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(1)

Homomorphism on

T

– Fuzzy Ideal of

ℓ-

Near-Rings

1

G. Chandrasekaran,

2

B.Chellappa and

3

M. Jeyakumar

1

Assistant Professor, Dept. of Mathematics, Sethupathi Govt Arts College, Ramanathapuram – 623 502, Tamilnadu, India 2Principal, Nachiappa Suvamical Arts and Science College, Karaikudi - 630 003, Tamilnadu, India

3

Assistant Professor, Dept. of Mathematics, Alagappa University Evening College,Rameswaram – 623 526, Tamilnadu, India

Abstract: In this paper, we made an attempt to study the properties of T-fuzzy ideal of a ℓ-near-ring and we introduce some theorems an onto homomorphic image, an epimorphic pre-image of a T-fuzzy ideal of a ℓ-near-ring.

Keywords: Fuzzy subset, T-fuzzy ideal,

homomorphism of T - fuzzy ideal, homomorphic image of T - fuzzy ideal, homomorphic pre-image T-fuzzy ideal and product of T- T-fuzzy ideal.

I. INTRODUCTION

The concept of fuzzy sets was initiated by Zadeh. L.A, [18] in 1965. After the introduction of fuzzy sets several researchers explored on the generalization of the concept of fuzzy sets. In Ayyappan,M., and Natarajan,R., [3] have introduced Lattice ordered near rings. In Dudek. W.A., and Jun. Y.B., [9] introduced Fuzzy subquasigroups over a t-norm. In 1971, Liu. W., [10] studied fuzzy ideals in rings. In Satyanarayana, Bh., and Syam Prasad, K

.

[12] introduced Gamma near-rings. In Srinivas, T., Nagaiah, T., and Narasimha Swamy, P., [14] studied anti fuzzy ideals of  near-rings. Dheena. P., and Mohanraaj. G., [8] have studied several properties of

T fuzzy ideals of rings and Akram, M., [2] have studied some results of Tfuzzy ideals of near-rings. We extended the results of Tfuzzy ideals of a near-rings.

In this paper we define, homomorphism and study the near-ring homomorphism. Wang, Z.D., [15] introduced the basic concepts of TL-ideals and Fuzzy invariant subgroups and fuzzy ideals. We introduced homomorphism in Tfuzzy ideals of

 near- ring. We discuss some of its properties. We have shown that homomorphism, homomorphic image of Tfuzzy ideal, homomorphic pre-image

T fuzzy ideal and product of Tfuzzy ideal of

 near-ring

II. DEFINITIONSAND EXAMPLES

Definition: 1

A mapping T: 0, 1

     

 0, 1  0, 1 is called a

triangular norm [tnorm] if and only if it satisfies the following conditions:

(i). T x

 

, 1 T

 

1, xx, for all x

 

0, 1

(ii). if x ,x* yy* thenT x y

,

T

x*, y*

(iii). T x y

,

T y x

,

, for all , yx

 

0, 1 .

(iv). T x T

,

 

y, z

T T

x, y , z .

Definition: 2

A fuzzy subset  of a ring R is called Tfuzzy right (resp. left) ideal if

(i) 

xy

T

   

x , y

min

   

x , y

(ii) 

 

xy

 

x

(resp. left 

 

xy

 

y

), for all

x y

,

in

R

Definition: 3

A fuzzy subset  of a near-ring R is called a

T fuzzy ideal, if the following conditions are satisfied,

(i) 

xy

T

   

x , y

, for all ,x yR

(ii) 

y x y

 

x

, for all x y, in R

(iii) 

 

xy 

 

y ;

 

xy 

 

x , for all ,x yR (iv) 

xz y

x y

 

z

, for all , ,x y z R

(v) 

xy

T

   

x , y

, for all ,x yR

(vi) 

xy

T

   

x , y

, for all ,x yR

Example: 1

Now

R

a b c, ,

,   , , ,

is a  near-ring. Consider the fuzzy subset

of the near-ring R

 

