Intuitionistic fuzzy G-modules relative with a t-norm

University of Mazandaran, Iran http://cjms.journals.umz.ac.ir ISSN: 1735-0611

CJMS.7(2)(2018), 122-135

Intuitionistic fuzzy G-modules relative with a t-norm

Behnam Talaee 1and Maryam Alinia 2

1 _{Department of Mathematics, Faculty of Basic Sciences,}

Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran Post Code:47148-71167

2 _{Department of Mathematics, Faculty of Basic Sciences,}

Babol Noshirvani University of Technology, Shariati Ave., Babol, Iran

Abstract. In this paper we study about intuitionistic fuzzy G

-modules and some properties of them. The relationship betweent -norms and intuitionistic fuzzyG-modules will be investigated. Let

Gbe a group andM be aG−module overK, which is a subfield of C. Then anintuitionistic fuzzy G-module(IFG−module) onM is an intuitionistic fuzzy subsetA= (µA, νA) ofM such that following conditions are satisfied:

1)µ_A(ax+by)≥µ_A(x)∧µ_A(y)

ν_A(ax+by)≤ν_A(x)∨ν_A(y), for everya, b∈Kandx, y∈M; 2)µ_A(gm)≥ µ_A(m) and ν_A(gm) ≤ν_A(m), for everyg ∈ Gand

m∈M.

We investigate the nature of this kind of intuitionistic fuzzyG -modules under some algebraic operators.

Keywords: IFG-modules,t-norms,t-conorm,C-annihilation.

2000 Mathematics subject classification: 03B52, 15A78; Secondary 16D80.

1. Introduction

Fuzzy sets were defined by Zadeh [12] at first. Then many authors de-fined some algebraic fuzzy structures. Pushkav dede-fined t-norm based

1_{Corresponding author: behnamtalaee@nit.ac.ir}

Received: 30 December 2017 Revised: 25 December 2017 Accepted: 29 July 2018

fuzzy submodules [11]. Consequently, (α, β)−fuzzy submodule with re-spect to a t-norm was introduced by Rahman and Saikia in [6].

The notion of intuitionistic fuzzy sets introduced by Atanassov [1] is one among them. Algebraic structures play a vital role in mathe-matics and numerous applications of these structures are seen in many disciplines such as computer sciences, information sciences, theoretical physics, control engineering and so on. This inspires researchers to study and carry out research in various concepts of abstract algebra in fuzzy setting.

Biswas [2] applied the concept of intuitionistic fuzzy sets to the theory of groups and studied intuitionistic fuzzy subgroups of a group.

Fuzzy submodules of a module M over a ring R were first introduced by Naevoita and Ralescu [9]. Since then different types of fuzzy sub-modules were investigated in the last two decades. Many more results have been obtained by other researchers on intuitionistic fuzzy modules (see [4, 7, 10]).

In This paper, we introduce the notion of intuitionistic fuzzyG−modules with respect to a t−norm.

Throughout article R means a commutative ring with unity and M denotes a unitary left module.

∨and ∧denote respectively the maximum and minimum in the unit interval [0, 1].

2. Intuitionistic fuzzy sets and submodules

By R we mean a ring and M denotes a left R-module. The zero elements ofR and M are 0 and θ, respectively.

Definition 2.1. Let X be a set, a map µ:X →[0,1] is called a fuzzy subsetof X. The complement of µ, denoted byµc, is a fuzzy subset of X denoted byµc(x) = 1−µ(x) for everyx∈X.

Definition 2.2. Let G be a group. A fuzzy subset µ of G is called a

fuzzy subgroup ofGif the following conditions hold for all x, y∈G 1)µ(xy)≥µ(x)∧µ(y);

2)µ(x−1)≥µ(x).

Definition 2.3. LetR be a ring andI :R→[0,1]. Then I is called a f uzzy ideal ofR ifI satisfies the following:

1)I(x−y)≥min{I(x), I(y)};∀x, y∈R; 2)I(xy)≥max{I(x), I(y)};∀x, y∈R.

