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

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, +, · ) and (F, +, · ) 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, +, · ) 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 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, +, · ) and (F, +, · ) 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, +, · ) 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. 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.

    REFERENCE

    1. Akram. M and Dar.K.H, On fuzzy d-algebras, Punjab University Journal of Mathematics, 37, 61-76, (2005).

    2. Anthony.J.M. and Sherwood.H, Fuzzy groups Redefined, Journal of mathematical analysis and applications, 69,124 -130 (1979 ).

    3. Azriel Rosenfeld, Fuzzy Groups, Journal of mathematical analysis and applications, 35, 512-517 (1971).

    4. Biswas.R, Fuzzy subgroups and Anti-fuzzy subgroups, Fuzzy sets and systems, 35,121-124 ( 1990 ).

    5. Biswas.R, Fuzzy fields and fuzzy linear spaces redefined, Fuzzy sets and systems, (1989) North Holland.

    6. Choudhury.F.P. and Chakraborty.A.B. and Khare.S.S. , A note on fuzzy subgroups and fuzzy homomorphism, Journal of mathematical analysis and applications ,131 ,537 -553 (1988 ).

    7. Kumbhojkar.H.V., and Bapat.M.S., Correspondence theorem for fuzzy ideals, Fuzzy sets and systems, (1991)

    8. Mustafa Akgul, Some properties of fuzzy groups, Journal of mathematical analysis and applications, 133, 93-100 (1988).

    9. Mohamed Asaad , Groups and fuzzy subgroups, fuzzy sets and systems (1991), North-Holland.

    10. Nanda. S, Fuzzy fields and fuzzy linear spaces, Fuzzy sets and systems, 19 (1986), 89-94.

    11. 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), 181-188.

    12. Palaniappan. N & K.Arjunan. 2007. Some properties of intuitionistic fuzzy subgroups, Aca Ciencia Indica, Vol.XXXIII (2): 321-328.

    13. Prabir Bhattacharya, Fuzzy Subgroups: Some Characterizations, Journal of Mathematical Analysis and Applications, 128, 241-252 (1987).

    14. Rosenfeld. A, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512- 517.

    15. Vasantha kandasamy.W.B, Smarandache fuzzy algebra, American research press, Rehoboth -2003.

    16. ZADEH.L.A, Fuzzy sets, Information and control, Vol.8, 338-353 (1965).

1Mathematics Wing, Annamalai University,

Annamalainagar- 608002, Chidambaram, India.

2Mathematics Wing, Annamalai University,

Annamalainagar- 608002, Chidambaram, India.