some p-fuzzifications of gamma soft subgroup structures ... · combine the strength of both soft...

Advances in Fuzzy Mathematics.

ISSN 0973-533X Volume 12, Number 6 (2017), pp. 1191-1204

© Research India Publications

http://www.ripublication.com

Some P-Fuzzifications of Gamma Soft Subgroup

Structures VIA MIN-Operations

M.Subha 1,G. Subbiah2* and M.Navaneethakrishnan 3

1.Assistant Professor in Mathematics, Sri K.G.S. Arts College, Srivaikuntam-628 619, Tamil Nadu, India.

2 *Associate Professor in Mathematics, Sri K.G.S. Arts College, Srivaikuntam-628 619, Tamil Nadu, India.

3 Associate Professor in Mathematics, Kamaraj College, Thoothukudi-628 003, Tamilnadu, India.

* Corresponding author

Abstract

In this paper, the notion of P-Fuzzy M-Gamma soft subgroups of group is introduced

and its basic properties are investigated. The study of the homomorphic and

pre-image of P-Fuzzy M-Gamma soft subgroups are persued. Using t-norm, the

notion of sensible P-Fuzzy M-Gamma soft subgroups in a group is introduced and

some related properties of M-Gamma soft subgroups are discussed.

Keywords: soft set, P– Fuzzy set, P - Fuzzy MΓ –soft subgroup, MΓ - group

homomorphism, Imaginable, t-norm, characterized P-Fuzzy soft set.

2000 AMS Subject Classification: 06F35, 03G25, 03E72, 20N25

Section-1 INTRODUCTION

Molodtsov[15] initiated the concept of soft sets that is free from the difficulties that

have troubled the usual theoretical approaches. Molodtsov pointed out several

directions for the applications of soft sets. Maji et.al [16] gave the operations of soft

sets and their properties. Furthermore, they [16] introduced fuzzy soft sets which

combine the strength of both soft sets and fuzzy sets. As a generalization of the soft

set theory, the fuzzy soft set theory makes description of the objective world more

realistic, practical precise in some cases, making it very promising. Since the notion

of soft groups was proposed by Aktas and Cagman[2], then the soft set theory is used

a new tool to discuss algebraic structures. Cagman et.al[6] studied on soft int-group,

which are different from the definitions of soft groups[2]. The new approach is based

1192 M.Subha ,G. Subbiah and M.Navaneethakrishnan

on the inclusion relation and intersection of sets. It brings the soft set theory, the set

theory, and the group theory together. On the basic of soft int-groups, Sezgin et.al[20]

introduced the concept of soft intersection near-rings (soft int-near rings) by using

intersection operation of sets and gave the applications of soft int near-rings to the

near-ring theory. By introducing soft intersection, union products and soft

characteristic functions, Sezer[20]made a new approach to the classical ring theory

via the soft set theory, with the concepts of soft union rings, ideals and bi-ideal. Jun

et.al[11] applied intersectional soft sets to BCK/BCI-algebras[12] an obtained many

results. Liu et.al [14] further the investigated isomorphisms and fuzzy isomorphisms

theorems of soft rings in [13] respectively. Soft sets were also applied to other

algebraic structures such as near-rings[17], Γ-modulus and BCK/BCI-algebras[10].

The concept of fuzzy sets was first introduced by Zadeh [21]. Rosenfeld [18]

used this concept to formulate the notion of Fuzzy groups. Since then, many other

fuzzy algebraic concepts based on Rosenfeld's fuzzy groups were developed.

Anthony and Sherwood [1] redefined fuzzy groups in terms of t-norms which is

replaced the min operations of Rosenfeld's definition. Using this concept, chang [5]

generalized some of the basic concepts of general topology, many Researchers [3] and

[4] applied the concept of fuzzy sets to the elementary theory of Γ - rings. In [3]

Booth introduced the concept of Γ- near rings which is due to satyanarayana [19].

Also Booth [4] studied radical theory of a Γ - near ring and introduced the notion of

M Γ -group. The notion of Intuitionistic P-Fuzzy semi primality in a semi group is

given by Kim[12]. In this paper, a new class of P-fuzzy M-Gamma soft subgroups of

group are introduced and characterization of some properties of M-Gamma soft

subgroups with respect to t-norms are discussed.

