of M. Then the products 𝜋𝑃 ∗ 𝜗𝑃 and 𝜗𝑃 ∗ 𝜋𝑃 are also Pythagorean fuzzy weak bi-ideals of 𝑀. Proof. Let 𝜋𝑃 and 𝜗𝑃 be a Pythagorean fuzzy weak bi-ideals of 𝑀 and let 𝛼, 𝛼1, 𝛼2 ∈ Γ. Then
- 5. 301 Pythagorean fuzzy weak bi-ideals of Γ- near ring (πP1 ⋆ πP2)(x − y) = sup x−y=aαb min{πP1(a), πP2(b)} ≥ sup x−y=a1α1b1−a2α2b2<(a1−a2)(b1−b2) min{πP1(a1 − a2), πP2(b1 − b2)} ≥ supmin{min{πP1(a1), πP1(a2)}, min{πP2(b1), πP2(b2)}} = supmin{min{πP1(a1), πP2(b1)}, min{πP1(a2), πP2(b2)}} ≥ min{ sup x=a1α1b1 min{πP1(a1), πP2(b1)}, sup y=a2α2b2 min{πP1(a2), πP2(b2)}} = min{(πP1 ⋆ πP2)(x), (πP1 ⋆ πP2)(y)}. (ϑP1 ⋆ ϑP2)(x − y) = inf x−y=aαb max{ϑP1(a), ϑP2(b)} ≤ inf x−y=a1α1b1−a2α2b2<(a1−a2)(b1−b2) max{ϑP1(a1 − a2), ϑP2(b1 − b2)} ≤ infmax{max{ϑP1(a1), ϑP1(a2)}, max{ϑP2(b1), ϑP2(b2)}} = infmax{max{ϑP1(a1), ϑP2(b1)}, max{ϑP1(a2), ϑP2(b2)}} ≤ max{ inf x=a1α1b1 max{ϑP1(a1), ϑP2(b1)}, inf y=a2α2b2 max{ϑP1(a2), ϑP2(b2)}} = max{(ϑP1 ⋆ ϑP2)(x), (ϑP1 ⋆ ϑP2)(y)}. It follows that 𝜋𝑃 ∗ 𝜗𝑃 is a Pythagorean fuzzy subgroup of 𝑀. Further, (𝜋𝑃 ⋆ 𝜗𝑃) ⋆ (𝜋𝑃 ⋆ 𝜗𝑃) ⋆ (𝜋𝑃 ⋆ 𝜗𝑃) = 𝜋𝑃 ⋆ 𝜗𝑃 ⋆ (𝜋𝑃 ⋆ 𝜗𝑃 ⋆ 𝜋𝑃) ⋆ 𝜗𝑃 ⊆ 𝜋𝑃 ⋆ 𝜗𝑃 ⋆ (𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃) ⋆ 𝜗𝑃 ⊆ 𝜋𝑃 ⋆ (𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃), since 𝑃 is a Pythagorean fuzzy weak bi-ideal of 𝑀 ⊆ 𝜋𝑃 ⋆ 𝜗𝑃. Therefore 𝜋𝑃 ∗ 𝜗𝑃 is a Pythagorean fuzzy weak bi-ideal of 𝑀. Similarly 𝜗𝑃 ∗ 𝜋𝑃 is a Pythagorean fuzzy weak bi-ideal of M. Lemma 2. Every Pythagorean fuzzy ideal of 𝑀 is a Pythagorean fuzzy bi-ideal of 𝑀. Proof. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy ideal of 𝑀. Then 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ M ⊆ 𝜋𝑃 ⋆ M ⊆ 𝜋𝑃, 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ M ⊇ 𝜗𝑃 ⋆ M ⊇ 𝜗𝑃, since 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy ideal of 𝑀. This implies that 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy bi-ideal of M.
- 6. 302 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 Theorem 2. Every Pythagorean fuzzy bi-ideal of 𝑀 is a Pythagorean fuzzy weak bi-ideal of 𝑀. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy bi-ideal of 𝑀. Then 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. We have 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃. This implies that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of M. Example 2. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation+ and 𝛤 = {𝛼} be a nonempty set of binary operations as the following tables: and Let 𝜋𝑃 : 𝑀 → [0,1] be a fuzzy set defined by 𝜋𝑃(𝑤) = 0.9, 𝜋𝑃(𝑥) = 0.4 = 𝜋𝑃(𝑦) and 𝜋𝑃(𝑧) = 0.6, and 𝜗𝑃(𝑤) = 0.1,𝜗𝑃(𝑥) = 0.5 = 𝜗𝑃(𝑦),𝜗𝑃(𝑧) = 0.3. Then 𝜋𝑃 is a fuzzy weak bi-ideal of 𝑀. But 𝜋𝑃 is not a fuzzy ideal and bi-ideal of 𝑀 and 𝜋𝑃(𝑧𝛾𝑦𝛾𝑧) = 𝜋𝑃(𝑦) = 0.4 > 0.6 = 𝑚𝑖𝑛{𝜋𝑃(𝑧), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼(𝑧 + 𝑤) − 𝑥𝛼𝑤) ≤ 𝜗𝑃(𝑧) = 0.5 ≤ 0.3 and 𝜗𝑃(𝑧𝛾𝑥𝛾𝑧) = 𝜗𝑃(𝑥) = 0.5 ≤ 0.3 = min{𝜗𝑃(𝑧),𝜗𝑃(𝑧)}. Theorem 3. Let {(𝜋𝑃𝑖 , 𝜗𝑃𝑖 )|𝑖 ∈ 𝛺} be family of Pythagorean fuzzy weak bi-ideals of a near-ring 𝑀, then ⋂𝑖∈𝛺 𝜋𝑃𝑖 and ∪𝑖∈𝛺 𝜗𝑃𝑖 are also a Pythagorean fuzzy weak bi-idea of 𝑀, where 𝛺 is any index set. Proof. Let {𝜋𝑖}𝑖∈𝛺 be a family of Pythagorean fuzzy weak bi-ideals of 𝑀. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀, 𝛼, 𝛽 ∈ 𝛤 and 𝜋 = ⋂𝑖∈𝛺 𝜋𝑖. Then,⋂𝑖∈𝛺 𝜋𝑃𝑖 (𝑥) = ⋂𝑖∈𝛺 𝜋𝑃𝑖 (𝑥) = (𝑖𝑛𝑓 𝑖∈𝛺 𝜋𝑃𝑖 )(𝑥) = 𝑖𝑛𝑓 𝑖∈𝛺 𝜋𝑃𝑖 (𝑥)and ∪𝑖∈𝛺 𝜗𝑃𝑖 (𝑥) =∪𝑖∈𝛺 𝜗𝑃𝑖 (𝑥) = (𝑠𝑢𝑝 𝑖∈𝛺 𝜗𝑃𝑖 )(𝑥) = 𝑠𝑢𝑝 𝑖∈𝛺 𝜗𝑃𝑖 (𝑥). ⋂ i∈Ω πPi (x − y) = inf i∈Ω πPi (x − y) + 𝐰 𝐱 𝐲 𝐳 𝐰 w x y z 𝐱 x w z y 𝐲 y z w x 𝐳 z y x w 𝛂 𝐰 𝐱 𝐲 𝐳 𝐰 w w w w 𝐱 w x w x 𝐲 w w y y 𝐳 w x y z
