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 Authors : B. Thenmozhi, S. Karthikeyan, S. Naganathan
 Paper ID : IJERTCONV4IS24003
 Volume & Issue : AMASE – 2016 (Volume 4 – Issue 24)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Some Properties of Intuitionistic LFuzzy Subnearrings of a Nearring
B. Thenmozhi*
Department of Mathematics, Syed ammal Engineering College Ramanathapuram
Tamil Nadu, India.
S. Karthikeyan
Department of Mathematics Velammal College of Engineering and Technology,
Madurai, Tamil Nadu, India.

Naganathan
Department of Mathematics, Government Arts College Pollachi, Udumalapet, Tamil Nadu, India.
AbstractIn this paper, we study some of the properties of intuitionistic Lfuzzy subnearring of a nearring and prove some results on these. With the expectations that these results may be applied in different fields.
KeywordsLfuzzy subset, intuitionistic Lfuzzy subset, intuitionistic Lfuzzy subnearring, intuitionistic Lfuzzy relation, Product of intuitionistic Lfuzzy subsets.

INTRODUCTION
After the introduction of fuzzy sets by L.A.Zadeh[15], several researchers explored on the generalization of the notion of fuzzy set. The concept of intuitionistic Lfuzzy subset was introduced by K.T.Atanassov[4,5], as a generalization of the notion of fuzzy set. Azriel Rosenfeld[6] defined a fuzzy groups. Asok Kumer Ray[3] defined a product of fuzzy subgroups. We introduce the concept of intuitionistic Lfuzzy subnearring of a nearring and established some results.

PRELIMINARIES
Definition 2.1. Let X be a nonempty set and L = (L, ) be a lattice with least element 0 and greatest element 1. A Lfuzzy subset A of X is a function A : X L.
Definition 2.2. Let (L, ) be a complete lattice with an involutive order reversing operation N : L L.
An intuitionistic Lfuzzy subset (ILFS) A in X is defined as an
(i) A(x y) A(x) A(y)

A(xy) A(x) A(y)

A(xy) A(x) A(y)

A(xy) A(x) A(y), for all x and y in R.
Definition 2.4. Let A and B be any two intuitionistic Lfuzzy subnearrings of nearrings R1 and R2 respectively. The product of A and B denoted by AxB is defined as AxB={(x, y),AxB( x, y), AxB( x, y)/for all x in R1 , y in R2}, where AxB(x,y) =A(x) B(y) and AxB(x, y) = A(x) B(y).
Definition 2.5. Let A be an intuitionistic Lfuzzy subset in a set S, the strongest intuitionistic Lfuzzy relation on S, that is an intuitionistic Lfuzzy relation on A is V given by
V( x, y) = A(x) A(y) and V(x, y) = A(x) A(y), for all x and y in S.
Definition 2.6. Let X and X be any two sets. Let f : X X be any function and A be a intuitionistic Lfuzzy subset in X, V be an intuitionistic Lfuzzy subset in f(X) = X, defined by
V(y) = sup A(x) and V(y) = inf A(x), for all
object of the form A={< x, A(x), A(x) > / x in X}, where
x f 1 ( y )
x f 1 ( y )
A : X L and A : X L define the degree of membership and the degree of nonmembership of the element xX respectively and for every xX satisfying A(x) N( A(x) ).
Definition 2.3. Let ( R, +, ) be a nearring. A intuitionistic Lfuzzy subset A of R is said to be an intuitionistic Lfuzzy subnearring (ILFSNR) of R if it satisfies the following axioms:
x in X and y in X. A is called a pre image of V under f and is denoted by f1(V).