Section- 2 PRELIMINARIES

2.1 Definition: A non-empty set ‘R’ with two binary operations ' + ' and ' • ' is called

a near-ring if it satisfies the following axioms.

1. (R, +) is a group

2. (R, •) is a semi group

3. (a+b)•c = a•c + b•c , for all a,b,c, ∈ R.

Precisely speaking it is a right near-ring. Because it satisfies the right distributive

law.

All near rings considered in this paper will be right distributive. A Γ-near ring is a

triple (M, +, Γ) Where,

(i) (M, +) is a group (not necessarily abelian)

(ii) Γ is a non-empty set of binary operators on M such that for each α ∈ Γ, (M,+, α)

Some P-Fuzzifications of Gamma Soft Subgroup Structures VIA MIN-Operations1193

is a near ring

(iii) a α (bβc) = (a α b)βc for all a,b,c, ∈ M, α,β ∈ Γ. If, in addition, it holds that

(iv) a α 0 = 0 for all a ∈ M, then the Γ-near ring M is said to be zero- symmetric.

2.2 Definition :A soft set 𝑓𝐴 over U is defined as 𝑓𝐴: E→ P(U) such that 𝑓𝐴(𝑥)=∅ if

𝑥 ∉ A. In other words, a soft set 𝑓𝐴 over U is a parameterized family of subsets of the

universe U. For all ε ∈A, 𝑓𝐴(ε) may be considered as the set of ε-approximate

elements of the soft set 𝑓𝐴. A soft set 𝑓𝐴 over U can be presented by the set of ordered

pair:

𝑓𝐴 = {(𝑥, 𝑓𝐴(𝑥)) /𝑥 ∈ E, 𝑓𝐴(𝑥)=P (U)} …………….. (1)

Clearly, a soft set is not a set. For illustration, Molodtsov consider several examples in

[15].

If 𝑓𝐴 is a soft set over U, then the image of 𝑓𝐴 is defined by

Im(𝑓𝐴) = {𝑓𝐴(𝑎)/𝑎 ∈ 𝐴}. The set of all soft sets over U will be denoted by S(U).

Some of the operations of soft sets are listed as follows.

2.3 Definition : Let 𝑓𝐴, 𝑓𝐵 ∈ S(U). If 𝑓𝐴(𝑥) ⊆ 𝑓𝐵(𝑥), for all 𝑥 ∈ E, then 𝑓𝐴is called a

soft subset of 𝑓𝐵and denoted by 𝑓𝐴 ⊆ 𝑓𝐵 .𝑓𝐴and𝑓𝐵 are called soft equal, denoted by

𝑓𝐴 = 𝑓𝐵 , if and only if 𝑓𝐴 ⊆ 𝑓𝐵 and 𝑓𝐵 ⊆ 𝑓𝐴.

2.4 Definition : Let 𝑓𝐴, 𝑓𝐵 ∈ S(U) and 𝜒 be a function from A to B. Then the soft anti-

image of 𝑓𝐴 under 𝜒,denoted by 𝜒𝑓𝐴 , is a soft set over U defined by,

𝜒𝑓𝐴(b) = {

∩ {𝑓𝐴(𝑎)

𝑎∈ 𝐴 , 𝜒(𝑎) = 𝑏} , 𝑖𝑓 𝜒−1(𝑏) ≠ ∅

0 , oterwise …………. (2)

for all b∈ B and the soft pre-image of 𝑓𝐵 under 𝜒 , denoted by 𝜒−1𝑓𝐵

, is a soft set over

U defined by 𝜒−1𝑓𝐵

(a) = 𝑓𝐵(𝜒(𝑎)), for all 𝑎 ∈ 𝐴 .

Note that the concept of level sets in the fuzzy set theory, Cagman et.al[6] initiated

the concept of lower inclusions soft sets which serves as a bridge between soft sets

and crisp sets.

2.5 Definition : Let G be an additive group. If, for all a,b ∈ M, α,β ∈ Γ and x ∈ G it

holds that

(i) a α x ∈ G


(ii) a α(bβc) = (a α b)βx

(iii) (a+b) α x = a α x + b α x, then G is called an M-Gamma group or MΓ -group.

In what follows, let M denotes the Γ -near ring and G denotes the MΓ -group

unless or otherwise specified.