- 7. 303 Pythagorean fuzzy weak bi-ideals of Γ- near ring ≥ inf i∈Ω min{πPi (x),πPi (y)} = min{inf i∈Ω πPi (x), inf i∈Ω πPi (y)} = min{⋂ i∈Ω πPi (x), ⋂ i∈Ω πPi (y)} ∪i∈Ω ϑPi (x − y) = sup i∈Ω πPi (x − y) ≤ sup i∈Ω max{ϑPi (x),ϑPi (y)} = max{sup i∈Ω ϑPi (x),sup i∈Ω ϑPi (y)} = max{∪i∈Ω ϑPi (x),∪i∈Ω ϑPi (y)}. And, ⋂ i∈Ω πPi (xαyβz) = inf i∈Ω πPi (xαyβz) ≥ inf i∈Ω min{πPi (x),πPi (y), πPi (z)} = min{inf i∈Ω πPi (x), inf i∈Ω πPi (y), inf i∈Ω πPi (z)} = min{⋂ i∈Ω πPi (x), ⋂ i∈Ω πPi (y), ⋂ i∈Ω πPi (z)} ∪i∈Ω ϑPi (xαyβz) = sup i∈Ω ϑPi (xαyβz) ≤ sup i∈Ω max{ϑPi (x), ϑPi (y),ϑPi (z)} = max{sup i∈Ω ϑPi (x), sup i∈Ω ϑPi (y),sup i∈Ω ϑPi (z)} = max{∪i∈Ω ϑPi (x),∪i∈Ω ϑPi (y),∪i∈Ω ϑPi (z)}. Hence the set ⋂𝑖∈𝛺 𝜋𝑃𝑖 and ∪𝑖∈𝛺 𝜗𝑃𝑖 are also a family of Pythagorean fuzzy weak bi-ideals of M. Theorem 4. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be a Pythagorean fuzzy subset of M. Then 𝑈(𝜋𝑃;𝑡) and 𝐿(𝜗𝑃;𝑠) is a Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃𝑡 is a weak bi-ideal of 𝑀, for all 𝑡 ∈ [0,1]. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑠, 𝑡 ∈ [0,1] such that 𝑥, 𝑦 ∈ 𝑈(𝜋𝑃;𝑡). Then 𝜋𝑃(𝑥) ≥ 𝑡 and 𝜋𝑃(𝑦) ≥ 𝑡, Then 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)} ≥ 𝑚𝑖𝑛{𝑡,𝑡} = 𝑡 and 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)} ≤ 𝑚𝑎𝑥{𝑠, 𝑠} = 𝑠.
- 8. 304 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 Thus 𝑥 − 𝑦 ∈ 𝑈(𝜋𝑃 𝑡).Let 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃𝑡 and 𝛼, 𝛽 ∈ 𝛤. This implies that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)} ≥ 𝑚𝑖𝑛{𝑡,𝑡, 𝑡} = 𝑡, and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} ≤ 𝑚𝑎𝑥{𝑠, 𝑠, 𝑠} = 𝑠. Therefore 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑈(𝜋𝑃; 𝑠). Hence 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is a weak bi-ideal of 𝑀. Conversely, assume that 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is a weak bi-ideal of 𝑀, for all 𝑠, 𝑡 ∈ [0,1]. Let 𝑥, 𝑦 ∈ 𝑀. Suppose 𝜋𝑃(𝑥 − 𝑦) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}and 𝜗𝑃(𝑥 − 𝑦) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. Choose 𝑡 such that 𝜋(𝑥 − 𝑦) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)}. This implies that 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡 and 𝜋𝑃(𝑥 − 𝑦) < 𝑡. Then we have 𝑥, 𝑦 ∈ 𝜋𝑃𝑡 but 𝑥 − 𝑦 ∈ 𝜋𝑃𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠 and 𝜗𝑃(𝑥 − 𝑦) > 𝑠, we have 𝑥, 𝑦 ∈ 𝜗𝑃𝑠 but 𝑥 − 𝑦 ∈ 𝜗𝑃𝑠 a contradiction. Thus 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. If there exist 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤 such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Choose 𝑡 such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}. Choose 𝑠 such that 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Then 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡, 𝜋𝑃(𝑧) > 𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠, 𝜗𝑃(𝑧) < 𝑠 and 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡. So, 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃 > but 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜋𝑃𝑡, and 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜗𝑃𝑠, which is a contradiction. Hence 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}, 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥), 𝜗𝑃(𝑦), 𝜗𝑃(𝑧)}. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of M. Theorem 5. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy weak bi-ideal of 𝑀 then the set 𝑀𝜋,𝜗 = {𝑥 ∈ 𝑀|𝜋𝑃(𝑥) = 𝜋𝑃(0) = 𝜗𝑃(𝑥)} is aweak bi-ideal of 𝑀. Proof. Let 𝑥, 𝑦 ∈ 𝑀(𝜋𝑃,𝜗𝑃). Then 𝜋𝑃(𝑥) = 𝜋𝑃(0),𝜋𝑃(𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥) = 0, 𝜗𝑃(𝑦) = 0 and 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} = min{𝜋𝑃(0),𝜋𝑃(0)} = 𝜋𝑃(0), and 𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦)} = max{𝜗𝑃(0), 𝜗𝑃(0)}