Definition 2.4. LetM be anR−module andµbe a fuzzy subset ofM. Then Ais called a f uzzy submoduleof M ifA satisfies the following:

1)µ(0) = 1;

2)µ(x+y)≥µ(x)∧µ(y);∀x, y∈M; 3)µ(rx)≥µ(x);∀x∈M, r∈R.

Definition 2.5. LetXbe a non-empty set. An intuitionistic f uzzy set (IF S) A ofX is an object of the formA={⟨x, µ_A(x), ν_A(x)⟩:x∈X}, where

µ_A :X→[0,1] and

ν_A :X →[0,1]

define the degree of membership and the degree of non-membership of the elementx∈X, respectively, and for anyx∈X, we have

µ_A(x) +ν_A(x)≤1 .

Definition 2.6. For any (IF S) A ={⟨x, µA(x), νA(x)⟩:x∈X} of X,

if

πA(x) = 1−µA(x)−νA(x)

for all x∈X. Then πA(x) is called the degree of indeterminacy of x in

A.

Definition 2.7. Let A={⟨x, µ_A(x), ν_A(x)⟩:x∈X} and B ={⟨x, µB(x), νB(x)⟩:x∈X} any twoIF S of X, then

1)A⊆B if and only ifµ_A(x)≤µ_B(x) andν_A(x)≥ν_B(x) for allx∈X; 2)A=B if and only ifµA(x) =µB(x) andνA(x) =νB(x) for allx∈X;

3) Ac = {⟨x,(µc

A),(νcA)(x)⟩ : x ∈ X} where (µcA)(x) = νA(x) and

(νc

A)(x) =µA(x) for allx∈X;

4) A∩B = {⟨x,(µA ∩µB)(x),(νA ∩νB)(x)⟩ : x ∈ X} where (µA ∩

µ_B)(x) = min{µ_A(x), µ_B(x)}=µ_A(x)∧µ_B(x) and (ν_A∩ν_B)(x) = max

{νA(x), νB(x)}=νA(x)∨νB(x);

5) A∪B = {⟨x,(µ_A ∪µ_B)(x),(ν_A ∪ν_B)(x)⟩ : x ∈ X} where (µ_A ∪ µB)(x) = max{µA(x), µB(x)} = µA(x) ∨µB(x) and (νA ∪ νB)(x) =

min{ν_A(x), ν_B(x)}=ν_A(x)∧ν_B(x)

Definition 2.8. LetG be a group. An IFSA= (µ_A(x), ν_A(x)) of Gis called anintuitionistic f uzzy subgroupofGif the following conditions hold for allx, y∈G

1)µA(xy)≥µA(x)∧µA(y);

2)ν_A(xy)≤ν_A(x)∨ν_A(y); 3)µA(x−

1_)≥µ

A(x)(consequently)µA(x−

1_{) =µ}

A(x);

Definition 2.9. Let R be a ring and A = (µ_A(x), ν_A(x)) an IFS of R. Then A is called an intuitionistic f uzzy ideal of R if A satisfies the following

1)µA(x−y)≥µA(x)∧µA(y), for everyx, y∈R; 2)µA(xy)≥µA(y), for every x, y∈R;

3)νA(x−y)≤νA(x)∨νA(y), for everyx, y∈R; 4)νA(xy)≤νA(y), for every x, y∈R.

Definition 2.10. Let M be an R-module and A = (µ_A(x), ν_A(x)) an IFS ofM. ThenA is called anintuitionistic f uzzy submodule(IF M) ofM if A satisfies the following

1)µ_A(0) = 1, ν_A(0) = 0

2)µA(x+y)≥µA(x)∧µA(y), for all x, y∈M;

ν_A(x+y)≤ν_A(x)∨nu_A(y), for allx, y∈M; 3)µA(rx)≥µA(x), for all x, y∈M and r∈R;

ν_A(rx)≤ν_A(x), for allx, y∈M and r ∈R.