2.6 Definition : A subgroup H of G for which a α h ∈ H for a ∈ M, α ∈ Γ, h∈ H is

called an MΓ-subgroup of G.

We now review some fuzzy logic concepts. A fuzzy set in a set G is a function

A : G → [0, 1 ]. We shall use the notation U(A;t), called a level subset of A for which

{x∈ G / A(x)≥t} where t ∈ [0,1].

2.7 Definition : Let P and G be a set and a group respectively. A mapping

A : G × P → [0, 1 ] is called P-fuzzy soft set in G.

2.8 Definition : A P-fuzzy soft set 'A' in G is called a P-fuzzy MΓ-soft subgroup of G

if

(i) A(x-y, p) ≥ min { A(x,p), A(y,p)}

(ii) A(a α x,p) ≥ A(x,p), for a ∈ M , p∈ P , x ∈G and α ∈ Γ.

2.9 Definition : By a t-norm T, we mean a function T : [0,1] × [0,1] → [0,1]

satisfying the following conditions

[T1] T(x,1) = x

[T2] T(x,y) ≤ T(x,z) if y ≤ z

[T3] T(x,y) = T(y,x)

[T4] T(x,T(y,z)) = T(T(x,y),z) for all x,y,z ∈ [0,1]

2.10 Proposition : For a t-norm, the following statement holds T(x,y) ≤ min {x,y} for

all x,y∈ [0,1]. For a t-norm T on [0,1], denoted by ∆T, the set of elements α ∈ [0,1]

such that T( α, α) = α (ie) ∆T = { α ∈ [ 0,1] / T( α, α) = α }.

2.11 Definition : Let T be a t-norm. A fuzzy set A in G is said to satisfy idempotent

property with respect to T if Im(A)⊆ ∆T.

In definition (2.8), we use T operator for min operation.


Section – 3 Properties of P-Fuzzy MΓ-soft subgroups of Group

In this section, the notion of P-fuzzy MΓ-soft subgroups of MΓ-group are discussed.

3.1 Proposition : Let T be a t-norm . If 'A' is idempotent P-fuzzy MΓ-soft subgroup

of G, then we have A(0,p) ≥ A(x,p) for all x ∈G, p∈P.

Proof: For every x ∈G and p∈P, we have

A(0,p) = A(x-x, p) ≥ T{A(x,p), A(x,p)} = A(x,p) .

This completes the proof.

3.2 Proposition : If 'A' is an idempotent P-fuzzy MΓ-soft subgroup of G, then the set

Gm = {x ∈G / A(x,p) ≥ A(m,p)} is an MΓ-softsubgroup of G.

Proof : Let x,y∈Gm, then A(x,p) ≥ A(m,p) and A(y,p) ≥ A(m,p). Since 'A' is an

P-fuzzy MΓ-subgroup of G, it follows that A(x-y,p) ≥ min {A(x,p), A(y,p)} ≥ min

{A(m,p), A(m,p)} = A(m,p). Now let a ∈ M, α ∈ Γ and h ∈ Gm.Then

A(a α h,p) ≥ A(h,p) ≥ A(m,p).Then we have A(x-y,p) ≥ A(m,p) and A(aαh,p) ≥

A(m,p) that x-y ∈ Gm and aαh ∈ Gm. This completes the proof.

3.3 Corollary :Let T be a t-norm. If 'A' is an idempotent P-fuzzy MΓ-soft subgroup

of G, then the set AG = {x ∈G / A(x,p) = A(0,p)} is an MΓ-soft subgroup of G.

Proof: From the proposition (3.1),

AG= {x ∈G / A(x,p) = A(0,p)}

= {x ∈G / A(x,p) ≥ A(m,p)}

Hence AG is an MΓ- soft subgroup of G from the Proposition (3.2).

3.4 Definition : Let G and G' be MΓ-groups. A map : G → G' is called a

MΓ-group homomorphism if (x+y) = (x) + (y) and (a α x) = a α (x) for all

a ∈ M, α ∈ T and x ∈ G.

3.5 Definition : Let : G → G' be an MΓ-group homomorphism of MΓ-groups.

For a fuzzy soft set A in G', we define a characterized P-fuzzy set A in G by

A (x,p) = A ( (x,p)) for all x ∈ G.