0.60.5 0.7

 

 

if x a

x if x b

if x c

(2)

Definition: 4

Let R1 and R2 be two near rings. Then the function f :R1R2 is called a  near ring homomorphism if satisfies the following conditions (i) f x

y

f x

 

f y

 

(ii) f xy

 

f x f y

   

(iii) f x

y

f x

 

f y

 

(iv) f x

y

f x

 

f y

 

, for all x y, in R

Example: 2

LetR

m n 2,for all m n Z, 

, R is a ring under usual addition and multiplication. Define

: 

f R R by f m n

 2

 m n 2 is near ring homomorphism, where Z is set of all integer

Definition: 5

A fuzzy set  of a  near ring R has the

supremum property if for any subset N ofR, there exists a a0N such that 

 

0 sup

 

a N

a a

Definition: 6

Let , be two fuzzy ideals of a near ring R, then the sum   is a fuzzy set of R defined by

 

 

x

sup min

   

y , z

,ifx y z, 0, otherwise, for all , ,x y zR.

Definition: 7

Let R1 and R2 be two near rings. A mapping

1 2

: 

f R R is called a near ring isomorphism if (i) f is one-to-one

(ii) f is onto, for all x y, in R

Definition: 8

Let M and N be any two sets and let f :MN

be any function. A fuzzy subset

of a M is called f-invariant if f x

   

f y  

   

x  y , for all ,x yM

Definition: 9

Let R be a near ring. Let  be a fuzzy set of a

T fuzzy ideals of a near ring R and f be a function defined on R , then the fuzzy set f in

 

f R is defined by

 

 

 

1

sup

 

 

f

n f y

y x , for all

 

y f R and is called the image of  under f

Definition: 10

Let R be a near ring. Let  be a fuzzy set of a

T fuzzy ideals of a near ring R and f be a function defined on R, if v is a fuzzy set in f R

 

, then  vf in R is defined by 

 

xv f x

 

, for allx R and is called the pre-image of vunder f .

III.THEOREMS

Theorem: 1

Every fuzzy ideal of a  near ring R is a

T fuzzy ideal of a near ringR.

Theorem: 2

An onto homomorphism image of a Tfuzzy ideal of a  near ring R with sup property is a

T fuzzy ideal of a near ring R.

Proof:

LetR and S be a  near rings

Let f R: S be an epimorphism and A be a

S fuzzy ideal of a  near ring R with sup

property. Let ,x yS

Let 1

 

0

x f x , 1

 

0

y f y and 1

 

0

z f z be such that

 

 

 

1

0 sup

 

 

n f x

x n ,

 

 

 

1

0 sup

 

 

n f y

y n and

 

 

 

1

0 sup

 

 

 

n f z

z n respectively, then

(i)

 

 

1

sup

 

  

 

f

z f x y

x y z

0 0

xy

   

0 0

min  ,

x y

   

 0 , 0

T x y

 

 

 

 

1 1

sup  , sup 

   

 

 

 

n f x n f y

T n n

 

 

 ,

f f

T x y

Therefore f

xy

T

f

 

x ,f

 

y

, for all , 

x y S

(ii) Since 

y x y



 

x

 

 

1

sup

 

   

  

f

z f y x y

y x y z

0 0 0

yxy

 

0 

x

 

 

1

sup 



n f x

(3)

 

f

x

Therefore f

y x y

f

 

x , for all ,x yS

(iii) Let be a Tfuzzy ideals of Rand let ,x yR

Since 

 

xy 

 

x and 

 

xy 

 

y

we have

 

 

 

1

sup

 

 

f

z f xy

xy z

0 0

x y

 

0 

x

 

 

1

sup 

  

n f x

n

 

f

x

Therefore f

 

xy f

 

x , for all ,x yS (iv) Since 

xz y

x y



 

z

 

 

 

1

sup

 