Definition 2.11. LetA={⟨x, µA(x), νA(x)⟩:x∈X}and

B ={⟨x, µ_B(x), ν_B(x)⟩:x∈X} any twoIF M‘sof X, then (A+B) = (µA+B(x), νA+B(x)), where

µA+B(x) =

∨

{µA(y)∧µB(z)|x=y+z;x, y, z∈M} νA+B(x) =

∧

{νA(y)∨νB(z)|x=y+z;x, y, z∈M}

Definition 2.12. LetA= (µA, νA) be an IF S of X. Define

A={(x, µA(x), µcA(x))|x∈X}

and

♢A={(x, ν_Ac(x), νA(x))|x∈X)}.

Clearly A and ♢A are IF S‘s of X. For an ordinary fuzzy set A =

{(x, µ(x), µc(x))|x∈X},

A=A=♢A.

If A = {(x, µA(x), νA(x))|x ∈ X} is an IFS such that 0 ≤ µA(x) + νA(x)<1 for some x∈X, then

A⊂A⊂ ♢A.

However IfA ={(x, µA(x), νA(x))|x∈X} is an IFS such thatµA(x) + νA(x) = 1 for allx∈X.

3. Fuzzy and intuitionistic fuzzy G− modules

Definition 3.1. LetGbe a group and M be a vector space over a field K. ThenM is called aG−moduleif for everyg∈Gandm∈M, there exists a product (called the action of G on M), g.m ∈ M satisfies the following axioms

1) 1G.m=m,∀m∈M (1Gbeing the identity of G); 2) (gh).m=g.(h.m),∀m∈M,g, h∈G;

3) g.(k1m1 +k2m2) = k1(g.m1) +k2(g.m2),∀k1, k2 ∈K; ∀m1, m2 ∈ M

and ∀g∈G.

Definition 3.2. Let G be a group andK be a subset of M. Then K is called a G-submodule of M if K is a submodule of M and also a G-module.

Definition 3.3. LetGbe a group andMbe aG−module overK, which is a subfield ofC. Then an intuitionistic fuzzyG-module (IFG−module)on M is an intuitionistic fuzzy set A = (µA, νA) of M such that following

conditions are satisfied

1)µ_A(ax+by)≥µ_A(x)∧µ_A(y)

νA(ax+by)≤νA(x)∨νA(y), for everya, b∈K and x, y∈M;

2)µ_A(gm)≥µ_A(m) andν_A(gm)≤ν_A(m), for everyg∈Gandm∈M.

Example 3.4. Let G = {−1,1}, M = Rn over R. Then M is a G-module. Define the intuitionistic fuzzy setA= (µA, νA) on M by

µA(x) =

{

1, if x= 0; 0.5 , if x̸= 0. and

νA(x) =

{

0, if x= 0; 0.25, if x̸= 0.

Where x = (x1, x2, ..., xn) ∈ Rn. Then A is an intuitionistic fuzzy G-module on M.

Example 3.5. Consider the G-moduleM =R(i) =C over the fieldR whereG={−1,1}. Define the intuitionistic fuzzy setA= (µA, νA) by

µA(z) =

  

1, z= 0;

0.5, z∈R− {0}; 0.25, z∈R(i)−R. and

νA(z) =

  

0, z= 0;

Then A is an intuitionistic fuzzyG-module on M.

Remark 3.6. In Example (3.4), if we take the group G={1,−1, i,−i}, then A is not IF G-module on M. Because the candition µA(gm) ≥ µA(m),

is not an satisfied, e. g. , takem= 3 and g=i, then

µA(3i) = 0.250.5 =µA(3) also

νA(3i) = 0.50.25 =νA(3).

4. t-norms and t-conorms

Definition 4.1. By a triangular norm or (t−norm) we mean a mapping T : [0,1]×[0,1]→[0,1],

which satisfies the following axioms for everya, b, c∈[0,1] (T1)T(a,1) =a (boundary condition);

(T2)b≤cimplies T(a, b)≤T(a, c) (monotonicity); (T3)T(a, b) =T(b, a) (commutativity);

(T4)T(a, T(b, c)) =T(T(a, b), c) (associativity).

