Homomorphism on
T
– Fuzzy Ideal of
ℓ-
Near-Rings
1
G. Chandrasekaran,
2B.Chellappa and
3M. 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 Tfuzzy ideals of near-rings. We extended the results of Tfuzzy 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 Tfuzzy ideals of
near- ring. We discuss some of its properties. We have shown that homomorphism, homomorphic image of Tfuzzy ideal, homomorphic pre-image
T fuzzy ideal and product of Tfuzzy ideal of
near-ring
II. DEFINITIONSAND EXAMPLES
Definition: 1
A mapping T: 0, 1
0, 1 0, 1 is called atriangular norm [tnorm] if and only if it satisfies the following conditions:
(i). T x
, 1 T
1, x x, 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 Tfuzzy right (resp. left) ideal if
(i)
xy
T
x , y
min
x , y
(ii)
xy
x
(resp. left
xy
y
), for allx y
,
inR
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 yRExample: 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
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 RExample: 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 integerDefinition: 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 yMDefinition: 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
x v 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 Tfuzzy 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
x y
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
y x y
0 x
1
sup
n f x
f
x
Therefore f
y x y
f
x , for all ,x yS(iii) Let be a Tfuzzy ideals of Rand let ,x yR
Since
xy
x and
xy
ywe 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
x z y x 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
x y
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
x y 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 Tfuzzy 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 Tfuzzy ideal of a
near ring is a Tfuzzy 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 Tfuzzy 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 x f 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
xy vf xy
v f xy
v f x f y
T v f x
vf x
x
Therefore
xy
x , for all ,x yR (iv) Since
xz y
x y
z
xz yx y vf xz yx y
v f xz yx y v f y z
,
T v f y f z v 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 x f y
,
T v f x v f y
,
T v f x v f y
,
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 x f y
,
T v f x v f y
,
T v f x v f y
,
T x y
Therefore
xy
T
x , y
, for all , x y R
Thus an epimorphic pre-image of a Tfuzzy ideal of a near ring is a Tfuzzy 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 x f 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 Tfuzzy ideal ofS.
Proof:
Let a b c S, , then there exists x y z, , R such that f x
a, f
y band f z
cSuppose is f invariant fuzzy ideal of a
near-ring R, then by Proposition 1
(i) we have f
a b
f
f x f y
f f xy
xy
,
T x y
,
T f a f b
Therefore f
a b
T f
a ,f b
, for all a b, Sand ,x yR(ii) Since
y x y
xWe 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
yWe have f
a b f
f x f y
f f xy
xy
x
f a
Therefore f
a b f
a , for all a b, Sand ,x yR
(iv) Since
xz y
xy
z
f a c b ab
f f x f z f y f x f y
xz yxy
z f
cTherefore f
a c b ab
f
c , for all , a b Sand ,x yR
(v) we have f
ab
f
f x f y
f f xy
xy
,
T x y
,
T f a f b
Therefore f
ab
T f
a , f b
, for all a b, Sand ,x yR(vi) we have f
ab
f
f x f y
f f xy
xy
,
T x y
,
T f a f b
Therefore f
ab
T f
a , f b
, for all a b, Sand ,x yRHence, f
is a Tfuzzy ideal of a near ringS.
Theorem: 5
Let f R: 1R2 be an onto homomorphism of a
near rings. If is Tfuzzy ideal of R1, then
f is a Tfuzzy ideal of R2.
Proof:
Let be a Tfuzzy ideal of a near ring R1
Let 1 f1
y1 and 2 f1
y2 , where1, 2 2
y y R are non-empty subsets of R2
Similarly, 3 f1
y1y2
Consider the set 1 2
a1a2/a11,a2
If x12, then xx1x2, for some x11,
22
x and so,
1 2
f x f x x f x
1 f x2 y1y2
1
1 2 3
x f y y
Thus123that is
x x/ f 1
y1y2
1
1
1 2/ 1 1 , 2 2
x x x f y x f 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 / ,
x x x f y x f y
1 1
1 2 1 1 2 2
sup min , / ,
x x x f y x f y
1 1
1 2 1 1 2 2
sup , / ,
T x x x f y x f y
1 1
1 1 1 2 2 2
sup / ,sup /
T x x f y x x f y
1 , 2
T f y f y
Therefore f
y1y2
T f
y1 ,f y2
, for all y1,y2R2(ii) Since
y x y
xWe 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 x f y x f y
1
1 1 1
sup /
x x f y
1 f y
Therefore f
y2y1y2
f
y1 , for all1, 2 2
y y R
(iii) We have
xy
x and
xy
yWe 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 y f x y f x
sup
x2 / y2f 1
x2
2 f x
Therefore f
x x1 2
f
x2 , for all1, 2 2
y y R
(iv) Since
xz y
xy
zWe have f
a c b ab
f f x f z f y f x f y
We have f
y1y3
y2y y1 2
1
1 3 2 1 2
sup /
x xf y y y y y
1 3 2 1 2sup /
y y y y y
1 1 1
1 1 , 2 2 , 3 3
x f y x f y x f y
1
3 3 3
sup /
3 f y
Therefore f
y1y3
y2y y1 2
f
y3 ,for all y y1, 2,y3R2
(v) f
y1y2
sup
x /xf1
y1y2
1 1
1 2 1 1 2 2
sup / ,
x x x f y x f y
1 2sup min , /
x x
1 1
1 1 , 2 2
x f y x f y
1 2sup , /
T x x
1 1
1 1 , 2 2
x f y x f y
11 1 1
sup / ,
T x x f y
1
2 2 2
sup x / x f y
1 , 2
T f y f y
Therefore
1 2
1 ,
2
f y y T f y f y , for all1, 2 2
y y R
(vi)f
y1y2
sup
x /xf1
y1y2
1 1
1 2 1 1 2 2
sup / ,
x x x f y x f y
1
1
1 2 1 1 2 2
sup min , / ,
x x x f y x f y
1 1
1 2 1 1 2 2
sup , / ,
T x x x f y x f y
1 1
1 1 1 2 2 2
sup / ,sup /
T x x f y x x f 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 aTfuzzy ideal of anear ring2
R .
Theorem: 6
Let and be Tfuzzy ideal of a near ring
R. Then is the smallest Tfuzzy ideal of R
containing both and .
Proof:
Let and be Tfuzzy 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
,
T x y
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
xWe 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
xy1xy2
1 2
x y xy
1 2
y y
1 2
1 2
y y y
y y
y
,
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
zWe 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
,
T x y
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
,
T x y
Therefore
xy
T
x ,
y
, for all , x y R
Thus is a Tfuzzy ideal of a near-ring
R.
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