   

  

f

z f x z y x y

x z y xy z

0 0 0 0 0

xz yx y

 

0

z

 

 

1

sup 



n f z

n

 

f

z

Therefore f

xz y

xy

f

 

z , for all , 

x y S

(v)

 

 

1

sup

 

  

 

f

z f x y

x y z

0 0

xy

   

0 0

min  , 

x y

   

 0 ,  0

T x y

 

 

 

 

1 1

sup  , sup 

   

 

 

 

n f x n f y

T n n

 

 

 , 

f f

T x y

Therefore f

xy

T

f

 

x ,f

 

y

, for all , 

x y S

(vi)

 

 

1

sup

 

  

 

f

z f x y

x y z

0 0

xy min

   

x0 , y0

   

 0 , 0

T x y

 

 

 

 

1 1

sup  , sup 

   

 

 

 

n f x n f y

T n n

 

 

 ,

f f

T x y

Therefore f

xy

T

f

 

x ,f

 

y

, for all , 

x y S

Thus an onto homomorphic image of a Tfuzzy ideal of a near ring R with sup property is a

T fuzzy ideal of a near-ring R .

Theorem: 3

An epimorphic pre-image of a Tfuzzy ideal of a

 near ring is a Tfuzzy ideal of a near ring

R.

Proof:

Let R and S be a near rings. Let f R: Sbe an epimorphism. Let v be a Tfuzzy ideal of a

 near ring S and  be the pre-image of v under

f for any , ,x y zR.

(i) we have   

x y

 

vf



xy

v f xy

 

 

v f xf y

 

 

,

T v f x v f y

   

  

,

T vf x vf y

   

 ,

T x y

Therefore 

xy

T

   

x , y

, for all , 

x y R

(ii) Since 

y x y

 

 

x

 



   y x y vf y x y

v f y x y

 

v f x

 

T v f x

 

T vf x

 

vf x

 

x

Therefore 

y x y



 

x , for all ,x yS

(iii) Since 

 

xy 

 

x

  

 

xyvf xy

 

v f xy

   

v f x f y

 

(4)

 

T vf x

 

vf x

 

x

Therefore 

 

xy 

 

x , for all ,x yR (iv) Since 

xz y

x y



 

z

 

xz yx yvf xz yx y

v f xz yx yv f y z

 

   

,

T v f y f zv f

 

z

  

T vf z

 

vf z

 

z

Therefore 

xz y

xy



 

z , for all ,x yS

(v) we have   

x y

 

v f



xy

v f xy

 

 

v f xf y

 

 

,

T v f x v f y

  

 

,

T v fx v fy

   

 ,

T x y

Therefore 

xy

T

   

x , y

, for all , 

x y R

(vi) we have  

x y

 

v f



xy

v f xy

 

 

v f xf y

 

 

,

T v f x v f y

   

  

,

T v fx v fy

   

 ,

T x y

Therefore 

xy

T

   

x , y

, for all , 

x y R

Thus an epimorphic pre-image of a Tfuzzy ideal of a near ring is a Tfuzzy ideal of a  near-ring R.

Proposition: 1

Let Rand S be near rings R and let f R: S

be a homomorphism. Let  be f invariant fuzzy ideal of a  near-ring R . If xf a

 

, then

  



 

f a x a , for all aR.

Theorem: 4

Let f R: S be an epimorphism of near rings

R andS . If  is f invariant fuzzy ideal of a

 near-ring R, then f

 

 is a Tfuzzy ideal of

S.