- 9. 305 Pythagorean fuzzy weak bi-ideals of Γ- near ring = 𝜗𝑃(0). So 𝜋𝑃(𝑥 − 𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥 − 𝑦) = 𝜗𝑃(0). Thus 𝑥 − 𝑦 ∈ 𝑀𝜋𝑃 , 𝑥 − 𝑦 ∈ 𝑀𝜗𝑃 . For every 𝑥, 𝑦, 𝑧 ∈ 𝑀𝜋𝑃 and 𝛼, 𝛽 ∈ 𝛤. We have πP(xαyβz) ≥ min{πP(x),πP(y), πP(z)} = min{πP(0), πP(0), πP(0)} = πP(0). And ϑP(xαyβz) ≤ max{ϑP(x),ϑP(y),ϑP(z)} = max{ϑP(0),ϑP(0), ϑP(0)} = ϑP(0). Thus 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜋𝑃 , 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜗𝑃 . Hence 𝑀(𝜋𝑃,𝜗𝑃) is a weak bi-ideal of M. 4| Homomorphism of Pythagorean Fuzzy Weak Bi-Ideals of 𝚪-Near- Rings In this section, we characterize Pythagorean fuzzy weak bi-ideals of 𝛤-near-rings using homomorphism. Definition 6. Let 𝑓 be a mapping from a set 𝑀 to a set 𝑆. Let 𝑓 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy subsets of 𝑀 and 𝑆, resp. then 𝑓 is image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑆 defined by f(πP)(y) = { sup x∈f−1(y) πP(x) if f −1 (y) ≠ ∅ 0 otherwise. f(ϑP)(y) = { inf x∈f−1(y) ϑP(x) if f −1 (y) ≠ ∅ 1 otherwise And the pre-image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑀 defined by 𝑓−1 (𝜋𝑃(𝑥)) = 𝜋𝑃(𝑓(𝑥)), 𝑓−1 (𝜗𝑃(𝑥)) = 𝜗𝑃(𝑓(𝑥)) for all 𝑥 ∈ 𝑀 and 𝑓−1 (𝑦) = {𝑥 ∈ 𝑀|𝑓(𝑥) = 𝑦}. Theorem 6. Let 𝑓: 𝑀 → 𝑆 be a homomorphism between 𝛤-near-rings 𝑀 and S. If 𝑃 = (𝜋𝑃,𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑆, then 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃,𝜗𝑃)] is a Pythagorean fuzzy weak bi-ideal of 𝑀. Proof. Let 𝑓 be a Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. Then f−1 (πP)(x − y) = πP(f(x − y)) = πP(f(x) − f(y))
- 10. 306 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 ≥ min{πP(f(x)), πP(f(y))} = min{f−1 (πP(x)), f −1 (πP(y))}. f −1 (ϑP)(x − y) = ϑP(f(x − y)) = ϑP(f(x) − f(y)) ≤ max{ϑP(f(x)), ϑP(f(y))} = max{f−1 (ϑP(x)), f−1 (ϑP(y))}. f −1 (πP)(xαyβz) = πP(f(xαyβz)) = πP(f(x)αf(y)βf(z)) ≥ min{πP(f(x)), πP(f(y)), πP(f(z))} = min{f−1 (πP(x)), f −1 (πP(y)), f−1 (πP(z))}. f −1 (ϑP)(xαyβz) = ϑP(f(xαyβz)) = ϑP(f(x)αf(y)βf(z)) ≤ max{ϑP(f(x)), ϑP(f(y)),ϑP(f(z))} = max{f−1 (ϑP(x)), f−1 (ϑP(y)),f −1 (ϑP(z))}. Therefore 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃, 𝜗𝑃)] is a Pythagorean fuzzy weak bi-ideal of M. We can also state the converse of the Theorem 7 by strengthening the condition on 𝑓 as follows. Theorem 7. Let 𝑓: 𝑀 → 𝑆 be an onto homomorphism of 𝛤-near-rings 𝑀 and S. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be a Pythagorean fuzzy subset of S. If 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃),𝑓−1 (𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀, then 𝑃 = (𝜋𝑃,𝜗𝑃) is a Pythagorean fuuzy weak bi-ideal of 𝑆. Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑆. Then 𝑓(𝑗) = 𝑥, 𝑓(𝑘) = 𝑦 and 𝑓(𝑙) = 𝑧 for some 𝑗, 𝑘, 𝑙 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. It follows that πP(x − y) = πP(f(j) − f(k)) = πP(f(j − k)) = f−1 (πP)(j − k) ≥ min{f−1 (πP)(j), f−1 (πP)(k)} = min{πP(f(j)), πP(f(k))} = min{πP(x),πP(y)}. ϑP(x − y) = ϑP(f(j) − f(k))
- 11. 307 Pythagorean fuzzy weak bi-ideals of Γ- near ring = ϑP(f(j − k)) = f−1 (ϑP)(j − k) ≤ max{f−1 (ϑP)(j),f −1 (ϑP)(k)} = max{ϑP(f(j)), ϑP(f(k))} = max{ϑP(x),ϑP(y)}. And πP(xαyβz) = πP(f(j)αf(k)βf(l)) = πP(f(jkl)) = f−1 (πP)(jkl) ≥ min{f−1 (πP)(j), f−1 (πP)(k), f −1 (πP)(l)} = min{πP(f(j)), πP(f(k)), πP(f(l))} = min{πP(x),πP(y), πP(z)}. ϑP(xαyβz) = ϑP(f(j)αf(k)βf(l)) = ϑP(f(jkl)) = f−1 (ϑP)(jkl) ≤ max{f−1 (ϑP)(j),f −1 (ϑP)(k),f−1 (ϑP)(l)} = max{ϑP(f(j)), ϑP(f(k)), ϑP(f(l))} = max{ϑP(x),ϑP(y),ϑP(z)}. Hence 𝑃 is a Pythagorean fuzzy weak bi-ideal of S. Theorem 8. Let 𝑓 : 𝑀 → 𝑆 be an onto 𝛤-near-ring homomorphism. If 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀, then 𝑓(𝑃) = 𝑓(𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑀. Proof. Let 𝑃 be a Pythagorean fuzzy weak bi-ideal of 𝑀. Since 𝑓(𝜋𝑃)(𝑥′) = 𝑠𝑢𝑝 𝑓(𝑥)=𝑥′ (𝜋𝑃(𝑥)) , for 𝑥′ ∈ 𝑆 and 𝑓(𝜗𝑃)(𝑥′) = 𝑖𝑛𝑓 𝑓(𝑥)=𝑥′ (𝜗𝑃(𝑥)) , for 𝑥′ ∈ 𝑆hence 𝑓(𝑃) is nonempty. Let 𝑥′, 𝑦′ ∈ 𝑆 and 𝛼, 𝛽 ∈ 𝛤. Then we have {𝑥|𝑥 ∈ 𝑓−1 (𝑥′ − 𝑦) ⊇ {𝑥 − 𝑦|𝑥 ∈ 𝑓−1 (𝑥′) and 𝑦 ∈ 𝑓−1 (𝑦′)} and {𝑥|𝑥 ∈ 𝑓−1 (𝑥′𝑦′)} ⊇ {𝑥𝛼𝑦|𝑥 ∈ 𝑓−1 (𝑥′) and 𝑦 ∈ 𝑓−1 (𝑦′)}. f(πP)(x′ − y′) = sup f(z)=(x′−y′) {πP(z)} ≥ sup f(x)=x′,f(y)=y′ {πP(x − y)}
