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 Total Downloads : 353
 Authors : M. Vasu & D. Sivakumar
 Paper ID : IJERTV2IS90338
 Volume & Issue : Volume 02, Issue 09 (September 2013)
 Published (First Online): 02102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Lower Level Subsets of Anti LFuzzy Subfield of a Field
M. Vasu & D. Sivakumar
ABSTRACT: In this paper, we made an attempt to study the algebraic nature of lower level subsets of anti Lfuzzy subfield of a field under homomorphism.
2000 AMS Subject classification: 03F55, 06D72, 08A72.
KEY WORDS: Lfuzzy set, anti Lfuzzy subfield, anti Lfuzzy (a,b)coset, lower level subset pseudo anti Lfuzzy coset.
INTRODUCTION: After the introduction of fuzzy sets by L.A.Zadeh[16], several researchers explored on the generalization of the concept of fuzzy sets. The notion of fuzzy subgroups, antifuzzy subgroups, fuzzy fields and fuzzy linear spaces was introduced by Biswas.R[4, 5 ]. In this paper, we introduce the some theorems in lower level subsets of anti Lfuzzy subfield of a field under homomorphism.

PRELIMINARIES:

Definition: Let X be a nonempty set and L be a complete lattice. A Lfuzzy subset A of X is a function A : X L.

Definition: Let ( F, +, ) be a field. A Lfuzzy subset A of F is said to be an anti Lfuzzy subfield(ALFSF) of F if the following conditions are satisfied:

A( x+y ) A(x) A(y), for all x and y in F,

A( x ) A( x ), for all x in F,

A( xy ) A(x) A(y), for all x and y in F,

A( x1 ) A( x ), for all x in F{0}, where 0 is the additive identity element of F.


Definition: Let ( F, +, ) and ( F, +, ) be any two fields. Let f : F F
be any function and A be an antifuzzy subfield in F, V be an anti Lfuzzy
subfield in f(F) = F, defined by V(y) = inf
x f 1 ( y )
A(x), for all x in F and y in F.
Then A is called a preimage of V under f and is denoted by f 1(V).

Definition: Let A be an anti Lfuzzy subfield of a field ( F, +, ). For any a and b0 in F, aAb is defined by (a+A)(x) = A(a+x), for all x in F and (bA)(x) = A(b1x), for all x in F, is called an anti Lfuzzy (a,b )coset of F.

Definition: Let A be an anti Lfuzzy subfield of a field (F, +, ) and a in
F. Then the pseudo anti Lfuzzy coset (aA)p is defined by ((aA)p)(x) = p(a)A(x), for every x in F and for some p in P.

Definition: Let A be a fuzzy subset of X. For in L, the lower level subset of A is the set A = { xX: A(x) }.


PROPERTIES OF ANTI LFUZZY SUBFIELDS:

Theorem: Let (F, +, Â· ) and (F, +, Â· ) be any two fields. The homomorphic image of an anti Lfuzzy subfield of F is an anti Lfuzzy subfield of F.
Proof: Let (F, +, Â· ) and (F, +, Â· ) be any two fields and f : FF be a
homomorphism. That is f(x+y) = f(x)+f(y), for all x and y in F and f(xy) = f(x)f(y), for all x and y in F. Let V= f(A), where A is an anti Lfuzzy subfield of F. We have to prove that V is an anti Lfuzzy subfield of F. Now, for f(x) and f(y) in F, we have V( f(x)f(y) ) = V( f(xy) ) A(xy)
A(x)A(y), which implies that V( f(x)f(y) ) V(f(x)) V(f(y)), for all f(x)
and f(y) in F. And V( f(x)( f(y) )1 ) = V( f(xy1) ) A(xy1) A(x) A(y),
which implies that V( f(x)( f(y) )1 ) V(f(x)) V( f(y) ), for all f(x) and f(y)
01 in F. Hence V is an anti Lfuzzy subfield of a field F.