3.6 Propositions : Let : G → G' be an MΓ-group homomorphism of MΓ-groups.


If 'A' is P-fuzzy MΓ-soft subgroup of G', then A is an P-fuzzy MΓ-soft subgroup of

G.

Proof: For any x,y ∈ G and p∈P, we have

A (x-y,p) = A ( (x-y,p))

= A (( (x) - (y)),p)

≥ T{A ( (x,p)), A ( (y,p)

≥ T{A(x,p), A(y,p)}

Let a ∈ M, α ∈ T and x ∈ G, then

A (a α x,p) = A ( (a α x,p))

= A (a α (x,p))

≥ A ( (x,p))

≥ A(x,p)


3.7 Proposition : Let I be an MΓ-subgroup of G and let 'A' be P-fuzzy soft set in G

defined by

A(x,p) = {(𝑎, 𝑝) 𝑖𝑓 𝑥 ∈ 𝐼(𝑏, 𝑝) 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

}

for all x ∈ G, where a,b ∈ [0,1] with a > b, then 'A' is P-fuzzy MΓ-soft subgroup of

G where min{a,b} = max {a+b-1, 0} for all a,b ∈ [0,1].

Proof :Let x , y ∈ G. If x, y ∈ I, then

min {A(x,p), A(y,p)} = min {a,a} = max {2a-1,0}

= {2𝑎 𝑖𝑓 𝑎 ≥ 1/2𝑏 𝑖𝑓 𝑎 < 1/2

}

≤ a = A(x-y, p)

and for all m∈ M and α ∈ Γ, we have

A(mαx, p) = A(x,p) = a.

If y ∈ I and x ∉ I (or x ∈ I and y ∉ I), then

min {A(x,p), A(y,p)} = min {a,b} =max {a+b-1, 0}


= {𝑎 + 𝑏 − 1 𝑖𝑓 𝑎 + 𝑏 ≥ 1/2𝑏 𝑖𝑓 𝑎 + 𝑏 < 1/2

}

≤ b = A(x-y, p)

and for all m∈ M and α∈ T, we have

A(mαx, p) ≥ b = A(y,p).

If y ∉ I and x ∉ I, then

min {A(x,p), A(y,p)} = A(b,b) = max {2b-1, 0}

= {2𝑏 − 1 𝑖𝑓 𝑏 ≥ 1/2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

}

≤ b = A(x-y, p)

and for all m∈ M and α∈ T, we have

A(mαx, p) ≥ b = A(x,p).

Hence 'A' is P-fuzzy MΓ –soft subgroup of G.

For any subset I of MΓ-subgroup of G, Φ denotes the characteristic function of I.

3.8 Corollary : I ⊆ G, then I is an MΓ- subgroup of G if and only if Φ is P-fuzzy

MΓ-soft subgroup of G.

Proof : Let I be an MΓ-subgroup of G. Then it is easy to show that Φ is P-fuzzy MΓ-

soft subgroup of G.

In fact, let x,y ∈ I and so x-y ∈ I.

Hence we have Φ(x-y,p) = 1 = T{ Φ(x,p), Φ(y,p)} = min{1,1}.

Assume that x ∈ I and y∈ I or x ∉ I and y∈ I.

Then Φ(x,p) = 1 >0 = Φ(y,p) (or Φ(x,p)=0<1= Φ(y,p)).

It follows that Φ(x-y,p)≥ T{ Φ(x,p), Φ(y,p)} = min{1,0} = 0.

Now let a∈ M and α ∈ Γ.

If y∈ I, then we have a α x ∈ I. Hence Φ( a α x,p) = 1 = Φ(y,p).

If y ∉ I, then Φ(a α x,p)≥ Φ(y,p).

Conversly , let Φ be P-fuzzy MΓ-soft subgroup of G.

Let x,y∈ I.


Then we have Φ(x-y,p)≥ T{ Φ(x,p), Φ(y,p)} = min{1,1} = 1 and so x-y ∈ I.

Now let a∈ M ,α ∈ Γ and y∈ I. Hence Φ( a α x,p) ≥ Φ(y,p) = 1 and so a α x ∈ I.