SOME PROPERTIES OF INTUITIONISTIC L FUZZY SUBNEARRINGS OF A NEARRING
Theorem 3.1. Intersection of any two intuitionistic Lfuzzy subnearrings of a nearring R is a intuitionistic Lfuzzy subnearring of R.
Proof. Let A and B be any two intuitionistic Lfuzzy subnearrings of a nearring R and x and y in R. Let A = {(x,A(x),A(x))/xR} and B = { (x, B(x), B(x) ) /
xR} and also let C = AB = { ( x, C(x), C(x) ) / xR}, where A(x) B(x) = C(x) and A(x) B(x) = C(x).
Now, C( xy) [A(x) A(y)] [B(x) B(y)] = [A(x)
B(x)] [A(y)B(y)] = C(x) C(y). Therefore, C( x
y) (x) (y), for all x and y in R.
Proof. Let AxB be an intuitionistic Lfuzzy subnearring of R1xR2. By contraposition, suppose that none of the statements
(i) and (ii) holds. Then we can find a in R1 and b in R2 such that A(a) B(e), A(a) B(e) and B(b) A(e), B(b)
C C (e). We have,
(a, b) (e) (e) = (e) (e)
And, C(xy) [A(x) A(y)] [B(x) B(y)] = A
AxB
B A A B
C C
C C
[ (x) (x)] [ (y) (y)] = (x) (y). Therefore,
= AxB(e, e). And, AxB ( a, b) B(e) A(e) = A(e)
A B A B
B(e) = AxB (e, e). Thus AxB is not an intuitionistic Lfuzzy
C(xy) C(x) C(y), for all x and y in R. Also, C(x y)
[A(x)A(y)][B(x)B(y)]=[A(x)B(x)][A(y)B(y)]
subnearring of R1xR2. Hence either B(e) A(x) and B(e)
(x), for all x in R or (e) (y) and (e) (y), for
= C(x) C(y). Therefore, C(xy) C(x) C(y), for all x and y in R. And, C(xy) [A(x) A(y)] [B(x) B(y)] =
A
all y in R2.
1 A B A B
[A(x) B(x)] [A(y) B(y)] = C(x) C(y). Therefore,C(xy) C(x) C(y), for all x and y in R. Therefore, C is an intuitionistic Lfuzzy subnearring of a nearring R.
Thorem 3.2. Let ( R, +, ) is a nearring. The intersection of a family of intuitionistic Lfuzzy subnearrings of R is an intuitionistic Lfuzzy subnearring of R.
Proof. It is trivial.
Theorem 3.3. If A and B are any two intuitionistic Lfuzzy subnearrings of the nearrings R1 and R2 respectively, then AxB is an intuitionistic Lfuzzy subnearring of R1xR2..
Proof. Let A and B be two intuitionistic Lfuzzy subnearrings of the nearrings R1 and R2 respectively. Let x1 and x2 be in R1 and y1, y2 be in R2. Then (x1, y1) and (x2, y2) in R1xR2. Now, AxB[ (x1, y1) (x2, y2) ] = A( x1x2) B(y1y2)
[A(x1)A(x2)][B(y1)B(y2)]=[A(x1)B(y1)] [A(x2)B(y2)] = AxB(x1, y1) AxB(x2, y2). Therefore,
AxB [(x1, y1) (x2, y2)] AxB ( x1, y1) AxB(x2, y2), for all (x1, y1) and (x2, y2) in R1xR2. Also, AxB[ (x1, y1)(x2, y2)] =
A(x1x2)B(y1y2) [ A(x1)A(x2)] [ B(y1)B(y2)] =
[A(x1) B(y1)][A(x2)B(y2)]= AxB(x1, y1)AxB(x2, y2). Therefore, AxB[(x1, y1)(x2, y2)] AxB(x1, y1)AxB(x2, y2), for all (x1, y1), (x2, y2 ) in R1xR2. And, AxB[(x1, y1)(x2, y2)]=
A( x1 x2) B( y1 y2) [A(x1) A(x2) ][B(y1) B(y2)
]=[A(x1) B(y1)][ A(x2)B(y2)] = AxB (x1, y1)AxB (x2, y2). Therefore,AxB[(x1, y1)(x2, y2)] AxB( x1, y1) AxB (x2, y2) , for all (x1, y1), (x2, y2) in R1xR2. Also, AxB [(x1, y1)(x2,
y2)] = A(x1x2) B(y1y2) [A(x1) A(x2) ][ B(y1)
B(y2)] = [A(x1)B(y1) ] [ A(x2) B(y2)]= AxB(x1, y1)
AxB (x2, y2). Therefore, AxB [(x1, y1)(x2, y2)] AxB (x1, y1)
AxB(x2, y2), for all (x1, y1), (x2, y2) in R1xR2. Hence AxB is an intuitionistic Lfuzzy subnearring of R1xR2.
Theorem 3.4. Let A and B be intuitionistic Lfuzzy subnearrings of the nearrings R1 and R2 respectively. Suppose that e and e are the identity element of R1 and R2 respectively. If AxB is an intuitionistic Lfuzzy subnearring of R1xR2, then at least one of the following two statements must hold.