Theorem: Let (F, +, Â· ) and (F, +, Â· ) be any two fields. The homomorphic preimage of an anti Lfuzzy subfield of F is an anti Lfuzzy subfield of F. Proof: Let (F, +, Â· ) and (F, +, Â· ) be any two fields and f : FF be a homomorphism. That is f(x+y) = f(x)+f(y), for all x and y in F and f(xy) = f(x)f(y), for all x and y in F. Let V = f(A), where V is an anti Lfuzzy subfield of F. We have to prove that A is an anti Lfuzzy subfield of
F. Let x and y in F. Then, A(xy)= V( f(xy) ) = V( f(x)f(y) ) V(f(x)) V(f(y)) = A(x) A(y), which implies that A(xy) A(x) A(y), for all x and y in F. And, A(xy1) = V( f(xy1) ) = V( f(x)f(y1) ) = V( f(x)(f(y) )1) V(f(x))
V(f(y)) = A(x) A(y), which implies that A(xy1) A(x) A(y), for all x
and y 0 in F. Hence A is an anti Lfuzzy subfield of a field F.
In the following Theorem is the composition operation of functions :

Theorem: Let A be an anti Lfuzzy subfield of a field H and f is an isomorphism from a field F onto H. Then Af is an anti Lfuzzy subfield of F. Proof: Let x and y in F and A be an anti Lfuzzy subfield of a field H. Then we have (Af )( xy) = A(f( xy ) ) = A( f(x)+f(y)) = A( f(x)f(y) ) A(f(x))
A(f(y)) (Af )(x) (Af )(y), which implies that (Af)(xy) (Af )(x)
(Af)(y), for all x and y in F. And, (Af )( xy1) = A( f( xy1)) = A( f(x)f(y1) )
= A( f(x)(f(y)) 1) A(f(x)) A(f(y)) (Af )(x) (Af )(y), which implies that (Af )(xy1) (Af )(x) (Af )(y), for all x and y 0 in F. Therefore (Af) is an anti Lfuzzy subfield of a field F.

Theorem: If A is an anti Lfuzzy subfield of a field (F, +, . ), then the pseudo anti Lfuzzy coset (aA)p is an anti Lfuzzy subfield of a field F, for every aF and p in P.
Proof: Let A be an anti Lfuzzy subfield of a field ( F, +, . ). For every x and y in F, we have( (aA)p )(xy ) = p(a)A( xy) p(a){A(x) A(y)}= p(a)A(x)
p(a)A(y) = ((aA)p )(x) ((aA)p )(y). Therefore, ((aA)p)(xy) ( (aA)p )(x) ((aA)p )(y), for all x and y in F. And for every x and y 0 in F,((aA)p )( xy1) = p(a)A(xy1) p(a){A(x)A(y)}= p(a)A(x) p(a)A(y) = ((aA)p)(x)((aA)p )(y). Therefore, ((aA)p )(xy1) ((aA)p )(x) ((aA)p )(y), for all x and y 0 in F. Hence (aA)p is an anti Lfuzzy subfield of a field F.

Theorem: Let A be an anti Lfuzzy subfield of a field (F, +, . ), then the anti Lfuzzy (0, 1 )coset 0A1 is an anti Lfuzzy subfield of a field F, where 0 and 1 are identity elements of F.
Proof: Let A be an anti Lfuzzy subfield of a field ( F, +, . ). For every x and y in F, we have, (0+A)( xy ) = A(0+ xy ) = A( xy ) A(x) A(y). Therefore (0+A)( xy ) A(x) A(y), for all x and y in F. And for x and y 0 in F, we have (1A)(xy1) = A(1.xy1 ) = A(xy1) A(x) A(y). Therefore (1A)(xy1) A(x) A(y), for all x and y 0 in F. Hence the anti Lfuzzy (0, 1 )coset 0A1 is an anti Lfuzzy subfield of a field F.

Theorem: Let A be an anti Lfuzzy subfield of a field ( F, +, . ). Then for

in L such that A(0), A(1), A is a subfield of F, where 0 and 1 are identity elements of F.
Proof: For all x and y in A , we have, A(x) and A(y) . Now, A(xy) A(x) A(y) = , which implies that, A(xy) . And also, A(xy1) A(x) A(y) = , which implies that, A(xy 1) . Therefore, A(xy)
, A(xy 1) , we get xy, xy1 in A. Hence A is a subfield of F.
2.1 Definition: Let A be an anti Lfuzzy subfield of a field (F, +, . ). The lower level subset A, for in L such that A(0), A(1), is called lower level subfield of A.