- 12. 308 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 ≥ sup f(x)=x′,f(y)=y′ {min{πP(x), πP(y)}} = min{ sup f(x)=x′ {πP(x)}, sup f(y)=y′ {πP(y)}} = min{f(πP)(x′), f(πP)(y′)}. And f(ϑP)(x′ − y′) = inf f(z)=x′−y′ {ϑP(z)} ≤ inf f(x)=x′,f(y)=y′ {ϑP(x − y)} ≤ inf f(x)=x′,f(y)=y′ {max{ϑP(x), ϑP(y)}} = max{ inf f(x)=x′ {ϑP(x)}, inf f(y)=y′ {ϑP(y)}} = max{f(ϑP)(x′), f(ϑP)(y′)}. Next, f(πP)(x′αy′βz′) = sup f(w)=x′αy′βz′ {πP(w)} ≥ sup f(x)=x′,f(y)=y′,f(z)=z′ {πP(xαyβz)} ≥ sup f(x)=x′,f(y)=y′,f(z)=z′ {min{πP(x),πP(y), πP(z)}} = min{ sup f(x)=x′ {πP(x)}, sup f(y)=y′ {πP(y)}, sup f(z)=z′ {πP(z)}} = min{f(πP)(x′),f(πP)(y′),f(πP)(z′)}. And f(ϑP)(x′αy′βz′) = inf f(w)=x′αy′βz′ {ϑP(w)} ≤ inf f(x)=x′,f(y)=y′,f(z)=z′ {ϑP(xαyβz)} ≤ inf f(x)=x′,f(y)=y′,f(z)=z′ {max{ϑP(x),ϑP(y),ϑP(z)}} = max{ inf f(x)=x′ {ϑP(x)}, inf f(y)=y′ {ϑP(y)}, inf f(z)=z′ {ϑP(z)}} = max{f(ϑP)(x′), f(ϑP)(y′), f(ϑP)(z′)}. Therefore 𝑓(𝑃) is a Pythagorean fuzzy weak bi-ideal of S.
- 13. 309 Pythagorean fuzzy weak bi-ideals of Γ- near ring 5| Interval Valued Pythagorean Fuzzy Weak Bi-Ideals of 𝚪-Near- Rings In this section, we initiate the notion of interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 and discuss some of its properties. Definition 7. An interval valued Pythagorean fuzzy set 𝑃 = (𝜋𝑃,𝜗𝑃) of 𝑀 is called an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, if I. 𝜋𝑃(𝑥 − 𝑦) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}. II. 𝜗𝑃(𝑥 − 𝑦) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. III. 𝜋𝑃(𝑥𝛾𝑦𝛾𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}. IV. 𝜗𝑃(𝑥𝛾𝑦𝛾𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} for all 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. Example 3. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation ` + ′ and 𝛤 = {𝛾} be a nonempty set of binary operations as the following tables: and Let 𝜋𝑃 : 𝑀 → 𝐷[0,1] and 𝜗𝑃 : 𝑀 → 𝐷[0,1] be aninterval valued fuzzy subsets defined by 𝜋𝑃(𝑤) = [0.6,0.7], 𝜋𝑃 (𝑥) = [0.5,0.6], 𝜋𝑃(𝑦) = 𝜋𝑃(𝑧) = [0.4,0.5]. And 𝜗𝑃(𝑤) = [0.2,0.3], 𝜗𝑃(𝑥) = [0.4,0.5], 𝜗𝑃(𝑦) = [0.7,0.8] = 𝜗𝑃(𝑧). Then 𝑃 = (𝜋𝑃,𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Theorem 9. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be an interval valued Pythagorean fuzzy subgroup of M. Then 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Let 𝑥, 𝑦, 𝑧, 𝑦1, 𝑦2 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤 such that 𝑥 = 𝑦𝛼𝑧 and 𝑦 = 𝑦1𝛽𝑦2. Then (πP ⋆ πP ⋆ πP)(x) = sup x=yαz {min{(πP ⋆ πP)(y),πP(z)}} = sup x=yαz {min{ sup y=y1βy2 min{πP(y1), πP(y2)}, πP(z)}} = sup x=yαz sup y=y1βy2 {min{min{πP(y1), πP(y2)}, πP(z)}} + 𝐰 𝐱 𝐲 𝐳 𝐰 w x y z 𝐱 x w z y 𝐲 y z w x 𝐳 z y x w 𝛄 𝐰 𝐱 𝐲 𝐳 𝐰 w x w x 𝐱 w x w x 𝐲 w x y z 𝐳 w x y z
- 14. 310 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 = sup x=y1βy2αz {min{πP(y1), πP(y2), πP(z)}} πP(y1βy2αz) ≥ min{πP(y1), πP(y2), πP(z)} ≤ sup x=y1βy2αz πP(y1βy2αz) = πP(x). And (ϑP ⋆ ϑP ⋆ ϑP)(x) = inf x=yαz {min{(ϑP ⋆ ϑP)(y),ϑP(z)}} = inf x=yαz {max{ inf y=y1βy2 min{ϑP(y1), ϑP(y2)}, ϑP(z)}} = inf x=yαz inf y=y1βy2 {max{max{ϑP(y1), ϑP(y2)}, ϑP(z)}} = inf x=y1βy2αz {max{ϑP(y1), ϑP(y2), ϑP(z)}} ϑP(y1βy2αz) ≤ max{ϑP(y1), ϑP(y2), ϑP(z)} ≥ inf x=y1βy2αz ϑP(y1βy2αz) = ϑP(x). Since 𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, If 𝑥 can not be expressed as 𝑥 = 𝑦𝛼𝑧, then (𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃)(𝑥) = 