Definition 4.2. A fuzzy union or (t−conorm) S is a mapping S: [0,1]×[0,1]→[0,1];

which satisfies the following axioms for every a, b, c∈[0,1] (S1)S(a,0) =a(boundary condition);

(S2)b≤c impliesS(a, b)≤S(a, c) (monotonicity); (S3)S(a, b) =S(b, a) (commutativity);

(S4)S(a, S(b, c)) =S(S(a, b), c) (associativity).

Definition 4.3. LetM be a module over a ringRandµa fuzzy subset of M. Then µ is called a f uzzy submodule of M with respect to a t-norm T if for everyx, y∈M and r∈Rthe following conditions hold: 1)µ(θ) = 1;

2)µ(x+y)≥T(µ(x), µ(y)); 3)µ(rx)≥µ(x).

If T is the standard t-norm min, then µis called a fuzzy submodule ofM.

At-normTand at-conormST are called dual with respect to standard fuzzy complement if

(D1) 1−T(a, b) =ST(1−a,1−b), (D2) 1−ST(a, b) =T(1−a,1−b), for everya, b∈[0,1].

Definition 4.4. Let M be a module over a ring R, A = (µA, νA) be

is said to be an intuitionistic fuzzy submodule of Mwith respect to the

t−norm T if and only if for every x, y ∈ M and r ∈ R; the following axioms hold:

(M1) µA(θ) = 1;

(M2) µA(x+y)≥T(µA(x), µA(y));

(M3) µA(rx)≥µA(x);

(M4) νA(x+y)≤ST(νA(x), νA(y));

(M5) νA(rx)≤νA(x).

Since 0≤µ_A(x) +ν_A(x) ≤1 for all x ∈M and µ_A(θ) = 1, we must have νA(θ) = 0.

If T andST are the standard t−normmin and t-conorm max, then everyIF S Asatisfied to above conditions is an intuitionistic fuzzy sub-module ofM.

Example 4.5. Consider M = Z8 over Z. Define µA, νA ∈ [0,1]Mas follows:

µ_A(x) =

  

1, x∈ {0,4}; 0.7, x∈ {2,6}; 0, otherwise.

ν_A(x) =

  

0, x∈ {0,4}; 0.2, x∈ {2,6}; 1, otherwise.

Then A = (µA, νA) is an IF fuzzy submodule of M with respect to the following pairs of t-norms and t-conorms:

1) min(a, b),max(a, b); 2)ab, a+b−ab;

3) max(0, a+b−1),min(1, a+b).

Definition 4.6. [4] The nilpotent minimum t-norm Tmo is defined for everya, b∈[0,1] as follows:

Tmo(a, b) =

{

0, b≤1−a; min(a, b), otherwise.

}

.

Definition 4.7. The C-annihilation of T is denoted by T_(c) and it is defined as follows:

T_(c)(a, b) : [0,1]×[0,1]→[0,1]

T(c)(a, b) = {

Theorem 4.8. For every continuous t-norm T,T(c) is a t-norm if only if T_(c) is isomorphic either to T_(mo)

or to TL(a, b) = max(0, a+b−1)(Lukasiewics t−norm) or to

TJ(a, b) =

  

0, a≤1−b;

1

3 +a+b−1, a, b∈[ 1 3,

2

3]and a >1−b;

min(a, b) otherwise.

for everya, b∈[0,1].

Proof. It follows immediacy from [7].

Theorem 4.9. LetRbe a field,M be a module overRandA= (µA, νA)

be an IFS of M. Then for every 0 ̸= α ∈ R, µA(αx) = µA(x) and νA(αx) =νA(x).

Proof. Let α̸= 0∈R. ThenνA(αx)≤νA(x) =νA(α−1(αx) ≤νA(αx). Therefore, νA(αx) =νA(x). Similarly,µA(αx) =µA(x).