3.9 Proposition : Let T be a t-norm. Then every idempotent P-fuzzy MΓ-soft

subgroup of G is a fuzzy soft ideal of G.

Proof : Let 'A' be an idempotent P-fuzzy MΓ-soft subgroup of G. Then

A(x-y,p) ≥ T{A(x,p), A(y,p)} for all x,y ∈ G. Since 'A' satisfies the idempotent

property. We have,

min {A(x,p), A(y,p)} = T{min {A(x,p), A(y,p)}, min {A(x,p), A(y,p)}}

≤ T{A(x,p), A(y,p)}

≤ min{A(x,p), A(y,p)}.

It follows that A(x-y,p) ≥ T{A(x,p), A(y,p)} = min{A(x,p), A(y,p)}.So that 'A' is a

fuzzy soft ideal of G.

3.10 Proposition : The family of P-fuzzy MΓ-soft subgroups of G is completely

distributive lattice with respect to meet '⋀' and join '⋁'.

Proof : Since [0,1] is a completely distributive lattice with respect to the usual

ordering in [0,1], it is sufficient to show that ⋁α∈∧ A⍺ and ⋀α∈∧A⍺ are P-fuzzy

MΓ-soft subgroups of G for a family of P-fuzzy MΓ- soft subgroups {A⍺ / ⍺∈∧}.

For any x,y ∈ G, we have

(⋁α ∈∧ A⍺) (x-y,p) = sup {A⍺ (x-y,p) / ⍺∈⋀}

≥ sup {T(A⍺ (x,p) , A⍺ (y,p)) / ⍺∈⋀}

≥ T{sup {T(A⍺ (x,p)) / ⍺∈⋀} , sup{T(A⍺ (y,p)) / ⍺∈⋀}}

= T{(⋁α∈∧ A⍺ ) (x,p), (⋁α∈∧ A⍺ ) (y,p)}

(⋀α∈∧ A⍺) (x-y,p) = inf {A⍺ (x-y,p) / ⍺∈⋀}

≥ inf {T(A⍺ (x,p) , A⍺ (y,p)) / ⍺∈⋀ }

≥ T{inf {T(A⍺ (x,p) )/ ⍺∈⋀} ,inf{T(A⍺ (y,p)) / ⍺∈⋀ }}

= T{(⋀α∈∧ A⍺ ) (x,p), (⋀α∈∧ A⍺ ) (y,p)}


Now let a∈ M , y∈ I and α ∈ Γ.Then

(⋁α∈∧ A⍺ ) (a α x,p) = sup{A⍺ (a α x,p) / ⍺∈⋀}

≥ sup {A⍺ (x,p) / ⍺∈⋀}

= (⋁α ∈∧ A⍺ ) (x,p)

(⋀α∈∧ A⍺ ) (a α x,p) = inf {A⍺ (a α x,p) / ⍺∈⋀}

≥ inf {A⍺ (x,p) / ⍺∈⋀}

= (⋀α ∈∧ A⍺ ) (x,p)

Hence ⋁α∈∧ A⍺ and ⋀α∈∧ A⍺ are P-fuzzy MΓ-soft subgroups of G.


3.11 Proposition : Let 'A' be P-fuzzy MΓ-soft subgroup of G and let α ∈ Γ be such

that T(⍺,⍺) = ⍺.Then ⋃(A; ⍺) is either empty or an MΓ-subgroup of G for all x∈ G.

Proof : Let x, y∈⋃(A; ⍺).Then we have

A(x,p) ≥⍺ and A(y,p) ≥⍺ and so

A(x-y,p) ≥ T(A(x,p) , A(y,p) )≥ T(⍺, ⍺ ) =⍺

Which implies that x-y∈⋃(A; ⍺). Now let a∈ M, y∈⋃(A; ⍺) and γ ∈ Γ. Then we have

A(a γ x,p) ≥ A(x,p) ≥⍺. So a γ x ∈⋃(A; ⍺). Hence ⋃ (A; ⍺) is MΓ-subgroup of G.

Since T(1,1) = 1 we have the following corollary.

3.12 Corollary : If 'A' is P-fuzzy MΓ-soft subgroup of G ,then ⋃(A; 1) is either

empty or an MΓ-subgroup of G.