0 ≤ 𝜋𝑃(𝑥) and (𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃)(𝑥) = 0 ≥ 𝜗𝑃(𝑥). In both cases 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Conversely, assume that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃. For 𝑥′, 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽, 𝛼1, 𝛽1 ∈ Γ. Let 𝑥′ be such that 𝑥′ = 𝑥𝛼𝑦𝛽𝑧. Then 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) = 𝜋𝑃(𝑥′) ≥ (𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃)(𝑥′) = sup x′=pα1q {min{(πP ⋆ πP)(p), πP(q)}} = sup x′=pα1q {min{ sup p=p1β1p2 min{πP(p1), πP(p2)}, πP(q)}} = sup x′=p1β1p2α1q {min{πP(p1), πP(p2), πP(q)}} ≥ min{πP(x),πP(y), πP(z)}. ϑP(xαyβz) = ϑP(x′) ≤ (ϑP ∗ ϑP ∗ ϑP)(x′)
- 15. 311 Pythagorean fuzzy weak bi-ideals of Γ- near ring = inf x′=pα1q {max{(ϑP ∗ ϑP)(p),ϑP(q)}} = inf x′=pα1q {max{ inf p=p1β1p2 min{ϑP(p1), ϑP(p2)}, ϑP(q)}} = inf x′=p1β1p2α1q {max{ϑP(p1), ϑP(p2), ϑP(q)}} ≤ max{ϑP(x),ϑP(y),ϑP(z)}. Hence 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Lemma 3. Let 𝜋𝑃 = (𝜋𝑃1, 𝜋𝑃2) and 𝜗𝑃 = (𝜗𝑃1, 𝜗𝑃2) be an interval valued Pythagorean fuzzy weal bi-ideals of M. Then the products 𝜋𝑃 ∗ 𝜗𝑃 and 𝜗𝑃 ∗ 𝜋𝑃 are also interval valued Pythagorean fuzzy weak bi-ideals of 𝑀. Proof. Let 𝜋𝑃 and 𝜗𝑃 be an interval valued Pythagorean fuzzy weak bi-ideals of 𝑀 and let 𝛼, 𝛼1, 𝛼2 ∈ 𝛤. Then (π1 ⋆ π2)(x − y) = sup x−y=aαb min{π(a),π(b)} ≥ sup x−y=a1α1b1−a2α2b2<(a1−a2)(b1−b2) min{π(a1 − a2), π(b1 − b2)} ≥ sup min{min{π(a1), π(a2)}, min{π(b1), π(b2)}} = supmin{min{π(a1), π(b1)}, min{π(a2), π(b2)}} ≥ min{ sup x=a1α1b1 min{π(a1), π(b1)}, sup y=a2α2b2 min{π(a2), π(b2)}} = min{(π ⋆ π)(x), (π ⋆ π)(y)}. (ϑ1 ⋆ ϑ2)(x − y) = inf x−y=aαb max{ϑ(a),ϑ(b)} ≤ inf x−y=a1α1b1−a2α2b2<(a1−a2)(b1−b2) max{ϑ(a1 − a2), ϑ(b1 − b2)} ≤ infmax{max{ϑ(a1), ϑ(a2)}, max{ϑ(b1), ϑ(b2)}} = infmax{max{ϑ(a1), ϑ(b1)}, max{ϑ(a2), ϑ(b2)}} ≤ max{ inf x=a1α1b1 max{ϑ(a1), ϑ(b1)}, inf y=a2α2b2 max{ϑ(a2), ϑ(b2)}} = max{(ϑ ⋆ ϑ)(x), (ϑ ⋆ ϑ)(y)}. It follows that 𝜋 ⋆ 𝜗 is an interval valued Pythagorean fuzzy subgroup of 𝑀. Further, (𝜋 ⋆ 𝜗) ⋆ (𝜋 ⋆ 𝜗) ⋆ (𝜋 ⋆ 𝜗) = 𝜋 ⋆ 𝜗 ⋆ (𝜋 ⋆ 𝜗 ⋆ 𝜋) ⋆ 𝜗 ⊆ 𝜋 ⋆ 𝜗 ⋆ (𝜗 ⋆ 𝜗 ⋆ 𝜗) ⋆ 𝜗 ⊆ 𝜋 ⋆ (𝜗 ⋆ 𝜗 ⋆ 𝜗), since 𝜗 is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 ⊆ 𝜋 ⋆ 𝜗.
- 16. 312 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 Therefore 𝜋𝑃 ⋆ 𝜗𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Similarly 𝜗𝑃 ⋆ 𝜋𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of M. Lemma 4. Every interval valued Pythagorean fuzzy ideal of 𝑀 is an interval valued Pythagorean fuzzy bi-ideal of 𝑀. Proof. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy ideal of 𝑀. Then 𝜋𝑃 ⋆ M ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ M ⋆ M ⊆ 𝜋𝑃 ⋆ M ⊆ 𝜋𝑃, 𝜗𝑃 ⋆ M ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ M ⋆ M ⊇ 𝜗𝑃 ⋆ M ⊇ 𝜗𝑃, since 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy ideal of 𝑀. This implies that 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy bi-ideal of M. Theorem 10. Every interval valued Pythagorean fuzzy bi-ideal of 𝑀 is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy bi-ideal of 𝑀. Then 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. We have 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃. This implies that 𝜋𝑃 ⋆ 𝜋𝑃 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 ⋆ 𝑀 ⋆ 𝜋𝑃 ⊆ 𝜋𝑃 and 𝜗𝑃 ⋆ 𝜗𝑃 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃 ⋆ 𝑀 ⋆ 𝜗𝑃 ⊇ 𝜗𝑃. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of M. Example 4. Let 𝑀 = {𝑤, 𝑥, 𝑦, 𝑧} be a nonempty set with binary operation+ and 𝛤 = {𝛼} be a nonempty set of binary operations as the following tables: and Let 𝑃 : 𝑀 → 𝐷[0,1] be aninterval valued Pythagorean fuzzy set defined by 𝜋𝑃(𝑤) = [0.8,0.9], 𝜋𝑃(𝑥) = [0.3,0.4] = 𝜋𝑃(𝑦) and 𝜋𝑃(𝑧) = [0.5,0.6], and 𝜗𝑃(𝑤) = [0,0.1],𝜗𝑃(𝑥) = [0.4,0.5] = 𝜗𝑃(𝑦), 𝜗𝑃(𝑧) = [0.2,0.3]. Then 𝜋𝑃 is an interval valued fuzzy weak bi-ideal of 𝑀. But 𝜋𝑃 is not a fuzzy ideal and bi-ideal of 𝑀 and + 𝐰 𝐱 𝐲 𝐳 𝐰 w x y z 𝐱 x w z y 𝐲 y z w x 𝐳 z y x w 𝛂 𝐰 𝐱 𝐲 𝐳 w w w w w x w x w x y w w y y z w x y z