5. Intuitionistic fuzzy G− modules with respect to a t−norm Definition 5.1. LetM be a G−module overK, which is a subfield of

C. Let

A={(x, µA(x), νA(x));x∈M}

be an (IF S) ofM,T be at−norm andST be its dual t−conorm. Then A is said to be an intuitionistic fuzzy G−submodule of M with respect to the t−norm T if for every x, y ∈ M and a, b ∈ K and g ∈ G, the following axioms hold:

(M1) µA(θ) = 1,νA(θ) = 0

(M2) µA(ax+by)≥T(µA(ax), µA(by));

(M4) νA(ax+by)≤ST(νA(ax), νA(by));

(M5) µA(gm)≥µA(m);

(M6) νA(gm)≤νA(m).

By above definition we have the following corollary.

Corollary 5.2. A fuzzy subset µ of M is a fuzzy G−submodule of

M with respect to a t−norm T if and only if A = (µ, µc) is an IF G−submodule of M with respect to thet−norm T.

Theorem 5.3. Let A be a non-empty subset of M. Then an A¯ = (µA, νA) is defined by

µA(x) =

{

1 x∈A;

α otherwise.

νA(x) =

{

0 x∈A;

where 0≤α ≤1, 0≤β ≤1 and α+β ≤1 is an IF G− submodule of

M with respect to a t− norm T if and only if A is a G− submodule of

M.

Proof. Let Abe a G−submodule ofM. Thenθ∈Aand so, µ_A(θ) = 1. Letx, y∈M.

Case(1):

ifx, y∈A, thenax+by∈A and soµ_A(ax+by) = 1≥1≥T(1,1) = T(µA(ax), µA(by)).

Also ν_A(ax+by) = 0≤0 =S_T(0,0) =S_T(ν_A(ax), ν_A(by)) Case(2):

ifx∈A, y /∈A, then

µ_A(ax+by) =α≥α=T(α,1) =T(µ_A(ax), µ_A(by))and νA(ax+by) =β≤β=ST(0, β) =ST(νA(ax), νA(by))

Case(3):

ifx /∈A, y /∈A, µA(ax+by)≥α=T(1, α)≥T(α, α) =T(µA(ax), µA(by))

and ν_A(ax+by)≤β=S_T(β, β) =S_T(ν_A(ax), ν_A(by)).

Thus for all casesµ_A(ax+by)≥T(µ_A, µ_A) and soµ_A(rx) = 1≥1 = µ_A(x) and

ν_A(rx) = 0 ≤ 0 = ν_A(x). If x /∈ A, then µ_A(x) ≥ α = µ_A(x) and νA(rx)≤β =νA(x).

Therefore, A = (µ_A, ν_A) is an IF G−submodule of M with respect

to at−norm T. The converse is obvious.

Corollary 5.4. Let A be a non-empty subset of a G−moduleM. Then

(χ_A, χc_A) is an IF G−submodule of M with respect to a t−norm T if and only if A is a G−submodule ofM.

Proof. It follows immediately from Theorem 5.3.

Theorem 5.5. Let R be a field, M be a G-module over R and A = (µA, νA) be an IFS of M. If there exists x∈M such that,A(x) = (1,0),

thenµA(θ) = 1(νA(θ) = 0).

Proof. µA(θ) =µA(ax−ax)≥T(µA(ax), µA(−ax)) =T(1, µA(−ax)) = µA(−ax)≥µA(x) = 1. Therefore, µA(θ) = 1.

Theorem 5.6. An IF subset A = (µA, νA) of a G-module M is an IF G-submodule ofM with respect to a t-norm T if and only ifAand♢A

both are IFG-submodules ofM with respect to the t-norm T.

obtained from our assumption thatA= (µA, νA) is anIF G−submodule ofM with respect to a t-normT. Therefore,Ais anIF G−submodule of M with respect to a t-norm T.Also,νA(θ) = 0 implies that ν_Ac(θ) = 1 Now, ν_Ac(ax+by) = 1−νA(ax+by) ≥ 1−ST(νA(ax), νA(by)) = T(1−νA(ax),1−νA(by)). Therefore,ν_Ac(ax+by)≥T(ν_Ac(ax), ν_Ac(by)). Moreover, ν_Ac(gx) = 1−νA(gx) ≥ 1−νA(x) = νAc(x). The other two conditions follow from our assumption thatA= (µA, νA) is anIF G− submodule of M with respect to a t-norm T. Therefore, ♢A is an IF G−submoduleof M with respect to a t-norm T. The converse part of

the theorem is obvious.