Proof:

Let a b c S, ,  then there exists x y z, , R such that f x

 

a, f

 

yband f z

 

c

Suppose  is f  invariant fuzzy ideal of a

 near-ring R, then by Proposition 1

(i) we have f

 

a b

f

   

f xf y

 

  

f f xy

xy

   

 ,

T x y

     

 , 

T f a f b

Therefore f

 

a b

T f

     

a ,fb

, for all a b, Sand ,x yR

(ii) Since 

y x y



 

x

We have

 

  

   

 

 

f b a b f f y f x f y

  

f f y x y

y x y

 

x

  

f a

Therefore f

 

b a b 

f

  

a , for all , 

a b Sand ,x yR

(iii) Since 

 

xy 

 

x and 

 

xy 

 

y

We have f

  

a bf

     

f x f y

   

f f xy

 

xy

 

x

  

f a

Therefore f

  

a bf

  

a , for all a b, S

and ,x yR

(iv) Since 

xz y

xy



 

z

  

f a c b ab

 

 

 

 

   

f f xf z f yf x f y

 

(5)

xz yxy

 

zf

  

c

Therefore f

  

a c b ab

f

  

c , for all , 

a b Sand ,x yR

(v) we have f

 

ab

f

   

f xf y

 

  

f f xy

xy

   

 ,

T x y

     

 , 

T f a f b

Therefore f

 

ab

T f

     

a , fb

, for all a b, Sand ,x yR

(vi) we have f

 

ab

f

   

f xf y

 

  

f f xy

xy

   

 ,

T x y

     

 , 

T f a f b

Therefore f

 

ab

T f

     

a , fb

, for all a b, Sand ,x yR

Hence, f

 

 is a Tfuzzy ideal of a near ring

S.

Theorem: 5

Let f R: 1R2 be an onto homomorphism of a

 near rings. If  is Tfuzzy ideal of R1, then

 

f is a Tfuzzy ideal of R2.

Proof:

Let  be a Tfuzzy ideal of a near ring R1

Let 1f1

 

y1 and 2f1

 

y2 , where

1, 2 2

y y R are non-empty subsets of R2

Similarly, 3f1

y1y2

Consider the set  12

a1a2/a11,a2

If x12, then xx1x2, for some x11,

22

x and so,

 

1 2

f x f x xf x

   

1 f x2 y1y2

1

1 2 3

  

x f y y

Thus123that is

x x/ f 1

y1y2

1

 

1

 

1 2/ 1 1 , 2 2

 

xx xf y xf y

Let y3R2, then (i) we have

 

 

1

1 2 sup / 1 2

f y y x x f y y

 

 

1 1

1 2 1 1 2 2

sup  /  , 

xx xf y xf y

   

 

 

1 1

1 2 1 1 2 2

sup min  , /  , 

x x xf y xf y

   

 

 

1 1

1 2 1 1 2 2

sup  , /  , 

T x x xf y xf y

 

 

 

 

1 1

1 1 1 2 2 2

sup  /  ,sup  / 

T x xf y x xf y

 

 

 

 

 1 ,  2

T f y f y

Therefore f

 

y1y2

T f

     

y1 ,fy2

, for all y1,y2R2

(ii) Since 

y x y



 

x

We have

 

 

1

2 1 2 sup / 2 1 2

   

f y y y x x f y y y

 

 

1 1

2 1 2 1 1 2 2

sup  /  , 

x  x x xf y xf y

 

 

1

1 1 1

sup  / 

x xf y

 

 

1

f y

Therefore f

 

y2y1y2

f

 

 

y1 , for all

1, 2 2

y y R

(iii) We have 

 

xy 

 

x and 

 

xy 

 

y

We have

 

 

1

1 2 sup / 1 2

f x x y y f x x

 

 

1 1

1 2 1 1 2 2

sup  /  , 

x x yf x yf x

sup

 

x2 / y2f 1

 

x2

 

 

2

f x

Therefore f

 

x x1 2

f

 

 

x2 , for all

1, 2 2

y y R

(iv) Since 

xz y

xy



 

z

We have f

  

a c b ab

 

 

 

 

   

f f xf z f yf x f y

We have f

 

y1y3

y2y y1 2

 

1

1 3 2 1 2

sup  / 

x xf yy yy y

1 3 2 1 2

sup  /

yy yy y

 

