# Lower Level Subsets of Anti L-Fuzzy Subfield of a Field

DOI : 10.17577/IJERTV2IS90338

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#### Lower Level Subsets of Anti L-Fuzzy 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 L-fuzzy subfield of a field under homomorphism.

2000 AMS Subject classification: 03F55, 06D72, 08A72.

KEY WORDS: L-fuzzy set, anti L-fuzzy subfield, anti L-fuzzy (a,b)-coset, lower level subset pseudo anti L-fuzzy 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, anti-fuzzy 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 L-fuzzy subfield of a field under homomorphism.

1. PRELIMINARIES:

1. Definition: Let X be a non-empty set and L be a complete lattice. A L-fuzzy subset A of X is a function A : X L.

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

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

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

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

4. A( x-1 ) A( x ), for all x in F{0}, where 0 is the additive identity element of F.

3. Definition: Let ( F, +, ) and ( F, +, ) be any two fields. Let f : F F

be any function and A be an anti-fuzzy subfield in F, V be an anti L-fuzzy

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).

4. Definition: Let A be an anti L-fuzzy 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(b-1x), for all x in F, is called an anti L-fuzzy (a,b )-coset of F.

5. Definition: Let A be an anti L-fuzzy subfield of a field (F, +, ) and a in

F. Then the pseudo anti L-fuzzy coset (aA)p is defined by ((aA)p)(x) = p(a)A(x), for every x in F and for some p in P.

6. 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) }.

1. PROPERTIES OF ANTI L-FUZZY SUBFIELDS:

1. Theorem: Let (F, +, Â· ) and (F, +, Â· ) be any two fields. The homomorphic image of an anti L-fuzzy subfield of F is an anti L-fuzzy 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 L-fuzzy subfield of F. We have to prove that V is an anti L-fuzzy 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(xy-1) ) A(xy-1) 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 L-fuzzy subfield of a field F.

2. Theorem: Let (F, +, Â· ) and (F, +, Â· ) be any two fields. The homomorphic pre-image of an anti L-fuzzy subfield of F is an anti L-fuzzy 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 L-fuzzy subfield of F. We have to prove that A is an anti L-fuzzy 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(xy-1) = V( f(xy-1) ) = V( f(x)f(y-1) ) = V( f(x)(f(y) )-1) V(f(x))

V(f(y)) = A(x) A(y), which implies that A(xy-1) A(x) A(y), for all x

and y 0 in F. Hence A is an anti L-fuzzy subfield of a field F.

In the following Theorem is the composition operation of functions :

3. Theorem: Let A be an anti L-fuzzy subfield of a field H and f is an isomorphism from a field F onto H. Then Af is an anti L-fuzzy subfield of F. Proof: Let x and y in F and A be an anti L-fuzzy 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 )( xy-1) = A( f( xy-1)) = A( f(x)f(y-1) )

= A( f(x)(f(y)) -1) A(f(x)) A(f(y)) (Af )(x) (Af )(y), which implies that (Af )(xy-1) (Af )(x) (Af )(y), for all x and y 0 in F. Therefore (Af) is an anti L-fuzzy subfield of a field F.

4. Theorem: If A is an anti L-fuzzy subfield of a field (F, +, . ), then the pseudo anti L-fuzzy coset (aA)p is an anti L-fuzzy subfield of a field F, for every aF and p in P.

Proof: Let A be an anti L-fuzzy 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 )( xy-1) = p(a)A(xy-1) p(a){A(x)A(y)}= p(a)A(x) p(a)A(y) = ((aA)p)(x)((aA)p )(y). Therefore, ((aA)p )(xy-1) ((aA)p )(x) ((aA)p )(y), for all x and y 0 in F. Hence (aA)p is an anti L-fuzzy subfield of a field F.

5. Theorem: Let A be an anti L-fuzzy subfield of a field (F, +, . ), then the anti L-fuzzy (0, 1 )-coset 0A1 is an anti L-fuzzy subfield of a field F, where 0 and 1 are identity elements of F.

Proof: Let A be an anti L-fuzzy 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)(xy-1) = A(1.xy-1 ) = A(xy-1) A(x) A(y). Therefore (1A)(xy-1) A(x) A(y), for all x and y 0 in F. Hence the anti L-fuzzy (0, 1 )-coset 0A1 is an anti L-fuzzy subfield of a field F.

6. Theorem: Let A be an anti L-fuzzy 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(xy-1) A(x) A(y) = , which implies that, A(xy -1) . Therefore, A(xy)

, A(xy -1) , we get xy, xy-1 in A. Hence A is a subfield of F.

2.1 Definition: Let A be an anti L-fuzzy 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.

1. Theorem: Let A be an anti L-fuzzy 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.

2. 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 L-fuzzy 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 xy-1 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(xy-1) 1=

1 2 = A(x) A(y), which implies that A(xy-1) 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 xy-1 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(xy-1)

2 = 12 = A(x) A(y), which implies that A(xy-1) A(x) A(y), for all x and y 0 in F. In all the cases, A is an anti L-fuzzy subfield of a field F.

3. Theorem: Let A be an anti L-fuzzy 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.

4. Theorem: Let A be an anti L-fuzzy 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.

5. Theorem: Let A be an anti L-fuzzy 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.

6. Theorem: Let A be an anti L-fuzzy 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.

7. Theorem: Any two different anti L-fuzzy 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 L-fuzzy 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 L-fuzzy subfields A and B have the same family of lower level subfields.

8. Theorem: Let (F, +, . ) be a finite field and A be an anti L-fuzzy subfield

of F. If , are elements of the image set of A such that

Proof: It is trivial.

A = A , then = .

9. 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 L-fuzzy subfield of F is a lower level subfield of an anti L-fuzzy 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 L-fuzzy subfield of F. Clearly V is an anti L-fuzzy 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, xy-1 in A. That is, A(x) and A(y) , A(xy) , A(xy-1) . 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(y-1) )

= V( f(xy-1) ) A(xy-1) , 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 L-fuzzy subfield V of a field F.

10. Theorem: Let (F, +, ) and (F, +, ) be any two fields. If f : F F is a homomorphism, then the homomorphic pre-image of a lower level subfield of an anti L-fuzzy subfield of F is a lower level subfield of an anti L-fuzzy 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 L-fuzzy subfield of F. Clearly A is an anti L-fuzzy 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(xy-1) = V(f(xy-1)) = V( f(x)f(y-1) ) = V( f(x)(f(y))-1 ) , which implies that A(xy-1) , for all x and y 0 in F. Therefore, A(xy) , A(xy-1) . Hence A is a lower level subfield of an anti L-fuzzy subfield A of F.

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#### 1Mathematics Wing, Annamalai University,

Annamalainagar- 608002, Chidambaram, India.

#### 2Mathematics Wing, Annamalai University,

Annamalainagar- 608002, Chidambaram, India.