- 17. 313 Pythagorean fuzzy weak bi-ideals of Γ- near ring 𝜋𝑃(𝑧𝛾𝑦𝛾𝑧) = 𝜋𝑃(𝑦) = [0.3,0.4] > [0.5,0.6] = 𝑚𝑖𝑛{𝜋𝑃(𝑧), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼(𝑧 + 𝑤) − 𝑥𝛼𝑤) ≤ 𝜗𝑃(𝑧) = [0.4,0.5] ≤ [0.2,0.3] and 𝜗𝑃(𝑧𝛾𝑥𝛾𝑧) = 𝜗𝑃(𝑥) = [0.4,0.5] ≤ [0.2,0.3] = min{𝜗𝑃(𝑧),𝜗𝑃(𝑧)}. Theorem 11. Let {(𝜋𝑃𝑖 , 𝜗𝑃𝑖 )|𝑖 ∈ 𝛺} be family of interval valued Pythagorean fuzzy weak bi-ideals of a near- ring 𝑀, then ⋂𝑖∈𝛺 𝜋𝑃𝑖 and ∪𝑖∈𝛺 𝜗𝑃𝑖 are also an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, where 𝛺 is any index set. Proof. Let {𝜋𝑃𝑖 , 𝜗𝑃𝑖 }𝑖∈Ω be a family of interval valued Pythagorean fuzzy weak bi-ideals of 𝑀. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀, 𝛼, 𝛽 ∈ Γ and 𝜋𝑃𝑖 = ⋂𝑖∈Ω 𝜋𝑖, 𝜗𝑃𝑖 =∪𝑖∈𝛺 𝜗𝑃𝑖 Then, 𝜋𝑃𝑖 (𝑥) = ⋂𝑖∈Ω 𝜋𝑃𝑖 (𝑥) = (inf 𝑖∈Ω 𝜋𝑃𝑖 )(𝑥) = inf 𝑖∈Ω 𝜋𝑃𝑖 (𝑥) and 𝜗𝑃𝑖 (𝑥) =∪𝑖∈𝛺 𝜗𝑃𝑖 (𝑥) = (sup 𝑖∈Ω 𝜗𝑃𝑖 )(𝑥) = sup 𝑖∈Ω 𝜗𝑃𝑖 (𝑥) . πPi (x − y) = inf i∈Ω πPi (x − y) ≥ inf i∈Ω min{πPi (x),πPi (y)} = min{inf i∈Ω πPi (x), inf i∈Ω πPi (y)} = min{⋂ i∈Ω πPi (x), ⋂ i∈Ω πPi (y)} = min{πPi (x), πPi (y)}. ϑPi (x − y) = sup i∈Ω ϑPi (x − y) ≤ sup i∈Ω max{ϑPi (x),ϑPi (y)} = max{sup i∈Ω ϑPi (x), sup i∈Ω ϑPi (y)} = max{∪i∈Ω ϑPi (x),∪i∈Ω ϑPi (y)} = max{ϑPi (x),ϑPi (y)}. And, πPi (xαyβz) = inf i∈Ω πPi (xαyβz) ≥ inf i∈Ω min{πPi (x),πPi (y), πPi (z)} = min{inf i∈Ω πPi (x), inf i∈Ω πPi (y), inf i∈Ω πPi (z)} = min{⋂ i∈Ω πPi (x), ⋂ i∈Ω πPi (y), ⋂ i∈Ω πPi (z)}
- 18. 314 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 = min{πPi (x),πPi (y), πPi (z)}. ϑPi (xαyβz) = sup i∈Ω ϑPi (xαyβz) ≤ sup i∈Ω max{ϑPi (x), ϑPi (y),ϑPi (z)} = max{sup i∈Ω ϑPi (x), sup i∈Ω ϑPi (y), sup i∈Ω ϑPi (z)} = max{∪i∈Ω ϑPi (x),∪i∈Ω ϑPi (y),∪i∈Ω ϑPi (z)} = max{ϑPi (x),ϑPi (y), ϑPi (z)}. Theorem 12. Let 𝑃 = (𝜋𝑃,𝜗𝑃) be an interval valued Pythagorean fuzzy subset of M. Then 𝑈(𝜋𝑃; 𝑡) and 𝐿(𝜗𝑃;𝑠) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 if and only if 𝜋𝑃𝑡 is a weak bi-ideal of 𝑀, for all 𝑡 ∈ [0,1]. Proof. Assume that 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑅. Let 𝑠, 𝑡 ∈ [0,1] such that 𝑥, 𝑦 ∈ 𝑈(𝜋𝑃; 𝑡). Then 𝜋𝑃(𝑥) ≥ 𝑡 and 𝜋𝑃(𝑦) ≥ 𝑡,then 𝜋𝑃(𝑥 − 𝑦) ≥ min{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} ≥ min{𝑡,𝑡} = 𝑡 and 𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)} ≤ 𝑚𝑎𝑥{𝑠,𝑠} = 𝑠. Thus 𝑥 − 𝑦 ∈ 𝑈(𝜋𝑃 𝑡).Let 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃𝑡 and 𝛼, 𝛽 ∈ Γ. This implies that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ min{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)} ≥ min{𝑡, 𝑡, 𝑡} = 𝑡, and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)} ≤ max{𝑠, 𝑠, 𝑠} = 𝑠. Therefore 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑈(𝜋𝑃; 𝑠). Hence 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Conversely, assume that 𝑈 (𝜋𝑃; t) and (𝜗𝑃;𝑠) is an interval valuedPythagorean fuzzy weak bi-ideal of 𝑀, for all 𝑠, 𝑡 ∈ [0,1]. Let 𝑥, 𝑦 ∈ 𝑀. Suppose 𝜋𝑃(𝑥 − 𝑦) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)}and 𝜗𝑃(𝑥 − 𝑦) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. Choose 𝑡 such that 𝜋𝑃(𝑥 − 𝑦) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}. This implies that 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡 and 𝜋𝑃(𝑥 − 𝑦) < 𝑡. Then we have 𝑥, 𝑦 ∈ 𝜋𝑃𝑡 but 𝑥 − 𝑦 ∈ 𝜋𝑃𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠 and 𝜗𝑃(𝑥 − 𝑦) > 𝑠, we have 𝑥, 𝑦 ∈ 𝜗𝑃𝑠 but 𝑥 − 𝑦 ∈ 𝜗𝑃𝑠 a contradiction. Thus 𝜋𝑃(𝑥 − 𝑦) ≥ min{𝜋𝑃(𝑥), 𝜋𝑃(𝑦)} and 𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦)}.