Theorem: Let A be an anti Lfuzzy subfield of a field (F, +, . ). Then two lower level subfields A1 and A2, 1 and 2 in L and 1 A(0), 2 A(0),
1 A(1), 2 A(1) with 2 1 of A are equal if and only if there is no x in
F such that 1 A(x) 2, where 0 and 1 are identity elements of F.
Proof: Assume that A1 = A2. Suppose there exists xF such that 1 A(x)
2. Then A1 A2, which implies that x belongs to A2, but not in A 1. This is contradiction to A1 = A 2. Therefore there is no xF such that 1 A(x)
2. Conversely, if there is no xF suc that 1 A(x) 2. Then A1 = A2.

Theorem: Let (F, +, . ) be a field and A be a fuzzy subset of F such that A be a lower level subfield of F. If in L satisfying A(0), A(1), then A is an anti Lfuzzy subfield of F, where 0 and 1 are identity elements of F. Proof: Let (F, +, . ) be a field. For x and y in F. Let A(x) = 1 and A(y) = 2. Case (i): If 1 2, then x and y in A1. As A1 is a lower level subfield of F, so x y and xy1 in A1. Now, A(x y) 1 = 1 2 = A(x) A(y), which implies that A(xy) A(x) A(y), for all x and y in F. Now, A(xy1) 1=
1 2 = A(x) A(y), which implies that A(xy1) A(x) A(y), for all x and
y 0 in F. Case (ii): If 1< 2, then x and y in A2. As A2 is a lower level subfield of F, so xy and xy1 in A2. Now, A(xy)2 = 12 = A(x) A(y), which implies that A(xy) A(x) A(y), for all x and y in F. Now, A(xy1)
2 = 12 = A(x) A(y), which implies that A(xy1) A(x) A(y), for all x and y 0 in F. In all the cases, A is an anti Lfuzzy subfield of a field F.

Theorem: Let A be an anti Lfuzzy subfield of a field (F, +, . ). If any two lower level subfields of A belongs to F, then their intersection is also lower level subfield of A in F.
Proof: For 1, 2 in L, 1 A(0) and 2 A(0), 1 A(1) and 2 A(1), where 0 and 1 are identity elements of F. Case (i): If 1> A(x) > 2, then A 2 A 1. Therefore, A1 A2 = A2 but A2 is a lower level subfield of A. Case (ii): If
1< A(x) < 2, then A 1 A2. Therefore, A1 A2 = A1, but A1 is a lower level subfield of A. Case (iii): If 1 = 2, then A1 = A2. In all cases, intersection of any two lower level subfields is a lower level subfield of A.

Theorem: Let A be an anti Lfuzzy subfield of a field (F, +, . ). If i in L,
i A(0), i A(1) and Ai, i in I, is a collection of lower level subfields of A, then their intersection is also a lower level subfield of A.
Proof: It is trivial.

Theorem: Let A be an anti Lfuzzy subfield of a field (F, +, . ). If any two lower level subfields of A belongs to F, then their union is also lower level subfield of A in F.
Proof: For 1, 2 in L, 1 A(0) and 2 A(0), 1 A(1) and 2 A(1), where 0 and 1 are identity elements of F. Case (i): If 1> A(x) > 2, then A2 A1. Therefore, A1A2 = A1, but A1 is a lower level subfield of A. Case (ii): If
1< A(x) < 2, then A 1 A2. Therefore, A1 A2 = A2, but A2 is a lower level subfield of A. Case (iii): If 1 = 2, then A1 = A2. In all cases, union of any two lower level subfields is a lower level subfield of A.

Theorem: Let A be an anti Lfuzzy subfield of a field ( F, +, . ). If i in L, i A(0), i A(1) and Ai, i in I, is a collection of lower level subfields of A, then their union is also a lower level subfield of A.
Proof: It is trivial.