Theorem 5.7. LetA=A1, A2, ..., An, whereAi = (µi, νi),i= 1,2, ..., n

be IF G−submodules of M with respect to a t−norm T. Then A1 ∩

A2∩...∩An is also anIF G−submodule ofM with respect to thet−norm

T.

Proof. Let A =A1 ∩A2 ∩...∩An. To show that A is an intuitionistic

fuzzy G−submodule of M with respect to the t−norm T, we will use induction on n.

Ifn= 1, thenA =A1, and so, Ais an IF G−submodule ofM with

respect to thet−normT.

We assume that the intersection of n−1 (or less) IF G−submodule of M with respect to the t−norm T is IF G−submodule of M with respect to thet−normT. By the induction hypothesis,

A2∩A3∩...∩Anis anIF G−module ofM with respect to thet−norm

T.

Therefore,

(µ₂ ∩µ₃...∩µn(gθ)) =Tn−1(µ2(θ), µ3(θ)...µn(θ) = 1.

Further

(µ₁∩µ₂∩...∩µn)(gθ) =Tn(µ1(gθ), ...µn(gθ))

=T(µ1(gθ)), Tn−1(µ2(gθ), ...µn(gθ) =T(1,1) = 1 (since µ1(gθ) = 1).

Then,

(µ1 ∩...∩µn)(gθ) =Tn(µ1(gθ), ..., µn(gθ)) = 1.

Letx, y∈M and r ∈R, g∈G. Then

(µ1 ∩...∩µn)(ax+by) =Tn(µ1(ax+by), ..., µn(ax+by)) =

T(µ1(ax+by), Tn−1(µ2)(ax+by), ..., µn(ax+by)))≥

=T(T(µ1)(by), µ1(ax)), T(Tn−1(µ2(ax), ..., µn(ax)), Tn−1(µ2(by), ..., µn(by)))) =

T(µ1(by), T(T(µ1(ax)), Tn−1(µ2(ax), ..., µn(ax))), Tn−1(µ2(by), ..., µn(by))))

=T(µ1(by), T(Tn(µ1(ax), ..., µn(ax)), Tn−1(µ2(by), ..., µn(by))))

=T(µ1(by), T(Tn−1(µ2(by), ..., µn(by)), Tn(µ1(ax), ..., µn(ax))))

=T(T(µ₁(by), T_n−1(µ₂(by), ..., µn(by))), Tn(µ1(ax), ..., µn(ax))) =

T(Tn(µ1(by), ..., µn(by)), Tn(µ1(ax), ..., µn(ax))) =

T(Tn(µ1(ax), ..., µn(ax)), Tn(µ1(by), ..., µn(by))) =T((µ1∩...∩µn)(ax),(µ1∩

...∩µn)(by))

Thus,

(µ₁∩...∩µn)(ax+by)≥T((µ1∩...∩µn)(ax),(µ1∩...∩µn)(by)).

LetST be the dualt−conorm of T. Then

(ν1 ∪...∪νn)(ax+by) =ST(ν1(ax+by), ..., νn(ax+by)) =

S_T(ν₁(ax+by), S_T

n−1(ν2(ax+by), ..., νn(ax+by)))≤

S_T(S_T(ν1(ax), ν1(by), ST(S_Tn−1(ν2(ax), ..., νn(ax)), ST_n−1(ν2(by), ..., νn(by))))

(sinceA2 ∩...∩An is anIF G−submodule ofM)

=S_T(S_T(ν1(by), ν1(ax)), ST(ST_n−1(ν2(ax), ..., νn(ax)), ST_n−1(ν2, ..., νn(by)))) =

ST(ν1(by), ST(ST(ν1(ax), ST

n−1(ν2(ax), ..., νn(ax))), STn−1(ν2(by), ..., νn(by)))).