 

 

1 1 1

1 1 , 2 2 , 3 3

  

  

x f y x f y x f y

 

 

1

3 3 3

sup  / 

(6)

 

 

3

f y

Therefore f

 

y1y3

y2y y1 2

f

 

 

y3 ,

for all y y1, 2,y3R2

(v) f

 

y1y2

sup

 

x /xf1

y1y2

 

 

1 1

1 2 1 1 2 2

sup  /  , 

xx xf y xf y

   

1 2

sup min  , /

x x

 

 

1 1

1 1 , 2 2

 

 

x f y x f y

   

1 2

sup  , /

T x x

 

 

1 1

1 1 , 2 2

 

 

x f y x f y

 

 

1

1 1 1

sup  /  ,

T x xf y

 

1

 

2 2 2

sup  x / xfy

 

 

 

 

 1 ,  2

T f y f y

Therefore

 

1 2

 

 

1 ,

 

 

2

f y y T f y f y , for all

1, 2 2

y y R

(vi)f

 

y1y2

sup

 

x /xf1

y1y2

 

 

1 1

1 2 1 1 2 2

sup  /  , 

xx xf y xf y

   

1

 

1

 

1 2 1 1 2 2

sup min  , /  , 

x x xf y xf y

   

 

 

1 1

1 2 1 1 2 2

sup  , /  , 

T x x xf y xf y

 

 

 

 

1 1

1 1 1 2 2 2

sup  /  ,sup  / 

T x xf y x xf y

 

 

 

 

 1 ,  2

T f y f y

Therefore

 

1 2

 

 

1 ,

 

 

2

f y y T f y f y , for all

1, 2 2

y y R

Hence f

 

 is aTfuzzy ideal of anear ring

2

R .

Theorem: 6

Let  and  be Tfuzzy ideal of a near ring

R. Then   is the smallest Tfuzzy ideal of R

containing both  and .

Proof:

Let  and  be Tfuzzy ideal of a near ring

Rand Let , ,x y zR

Then

(

x- y

)

=

(

a+b

) (

- c- d

)

(

a b

)

c d

= + -

-(

b a b

)

c

(

c b c

)

d

= + - - + + -

-e f

= + ,

where e=

(

b+a- b

)

- c f =

(

c+b- c

)

- d

(i)

 



 

 

  

  

 

x y e f

x y e f

  

  

,

, ,

   

   

 

    

x a b y c d

T b a b c T c b c d

   

   

,

, ,

   

   

 

x a b y c d

T a c T b d

   

   

,

, ,

   

   

 

x a b y c d

T a b T c d

   

 ,

   

,

   

 

   

x a b

y c d

T a b c d

  

 

  ,  

Txy

Therefore

 



xy

T

 

  

x ,  

 

y

, for all , 

x y R

For any x= a+b,

We have y+ x- y= y+a+b- y

(

y a y

) (

y b y

)

= + - + + - , for each

y+a- y= c+d

We have x = - y+c+d+ y

(

y c y

) (

y d y

)

= - + + + - + +

(

b a b

)

c

(

c b c

)

d

= + - - + + - -(ii) Since

 

y x y  x and

y x y

 

 

x

We have

 



   

,

   

   

y z y c d

y x y c d

 

,

 

 

    

x a b

y a y y b y

   

 

 

 

x a b

a b

 

 

x

Therefore

 



y x y

 

  

 

x , for all , 

x y R

(iii) Since 

   

xy  x and 

   

xy  y

 

  

x y   

xy1xy2

   

1 2

 

 

x yxy

   

1 2

 

 

yy

   

1 2

1 2

 

 

 

y y y

y y

 

 

  y

(7)

, 

x y R

If  is a fuzzy ideal of Rsuch that 

   

x  x and

   

x  x , for all xR

(iv) Since

 

xz yxy  z and

xz y

xy



 

z

We have

 

 

 

 

 

 