- 19. 315 Pythagorean fuzzy weak bi-ideals of Γ- near ring If there exist 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ Γ such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦), 𝜋𝑃(𝑧)} and 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦), 𝜗𝑃(𝑧)}. Choose 𝑡 such that 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡 < 𝑚𝑖𝑛{𝜋𝑃(𝑥),𝜋𝑃(𝑦),𝜋𝑃(𝑧)}. Choose 𝑠 such that 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) > 𝑠 > 𝑚𝑎𝑥{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Then 𝜋𝑃(𝑥) > 𝑡, 𝜋𝑃(𝑦) > 𝑡, 𝜋𝑃(𝑧) > 𝑡 and 𝜗𝑃(𝑥) < 𝑠, 𝜗𝑃(𝑦) < 𝑠, 𝜗𝑃(𝑧) < 𝑠 and 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) < 𝑡. So, 𝑥, 𝑦, 𝑧 ∈ 𝜋𝑃 > but 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜋𝑃𝑡, and 𝑥𝛼𝑦𝛽𝑧 ∈ 𝜗𝑃𝑠, which is a contradiction. Hence 𝜋𝑃(𝑥𝛼𝑦𝛽𝑧) ≥ min{𝜋𝑃(𝑥),𝜋𝑃(𝑦), 𝜋𝑃(𝑧)}, 𝜗𝑃(𝑥𝛼𝑦𝛽𝑧) ≤ max{𝜗𝑃(𝑥),𝜗𝑃(𝑦),𝜗𝑃(𝑧)}. Therefore 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of M. Theorem 13. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀 then the set 𝑀𝜋𝑃,𝜗𝑃 = {𝑥 ∈ 𝑀|𝜋𝑃(𝑥) = 𝜋𝑃(0) = 𝜗𝑃(𝑥)} is a interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Proof. Let 𝑥, 𝑦 ∈ 𝑀(𝜋𝑃,𝜗𝑃) . Then 𝜋𝑃(𝑥) = 𝜋𝑃(0), 𝜋𝑃(𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥) = 0, 𝜗𝑃(𝑦) = 0 and 𝜋𝑃(𝑥 − 𝑦) ≥ min{𝜋𝑃(𝑥),𝜋𝑃(𝑦)} = min{𝜋𝑃(0), 𝜋𝑃(0)} = 𝜋(0), and 𝜗𝑃(𝑥 − 𝑦) ≤ max{𝜗𝑃(𝑥), 𝜗𝑃(𝑦)} = max{𝜗𝑃(0),𝜗𝑃(0)} = 𝜗𝑃(0). So 𝜋𝑃(𝑥 − 𝑦) = 𝜋𝑃(0),𝜗𝑃(𝑥 − 𝑦) = 𝜗𝑃(0). Thus 𝑥 − 𝑦 ∈ 𝑀𝜋𝑃 , 𝑥 − 𝑦 ∈ 𝑀𝜗𝑃 . For every 𝑥, 𝑦, 𝑧 ∈ 𝑀𝜋𝑃 and 𝛼, 𝛽 ∈ Γ. We have πP(xαyβz) ≥ min{πP(x), πP(y),πP(z)}, = min{πP(0), πP(0), πP(0)} = πP(0), and ϑP(xαyβz) ≤ max{ϑP(x), ϑP(y),ϑP(z)} = max{ϑP(0),ϑP(0), ϑP(0)} = ϑP(0). Thus 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜋𝑃 , 𝑥𝛼𝑦𝛽𝑧 ∈ 𝑀𝜗𝑃 . Hence 𝑀(𝜋𝑃,𝜗𝑃) is an interval valuedPythagorean fuzzy weak bi-ideal of M. 6| Homomorphism of Interval Valued Pythagorean Fuzzy Weak Bi- Ideals of 𝚪-Near-Rings In this section, we characterize interval valued Pythagorean fuzzy weak bi-ideals of 𝛤-near-rings using homomorphism.
- 20. 316 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 Definition 8. Let 𝑓 be a mapping from a set 𝑀 to a set 𝑆. Let 𝑓 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy subsets of 𝑀 and 𝑆, resp. then 𝑓 is image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑆 defined by f(πP)(y) = { sup x∈f−1(y) πP(x) if f−1 (y) ≠ ∅ 0 otherwise. f(ϑP)(y) = { inf x∈f−1(y) ϑP(x) if f−1 (y) ≠ ∅ 1 otherwise And the pre-image of 𝜋𝑃 and 𝜗𝑃 under 𝑓 is a fuzzy subset of 𝑀 defined by 𝑓−1 (𝜋𝑃(𝑥)) = 𝜋𝑃(𝑓(𝑥)), 𝑓−1 (𝜗𝑃(𝑥)) = 𝜗𝑃(𝑓(𝑥)) for all 𝑥 ∈ 𝑀 and 𝑓−1 (𝑦) = {𝑥 ∈ 𝑀|𝑓(𝑥) = 𝑦}. Theorem 14. Let 𝑓: 𝑀 → 𝑆 be a homomorphism between 𝛤-near-rings 𝑀 and S. If 𝑃 = (𝜋𝑃,𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑆, then 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃,𝜗𝑃)] is an interval valued fuzzy weakbi-ideal of 𝑀. Proof. Let 𝑓 be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑆. Let 𝑥, 𝑦, 𝑧 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. Then f −1 (πP)(x − y) = πP(f(x − y)) = πP(f(x) − f(y)) ≥ min{πP(f(x)), πP(f(y))} = min{f−1 (πP(x)), f −1 (πP(y))}. f −1 (ϑP)(x − y) = ϑP(f(x − y)) = ϑP(f(x) − f(y)) ≤ max{ϑP(f(x)), ϑP(f(y))} = max{f−1 (ϑP(x)), f−1 (ϑP(y))}. f −1 (πP)(xαyβz) = πP(f(xαyβz)) = πP(f(x)αf(y)βf(z)) ≥ min{πP(f(x)), πP(f(y)), πP(f(z))} = min{f−1 (πP(x)), f −1 (πP(y)), f−1 (πP(z))}. f −1 (ϑP)(xαyβz) = ϑP(f(xαyβz)) = ϑP(f(x)αf(y)βf(z))