Proof : For a family {A⍺ / ⍺∈∧ }of P-fuzzy soft sets in G, define the join ⋁α ∈∧ A⍺

and the meet ⋀α ∈∧ A⍺ as follows

(⋁α∈∧ A⍺ ) (x,p) = sup {A⍺ (x,p) / ⍺∈⋀}

(⋀α∈∧ A⍺ ) (x,p) = inf {A⍺ (x,p) / ⍺∈⋀}


for all x∈ G, where ⋀ is any index set.

Hence the proof.

3.13 Proposition : Let T be a t-norm and let 'A' be a P-fuzzy set in G with

Im(A) = {⍺1, ⍺2, ... … …,⍺n} where ⍺i <⍺j where i > j. Suppose that there exists a

chain of MΓ-subgroups of G: G0 G1 ... ... ... … Gn = G such that A(Ḡk) = Gk,

where Ḡk = Gk/ Gk-1 and G1=0 for k=0,1,2,... ... ....,n .Then 'A' is P-fuzzy MΓ-soft

subgroup of G.

Proof : Let x, y ∈ G. If x and y belong to the some Ḡk, then A(x,p) = A(y,p) = Gk

and x-y ∈ Gk. Hence A(x-y,p) ≥ Gk = min {A(x,p) , A(y,p)}

≥ T{A(x,p), A(y,p)}

Let x ∈ Ḡi and y ∈ Ḡj for every i ≠ j.Without loss of generality ,we may assume that

i >j, then A(x,p) = Gi <Gj = A(y,p) and x-y∈Gi. It follows that

A(x-y,p) ≥Gi = min {A(x,p) , A(y,p)}

≥ T{A(x,p), A(y,p)}

Now let y∈ G, then there exists Gk such that y ∈ Ḡk for some k ∈ {0,1,2,...}. For

any a∈ M, x ∈ Ḡk and α ∈ Γ. We have a α x ∈ Gk and so

A(a α x,p) ≥ Gk ≥ A(x,p). Hence 'A' is P-fuzzy MΓ-soft subgroup of G.

3.14 Proposition : Let T be a t-norm. Then every imaginable P-fuzzy MΓ-soft

subgroup of G is a fuzzy MΓ-soft subgroup of G.

Proof : Assume 'A' is imaginable P-fuzzy MΓ-soft subgroup of G, then we have

A(x-y,p) ≥ T{A(x,p), A(y,p)} and A(a α x,p) ≥ A(x,p) for all x,y in G.

Since A is imaginable, we have

min{A(x,p), A(y,p)} = T{min{A(x,p), A(y,p)}, min{A(x,p), A(y,p)}}

≤ T{A(x,p), A(y,p)}

≤ min {A(x,p), A(y,p)}

and so T(A(x,p), A(y,p)) = min{A(x,p), A(y,p)}.

It follows that A(x-y,p) ≥ T{A(x,p), A(y,p)} = min{A(x,p), A(y,p)} for all x,y∈ G.

Hence 'A' is fuzzy MΓ-soft subgroup of G.


3.15 Proposition: If 'A' is P-fuzzy MΓ-soft subgroup of a MΓ-group G and is an

endomorphism of G, then A[] is P-fuzzy MΓ-soft subgroup of G.

Proof : For any x,y∈ G, we have

1. A[] (x-y,p) = A[ (x,p), (y,p)]

≥ T{A[ (x,p)], A[ (y,p)]}

= T{A[] (x,p), A[] (y,p)}

2. A[] (a α x,p) = A[ (a α x ,p)]

≥ A[ (x,p)]

= A[](x,p)

Hence A[]is P-fuzzy MΓ-soft subgroup of G.

3.16 Definition : Let f : G → G' be a group homomorphism and 'A' be P-fuzzy

MΓ-subgroup of G'. Then A (f(x,p)) = (Af)(x,p) = f-1(A)(x,p)

3.17 Proposition : Let f : G → G' be a group homomorphism and 'A' be P-fuzzy

MΓ-soft subgroup of G' .Then f-1(A) is P-fuzzy MΓ-soft subgroup of G.