 

x a b

x a b

   

 

 

 

x a b

a b

 

 

x a b

a b

 

z

Therefore

 

 

x 

 

z , for all , ,x y zR

(v) We have

 



 

 

  

  

 

x y e f

x y e f

  

  

,

, ,

   

   

 

    

x a b y c d

T b a b c T c b c d

   

   

,

, ,

   

   

 

x a b y c d

T a c T b d

   

   

,

, ,

   

   

 

x a b y c d

T a b T c d

   

 ,

   

,

   

 

   

x a b

y c d

T a b c d

  

 

  ,  

Txy

Therefore

 



xy

T

 

  

x ,  

 

y

, for all ,x yR

(vi) We have

 



 

 

  

  

 

x y e f

x y e f

  

  

,

, ,

   

   

 

    

x a b y c d

T b a b c T c b c d

   

   

,

, ,

   

   

 

x a b y c d

T a c T b d

   

   

,

, ,

   

   

 

x a b y c d

T a b T c d

   

 ,

   

,

   

 

   

x a b

y c d

T a b c d

  

 

  ,  

Txy

Therefore

 



xy

T

 

  

x ,  

 

y

, for all , 

x y R

Thus   is a Tfuzzy ideal of a  near-ring

R.

REFERENCES

[1] Abu Osman, M.T., On some product of fuzzy subgroups, Fuzzy Sets and Systems 24 (1987), 79-86.

[2] Akram, M., On T-fuzzy ideals in near-rings, Int. J. Math. Math. Sci. Volume 2007 (2007), Article ID 73514,14 pages [3] Ayyappan, M., and Natarajan, R., Lattice ordered near

rings, Acta Ciencia Indica, 1038(4), (2012) 727-738.

[4] Barnes, W.E., On the Γ-rings of Nobusawa, Pacific J. Math. 18 (1966), 411-422.

[5] Booth, G.L., A note on Γ near-rings, Studia. Sci. Math. Hungar. 23 (1988) 471-475.

[6] Coppage, W.E., and Luh, J., Radicals of gamma-rings, J. Math. Soc. Japan 23 (1971), 40-52.

[7] Chandrasekaran, G.,Chellappa, B., and Jeyakumar, M.,

Some theorems on T – anti-fuzzy ideals of a near-ring,

International Journal of Mathematics Trends and Technology. Volume 49, Number 5, 2017, pp 316-320. [8] Dheena, P., and Mohanraaj, G., T-fuzzy ideals in rings,

International Journal of Computational Cognition 2 (2011) 98-101.

[9] Dudek. W.A., and Jun. Y.B., Fuzzy subquasigroups over a

t-norm, Quasigroups and Related Systems 6 (1999),

pp.87-98

[10] Liu, W., Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139.

[11] Nobusawa, N., On a generalization of the ring theory, Osaka J. Math. 1 (1964), 81-89.

[12] Satyanarayana, Bh., and Syam Prasad, K., On Fuzzy cosets

of Gamma near-rings, Turkish J. Math. 29 (2005) 11-22.

[13] Schweizer, B., and Sklar, A., Statistical metric spaces, Pacific Journal of Mathematics. 10 (No. 1) (1963), 313-334. [14] Srinivas, T., Nagaiah, T., and Narasimha Swamy, P., Anti

fuzzy ideals of Γ near-rings, Ann. Fuzzy Math. Inform. 3(2)

(2012) 255-266.

[15] Wang-jin Liu., Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8(2) (1982) 133- 139.

[16] Yu, J., Mordeson, N., and Cheng, S.C., Elements of

L-algebra, Lecture Notes in Fuzzy Math. and Computer

Sciences, Creighton Univ., Omaha, Nebraska 68178, USA (1994).

[17] Yu, Y.-D., and Wang, Z.-D., TL-subrings and TL-ideals part1: basic concepts. Fuzzy Sets and Systems, 68 (1994), 93–103.

References

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