Then,(ν1∪...∪νn)(ax+by)≤ST((ν1∪...∪νn)(ax),(ν1∪...∪νn)(by)

(µ1∩...∩µn)(gx) =Tn(µ1(gx), ..., µn(gx))

=T₍µ₁(gx), T_n−1(µ₂(gx), ..., µn(gx)) ≥T(µ1(x), Tn−1(µ2(x), ..., µn(x))

(ν1∪...∪νn)(gx) =S_Tn(ν1(gx), ..., νn(gx)) =

S_T(ν₁(gx), S_T

n−1(ν2(gx), ..., µn(gx))

≤S_T(ν1(x), ST_n−1(ν2(x), ..., νn(x))(ν1 ∪...∪νn)(x).

Hence,A =A1∩....∩An is IF G−submodule ofM with respect to

thet−normT.

Theorem 5.8. Suppose that the C-annihilationT(c) of the t−normT provides a t−norm. IfA = (µA;νA) is an IF G-submodule of M with

respect to the t-norm T, then A and ♢A are IF G-submodules of M

with respect to the t-norm T(c).

Proof. Let T(c) be the C-annihilation of T. Then the dual ST(c) of T(c)

is given by

ST(c)(a, b) =

{

1, 1−a≤b;

1−T(1−a,1−b), otherwise

for every a, b∈[0,1]. Since A= (µA, νA) is an IFG−submodule of M with respect to the t-norm T.

For everyx, y∈M andr ∈R we have (M1) µA(θ) = 1;

(M2) µA(ax+by)≥T(µA(ax), µA(by))≥T(c)(µA(ax), µA(by)); (M3) µA(gx)≥µA(x);

(M4) Now we get

ST(c)(µ

c

A(ax), µcA(by)) =

{

1, µA(ax)≤1−µA(by); 1−T(1−µc_A(ax),1−µc_A(by)), otherwise. ST(c)(µ

c

A(ax), µcA(by))≥1−T(1−µcA(ax),1−µcA(by)) = 1−T(µA(ax), µA(by))≥1−µA(ax+by)

(sinceµ(ax+by)≥T(µA(ax), µA(by))) =µc_A(ax+by).

Then, we have,µc_A(ax+by)≤ST(c)(µ

c

A(ax), µcA(by)). (M5)

µc_A(gx) = 1−µA(gx)≤1−µA(x) =µcA(x).

Hence,Ais an IF G-submodule of M with respect to T(c).

Next we claim that♢Ais an IF G-submodule of M. (M1) Since λA(θ) = 0, λcA(θ) = 1.

(M2) Now

T_(c)(λc_A(ax), λc_A(by)) =

{

T(c)(λcA(ax), λcA(by)) ≤ T(λcA(ax), λcA(by)) = 1−ST(λA(ax), λA(by)) (by duality)

≤ 1−λA(ax+by), (since λA(ax +by) ≤ ST(λA(ax), λA(by))) = λc_A(ax+by).

Thus, we have, λc_A(ax+by)≥T(c)(λc

A(ax),λcA(by).

(M3) λcA(gx) = 1−λA(gx)≥1−λA(x) =λcA(x). (M4)

λA(ax+by)≤ST(λA(ax), λA(by)) = 1−T(1−λA(ax),1−λA(by))(by duality) ≤ST(c)(λA(ax), λA(by)).

Thus, we have λA(ax+by)≤ST(c)(λA(ax), λA(by)).

(M5) λA(gx)≤λA(x).

Hence,♢Ais an IF G-submodule of M with respect toT(c).

Corollary 5.9. Let A = (µA, λA) be an IF G-submodule of M with

respect to the t-norm minimum.

ThenA,♢A and A are IFG-submodules ofM with respect to the t-norm Tmo (the nilpotent minimum t-norm)

Proof. Follows from Theorem (3.8) and (4.9).

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