B(e) A(x) and B(e) A(x), for all x in R1,

A(e) B(y) and A(e ) B(y), for all y in R2.
Theorem 3.5. Let A and B be two intuitionistic Lfuzzy subsets of the nearrings R1 and R2 respectively and AxB is an intuitionistic Lfuzzy subnearring of R1xR2. Then the following are true:
if A(x) B(e) and A(x) B( e), then A is an intuitionstic Lfuzzy subnearring of R1.

if A(x) B(e) and A(x) B( e), then A is an intuitionistic Lfuzzy subnearring of R1.

if B(x) A(e) and B(x) A(e), then B is an intuitionistic Lfuzzy subnearring of R2.

either A is an intuitionistic Lfuzzy subnearring of R1 or B is an intuitionistic Lfuzzy subnearring of R2.
Proof. Let AxB be an intuitionistic Lfuzzy subnearring of R1xR2, x, y in R1and e in R2. Then (x, e) and (y, e) are in R1xR2. Now, using the property that A(x) B(e) and
A(x) B(e), for all x in R1, we get, A(xy) = AxB [(xy), (e+e)] AxB (x, e) AxB (y, e) = [A(x) B (e) ]
[ A(y)B(e)]=A(x) A(y) A(x) A(y). Therefore,A(x y) A(x)A(y), for all x and y in R1. Also, A(xy)
=AxB [(xy), (ee )] AxB(x, e) AxB(y, e) = [A(x)B(e]
[A(y)B(e)] = A(x)A(y). Therefore, A(xy)
A(x) A(y), for all x and y in R1. And, A(xy) = AxB
[(xy), (e+e)]AxB(x, e) AxB(y, e) = [A(x) B(e)] [A(y)B(e) ] = A(x) A(y) A(x)A(y). Therefore,A(xy) A(x) A(y), for all x and y in R1. Also,
A(xy)=AxB[(xy), (ee )] AxB(x, e) AxB(y, e) =
[A(x) B(e) ][A(y)B(e) ] = A(x) A(y). Therefore,A(xy) A(x) A(y), for all x and y in R1. Hence A is an intuitionistic Lfuzzy subnearring of R1. Thus (i) is proved. Now, using the property that B(x) A(e) and B(x)A(e), for all x in R2. Let x and y in R2 and e in R1. Then (e, x) and (e, y) are in R1xR2. We get, B(xy) = AxB[(e+e), (xy)]
AxB(e, x)AxB(e, y) = [A(e)B(x)][A(e)B(y)] =
B(x)B(y) B(x)B(y). Therefore, B(xy)
B(x)B(y), for all x and y in R2. Also, B(xy) = AxB[(ee ), (xy)] AxB(e, x)AxB(e, y) = [ A(e)B(x)][ A(e)
B(y)] = B(x)B(y). Therefore, B(xy)B(x)B(y), for all
x and y in R2. And, B(xy) = AxB[(e+e ), (xy)]AxB(e, x)
AxB(e, y)= [A(e) B(x)] [A(e) B(y)] = B(x)
B(y) B(x)B(y). Therefore, B(xy) B(x) B(y), for all x and y in R2. Also, B(xy)= AxB[(ee ), (xy)] AxB(e, x)
AxB(e, y)=[ A(e)B(x) ][ A(e) B(y) ] = B(x)
B(y). Therefore, B(xy) B(x) B(y), for all x and y in R2. Hence B is an intuitionistic Lfuzzy subnearring of a nearring R2. Thus (ii) is proved. (iii) is clear.
Theorem 3.6. Let A be an intuitionistic Lfuzzy subset of a nearring R and V be the strongest intuitionistic Lfuzzy relation of R. Then A is an intuitionistic Lfuzzy subnearring of R if and only if V is an intuitionistic Lfuzzy subnearring of of RxR.
Proof. Suppose that A is an intuitionistic Lfuzzy subnearring of a nearring R. Then for any x = (x1, x2) and y = (y1, y2) are in RxR. We have, V(xy) = V[( x1y1 , x2y2)] [A(x1)A(y1)] [A(x2) A(y2)] = [A(x1)
A(x2)][A(y1)A(y2)] = V(x1, x2) V(y1, y2)=
V(x)V(y). Therefore, V(xy) V(x)V (y), for all x and y in RxR. And, V(xy) = V[( x1y1, x2y2)] [A(x1) A(y1)] [A(x2)A(y2)] = [A(x1)A(x2)] [A(y1)A(y2)]= V(x1, x2) V(y1, y2)= V(x) V(y). Therefore, V(xy) V(x)
V(y), for all x and y in RxR. Also we have,
V(xy)=V[(x1y1, x2y2)] [A(x1)A(y1)] [A(x2)
A(y2)] = [A(x1) A(x2) ][A(y1) A(y2)] = V(x1, x2) V
(y1, y2) = V(x) V(y). Therefore, V(xy) V (x) V (y), for all x and y in RxR. And, V(xy)= V(x1y1, x2y2) [A(x1)
A(y1)][A(x2) A(y2)] = [A(x1) A(x2)] [A(y1)
(y )] = (x , x ) (y , y ) = (x) (y). Therefore,


CONCLUSION

We tried to prove some results to use in the field of Intuitionistic L fuzzy subnearring of a nearing
REFERENCES

Ajmal.N and Thomas.K.V., Fuzzy lattices, Information sciences 79 (1994), 271291.