- 21. 317 Pythagorean fuzzy weak bi-ideals of Γ- near ring ≤ max{ϑP(f(x)), ϑP(f(y)),ϑP(f(z))} = max{f−1 (ϑP(x)),f −1 (ϑP(y)),f −1 (ϑP(z))}. Therefore 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean fuzzy weak bi-ideal of M. We can also state the converse of the Theorem 7 by strengthening the condition on 𝑓 as follows. Theorem 15. Let 𝑓: 𝑀 → 𝑆 be an onto homomorphism of 𝛤-near-rings 𝑀 and S. Let 𝑃 = (𝜋𝑃, 𝜗𝑃) be an interval valued Pythagorean fuzzy subset of S. If 𝑓−1 (𝑃) = [𝑓−1 (𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, then 𝑃 = (𝜋𝑃, 𝜗𝑃) is a Pythagorean fuzzy weak bi-ideal of 𝑆. Proof. Let 𝑥, 𝑦, 𝑧 ∈ 𝑆. Then 𝑓(𝑗) = 𝑥, 𝑓(𝑘) = 𝑦 and 𝑓(𝑙) = 𝑧 for some 𝑗, 𝑘, 𝑙 ∈ 𝑀 and 𝛼, 𝛽 ∈ 𝛤. It follows that πP(x − y) = πP(f(j) − f(k)) = πP(f(j − k)) = f −1 (πP)(j − k) ≥ min{f−1 (πP)(j),f −1 (πP)(k)} = min{πP(f(j)), πP(f(k))} = min{πP(x), πP(y)}. ϑ(x − y) = ϑ(f(j) − f(k)) = ϑ(f(j − k)) = f −1 (ϑ)(j − k) ≤ max{f−1 (ϑ)(j), f−1 (ϑ)(k)} = max{ϑ(f(j)), ϑ(f(k))} = max{ϑ(x),ϑ(y)}. And πP(xαyβz) = πP(f(j)αf(k)βf(l)) = πP(f(jkl)) = f −1 (πP)(jkl) ≥ min{f−1 (πP)(j),f −1 (πP)(k), f−1 (πP)(l)} = min{πP(f(j)), πP(f(k)), πP(f(l))}
- 22. 318 Marynirmala and Sivakumar |J. Fuzzy. Ext. Appl. 2(3) (2021) 297-320 = min{πP(x),πP(y), πP(z)}. ϑ(xαyβz) = ϑ(f(j)αf(k)βf(l)) = ϑ(f(jkl)) = f−1(ϑ)(jkl) ≤ max{f−1 (ϑ)(j), f−1 (ϑ)(k),f −1 (ϑ)(l)} = max{ϑ(f(j)), ϑ(f(k)), ϑ(f(l))} = max{ϑ(x),ϑ(y),ϑ(z)}. Hence 𝑃 is an interval valued Pythagorean fuzzy weak bi-ideal of S. Theorem 16. Let 𝑓 : 𝑀 → 𝑆 be an onto 𝛤-near-ring homomorphism. If 𝑃 = (𝜋𝑃, 𝜗𝑃) is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀, then 𝑓(𝑃) = [𝑓(𝜋𝑃, 𝜗𝑃)] is an interval valued Pythagorean fuzzy weak bi-ideal of 𝑆. Proof. Let 𝑃 be an interval valued Pythagorean fuzzy weak bi-ideal of 𝑀. Since 𝑓(𝜋𝑃)(𝑥′) = 𝑠𝑢𝑝 𝑓(𝑥)=𝑥′ (𝜋𝑃(𝑥)) and 𝑓(𝜗𝑃)(𝑥′) = 𝑖𝑛𝑓 𝑓(𝑥)=𝑥′ (𝜗𝑃(𝑥)) , for 𝑥′ ∈ 𝑆 and hence 𝑓(𝑃) is nonempty. Let 𝑥′, 𝑦′ ∈ 𝑆 and 𝛼, 𝛽 ∈ 𝛤. Then we have {𝑥|𝑥 ∈ 𝑓−1 (𝑥′ − 𝑦) ⊇ {𝑥 − 𝑦|𝑥 ∈ 𝑓−1 (𝑥′) and 𝑦 ∈ 𝑓−1 (𝑦′)} and {𝑥|𝑥 ∈ 𝑓−1 (𝑥′𝑦′)} ⊇ {𝑥𝛼𝑦|𝑥 ∈ 𝑓−1 (𝑥′) and 𝑦 ∈ 𝑓−1 (𝑦′)}. f(πP)(x′ − y′) = sup f(z)=(x′−y′) {πP(z)} ≥ sup f(x)=x′,f(y)=y′ {πP(x − y)} ≥ sup f(x)=x′,f(y)=y′ {min{πP(x), πP(y)}} = min{ sup f(x)=x′ {πP(x)}, sup f(y)=y′ {πP(y)}} = min{f(πP)(x′), f(πP)(y′)}. And f(ϑP)(x′ − y′) = inf f(z)=x′−y′ {ϑP(z)} ≤ inf f(x)=x′,f(y)=y′ {ϑP(x − y)} ≤ inf f(x)=x′,f(y)=y′ {max{ϑP(x),ϑP(y)}} = max{ inf f(x)=x′ {ϑP(x)}, inf f(y)=y′ {ϑP(y)}}
- 23. 319 Pythagorean fuzzy weak bi-ideals of Γ- near ring = max{f(ϑP)(x′), f(ϑP)(y′)}. Next, f(π)(x′αy′βz′) = sup f(w)=x′αy′βz′ {π(w)} ≥ sup f(x)=x′,f(y)=y′,f(z)=z′ {π(xαyβz)} ≥ sup f(x)=x′,f(y)=y′,f(z)=z′ {min{π(x),π(y), π(z)}} = min{ sup f(x)=x′ {π(x)}, sup f(y)=y′ {π(y)}, sup f(z)=z′ {π(z)}} = min{f(π)(x′), f(π)(y′), f(π)(z′)}. And 𝑓(𝜗𝑃)(𝑥′𝛼𝑦′𝛽𝑧′) = inf 𝑓(𝑤)=𝑥′𝛼𝑦′𝛽𝑧′ {𝜗𝑃(𝑤)} ≤ inf 𝑓(𝑥)=𝑥′,𝑓(𝑦)=𝑦′,𝑓(𝑧)=𝑧′ {𝜗𝑃(𝑥𝛼𝑦𝛽𝑧)} ≤ inf 𝑓(𝑥)=𝑥′,𝑓(𝑦)=𝑦′,𝑓(𝑧)=𝑧′ {max{𝜗𝑃(𝑥), 𝜗𝑃(𝑦), 𝜗𝑃(𝑧)}} = max{ inf 𝑓(𝑥)=𝑥′ {𝜗𝑃(𝑥)}, inf 𝑓(𝑦)=𝑦′ {𝜗𝑃(𝑦)}, inf 𝑓(𝑧)=𝑧′ {𝜗𝑃(𝑧)}} = max{𝑓(𝜗𝑃)(𝑥′), 𝑓(𝜗𝑃)(𝑦′),𝑓(𝜗𝑃)(𝑧′)}. Therefore 𝑓(𝐵) is an interval valued Pythagorean fuzzy weak bi-ideal of S. 7| Conclusion In this paper, we discuss Pythagorean fuzzy weak ideal, Pythagorean fuzzy weak bi-ideal, Homomorphism of Pythagorean fuzzy weak ideal and weak bi-ideal. An interval valued Pythagorean fuzzy ideal, interval valued Pythagorean fuzzy weak bi-ideal, Homomorphism of interval valued Pythagorean fuzzy weak ideal and bi-ideal in gamma near ring are studied and investigated some properties with suitable examples. Acknowledgment This manuscript has been written with the financial support Maulana Azad National Fellowship under the University Grants Commission, New Delhi (F1-17.1/2016-17/MANF-2015-17-TAM 65281/(SA-III/Website)).
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