Proof : Let x,y∈ G. we have

f-1(A) (x-y,p) = (Af)(x-y,p) = A(f(x-y,p)) = A((f(x)-f(y)),p) ≥ T{A(f(x,p)),

A(f(y,p))} ≥ T{(Af)(x,p), (Af)(y,p)}≥ T{f-1(A)(x,p), f-1(A)(y,p)}.

f-1(A) (a α x,p) = (Af)(a α x,p) = A(f(a α x,p)) ≥ f-1(A)(x,p)

3.18 Proposition : Let 'A' be P-fuzzy MΓ-soft subgroup of G and A* be P-fuzzy

soft set in G defined by A*(x,p) = A(x,p)+1 – A(e,p). Then A* is P-fuzzy MΓ-soft

subgroup of G containing A.

Proof : For x,y∈ G, we have

A*(x-y,p) = A(x-y,p) + 1 – A(e,p)

≥ T{A(x,p), A(y,p)}+1 – A(e,p)

≥ T{A(x,p) +1 – A(e,p), A(y,p)+1 – A(e,p)}

≥ T{A*(x,p), A*(y,p)}

A*(a α x,p) = A(a α x,p) + 1 – A(e,p)


≥ A(x,p)+1 – A(e,p)

≥ A*(x,p)

Therefore A* is P-fuzzy MΓ-soft subgroup of G containing A.

3.19 Proposition : Let T be a continuous t-norm and let 'f' be a homomorphism on a

group G. If 'A' is P-fuzzy MΓ-soft subgroup of G, then Af is P-fuzzy MΓ-soft

subgroup of f(G).

Proof : Let A1 = f-1(y1,p), A2 = f-1(y2,p) and A12 = f-1(y12,p), where y1, y2∈ f(R), p∈P.

Consider the set

A1 - A2 = {x∈S / (x,p) = (a1,p) - (a2,p)} for some (a1,p) ∈A1 and (a2,p) ∈A2 .

If (x,p) ∈ A1 – A2, then (x,p) = (x1,p) = (x2,p) for some (x1,p) ∈A1 and (x2,p) ∈A2,so

that we have

f(x,p) = f(x1,p) – f(x2,p) = y1 – y2

(ie) (x,p) ∈ f-1((y1,p) – (y2,p)) = f-1(y1-y2,p) = A12

Thus A1 – A2 A12.

It follows that

(i) Af ( y1-y2,p) = sup {A(x,p) / (x,p) ∈ f-1[(y1,p) – (y2,p)]}

= sup {A(x,p) / (x,p) ∈ A12}

≥ sup {A(x,p) / (x,p) ∈ A1 – A2}

≥ sup {A((x1,p) - (x2,p)) / (x1,p) ∈A1 and (x2,p) ∈A2 }

≥ sup {T{A(x1,p), A(x2,p)} / (x1,p) ∈A1 and (x2,p) ∈A2 }

Since T is continuous. For every ε > 0, we see that if

sup{{A(x1,p) / (x1,p) ∈ A1 } - (x1*,p)} ≤ δ and

sup{{A(x2,p) / (x2,p) ∈ A2 } - (x2*,p)} ≤ δ

T{sup{A(x1,p) / (x1,p) ∈ A1 }, sup{A(x2,p) / (x2,p) ∈ A2 }} - T((x1*,p), (x2*,p))≤ ε

Choose (a1,p) ∈ A1 and (a2,p) ∈ A2 such that

sup{{A(x1,p) / (x1,p) ∈ A1 } - A(a1,p)} ≤ δ and

sup{{A(x2,p) / (x2,p) ∈ A2 } - A(a2,p)} ≤ δ then we have

T{sup{A(x1,p) / (x1,p) ∈ A1 }, sup{A(x2,p) / (x2,p) ∈ A2}} -T((a1,p),(a2,p)) ≤ ε

Consequently, we have


Af ( y1-y2,p) ≥ sup {T{A(x1,p), A(x2,p)} / (x1,p) ∈A1 and (x2,p) ∈A2 }

≥ T{sup {A(x1,p) / (x1,p) ∈A1 }, sup{A(x2,p) / (x2,p) ∈A2 }}

≥ T{Af(y1,p), Af(y2,p)}

Similarly, we can show Af ( a α x,p) ≥ Af (x,p). Hence Af is P-fuzzy MΓ-subgroup of

G.

CONCLUSION

In this paper, we investigated the concept of P-fuzzy MΓ-soft subgroups with respect

to t-norm and characterization of them.

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