Akram.M and Dar.K.H, On fuzzy dalgebras, Punjab university journal of mathematics, 37(2005), 6176.

Asok Kumer Ray, On product of fuzzy subgroups, fuzzy sets and systems, 105, 181183 (1999).

Atanassov.K., Intuitionistic fuzzy sets, fuzzy sets and systems, 20(1), 8796 (1986).

Atanassov.K., Stoeva.S., Intuitionistic Lfuzzy sets, Cybernetics and systems research 2 (Elsevier Sci. Publ., Amsterdam, 1984), 539540.

Azriel Rosenfeld, Fuzzy Groups, Journal of mathematical analysis and applications 35, 512517 (1971).

Chakrabarty, K., Biswas, R., Nanda, A note on union and intersection of intuitionistic fuzzy sets , Notes on intuitionistic fuzzy sets , 3(4), (1997).

Davvaz.B and Wieslaw.A.Dudek, Fuzzy nary groups as a generalization of rosenfeld fuzzy groups, ARXIV 0710.3884VI(MATH.RA)20 OCT 2007,116.

Goran Trajkovski, An approach towards defining Lfuzzy lattices, IEEE, 7(1998), 221225.

Mohammad M. Atallah, On the Lfuzzy prime ideal theorem of distributive lattices, The journal of fuzzy mathematics, Vol.9, No.4, 2001.

Palaniappan.N and Arjunan.K, The homomorphism, anti homomorphism of a fuzzy and anti fuzzy ideals, Varahmihir journal of mathematical sciences, Vol.6 No.1 (2006), 181188.

Palaniappan. N & K. Arjunan, Operation on fuzzy and anti fuzzy ideals , Antartica J. Math ., 4(1); 5964, 2007.

Palaniappan. N & K.Arjunan, Some properties of intuitionistic fuzzy
A 2 V 1 2
V 1 2 V V
subgroups, Acta Ciencia Indica, Vol.XXXIII (2); 321328, 2007.
V(xy) V(x) V(y), for all x and y in RxR. This proves that V is an intuitionistic Lfuzzy subnearring of RxR. Conversely assume that V is an intuitionistic Lfuzzy subnearring of RxR, then for any x=(x1, x2) and y=(y1, y2) are
in RxR, we have (x y ) (x y )= [(x , x ) (y ,

B.Thenmozhi,S.Naganathan,Some properties of Anti L Fuzzy Subnearring of a nearing, International Journal of Applied Research,3839.

B.Thenmozhi,S.Naganathan,Propertie on Anti L fuzzy Subnearring of a nearing in Homomorphism,National Conference in Fuzzy
A 1 1
A 2 2
V 1 2 1
Algebra, Paramakudi.
y2)]= V (xy) V(x) V(y)= V(x1, x2) V(y1, y2) = [A(x1) A(x2)] [A(y1) A(y2)]. If we put x2= y2= 0, we get, A(x1 y1) A(x1) A(y1), for all x1 and y1 in R. And,
A(x1y1) A(x2y2) = V[(x1, x2)(y1, y2)] = V(xy)
V(x)V(y)= V(x1, x2)V(y1, y2)=[A(x1)A(x2)] [A(y1)A(y2)]. If we put x2=y2=0, we get, A(x1y1)
A(x1)A(y1), for all x1,y1 in R. Also we have,
A(x1y1) A(x2y2) = V[(x1, x2) (y1, y2)]= V(xy)
V (x) V(y) = V(x1, x2) V(y1, y2) = [A(x1)A(x2)]
[A(y1) A(y2)]. If we put x2 = y2 = 0, we get,
A(x1y1)A(x1)A(y1), for all x1 and y1 in R. And,
A(x1y1)A(x2y2) = V[(x1, x2) (y1, y2)] = V (x y)
V(x)V(y)=V(x1,x2)V (y1, y2) = [A(x1)
A(x2)][A(y1) A(y2)]. If we put x2 = y2 = 0, we get,
A(x1y1) A(x1)A(y1), for all x1 and y1 in R. Hence A is a A is an intuitionistic Lfuzzy subnearring of a nearring R.

Rajesh Kumar, Fuzzy Algebra, Volume 1, University of Delhi Publication Division, July 1993.

Zadeh.L.A., Fuzzy sets , Information and control ,Vol.8, 338353 (1965).