Theorem: Any two different anti Lfuzzy subfields of a field may have identical family of lower level subfields.
Proof: We consider the following example: Consider the field F = Z5 = { 0, 1, 2, 3, 4 } with addition modulo 5 and multiplication modulo 5 operations. Define fuzzy subsets A and B of F by A = { 0, 0.1, 1, 0.4, 2, 0.4, 3, 0.4,
4, 0.4 } and B = {0, 0.2, 1, 0.3, 2, 0.3, 3, 0.3, 4, 0.3 }. Clearly A and B are two different anti Lfuzzy subfields of F. And, Im A = {0.1, 0.4}, then the lower level subfields of A are A0.1 = {0}, A0.4 = { 0, 1, 2, 3, 4 }= F. And,
Im B = {0.2, 0.3}, then the lower level subfields of B are B0.2= {0}, B0.3 =
{ 0, 1, 2, 3, 4 } = F. Thus the two anti Lfuzzy subfields A and B have the same family of lower level subfields.

Theorem: Let (F, +, . ) be a finite field and A be an anti Lfuzzy subfield
of F. If , are elements of the image set of A such that
Proof: It is trivial.
A = A , then = .

Theorem: Let (F, +, ) and (F, +, ) be any two fields. If f : F F is a homomorphism, then the homomorphic image of a lower level subfield of an anti Lfuzzy subfield of F is a lower level subfield of an anti Lfuzzy subfield of F.
Proof: Let (F, +, ) and (F, +, ) be any two fields and f : F F be a
homomorphism. That is, f(x+y) = f(x)+f(y), for all x and y in F and f(xy) = f(x)f(y), for all x and y in F. Let V = f(A), where A is an anti Lfuzzy subfield of F. Clearly V is an anti Lfuzzy subfield of F. If x and y in F, then f(x) and f(y) in F. Let A be a lower level subfield of A. Suppose x, y and xy, xy1 in A. That is, A(x) and A(y) , A(xy) , A(xy1) . We have to prove that f(A) is a lower level subfield of V. Now, V(f(x)) A(x) , implies that V(f(x)) ; V(f(y)) A(y) , implies that V(f(y)) , V(f(x)f(y)) = V(f(x)+f(y) ) = V(f(xy) ) A(xy ) , which implies that V( f(x)f(y)) , for all f(x) and f(y) in F. And V(f(x)(f(y) )1) = V(f(x)f(y1) )
= V( f(xy1) ) A(xy1) , which implies that V(f(x)(f(y))1) , for f(x) and
f(y) 0 in F. Therefore, V(f(x)f(y)) , V( f(x)( f(y))1) . Hence f (A) is a lower level subfield of an anti Lfuzzy subfield V of a field F.

Theorem: Let (F, +, ) and (F, +, ) be any two fields. If f : F F is a homomorphism, then the homomorphic preimage of a lower level subfield of an anti Lfuzzy subfield of F is a lower level subfield of an anti Lfuzzy subfield of F.
Proof: Let (F, +, ) and (F, +, ) be any two fields and f : FF be a
homomorphism. That is, f(x+y) = f(x)+f(y), for all x and y in F and f(xy) = f(x)f(y), for all x and y in F. Let V = f(A), where V is an anti Lfuzzy subfield of F. Clearly A is an anti Lfuzzy subfield of F. Let x and y in F. Let f(A) be a lower level subfield of V. Suppose f(x), f(y) and f(x)f(y), f(x)(f(y))1 in f(A). That is, V(f(x)) and V(f(y)) ; V(f(x)f(y)) ,
V( f(x)(f(y))1 ) . We have to prove that A is a lower level subfield of A. Now, A(x) = V(f(x)) , implies that A(x) ; A(y) = V(f(y)) , implies
that A(y) , we have A(xy) = V(f(xy)) = V(f(x)+f(y)) = V(f(x)f(y)) , which implies that A(xy) , for all x and y in F. And A(xy1) = V(f(xy1)) = V( f(x)f(y1) ) = V( f(x)(f(y))1 ) , which implies that A(xy1) , for all x and y 0 in F. Therefore, A(xy) , A(xy1) . Hence A is a lower level subfield of an anti Lfuzzy subfield A of F.
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1Mathematics Wing, Annamalai University,
Annamalainagar 608002, Chidambaram, India.
2Mathematics Wing, Annamalai University,
Annamalainagar 608002, Chidambaram, India.