Fuzzy and Anti-Fuzzy Normal HX Subring of a HX Ring

IJIRST –International Journal for Innovative Research in Science & Technology| Volume 3 | Issue 04 | September 2016 ISSN (online): 2349-6010

Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring R. Muthuraj Department of Mathematics H. H. The Rajah’s College, Pudukkottai – 622001, Tamilnadu, India

N. Ramila Gandhi Department of Mathematics PSNA College of Engineering and Technology, Dindigul-624 622, Tamilnadu, India

Abstract In this paper, we define the concept of a fuzzy and anti-fuzzy normal HX subring of a HX ring also discuss some related properties of it.

Keywords: HX Ring, Fuzzy HX Ring, Fuzzy Set, Fuzzy Normal HX Subring, Anti-Fuzzy Normal HX Subring _______________________________________________________________________________________________________ I.

INTRODUCTION

In 1965, Zadeh [15] introduced the concept of fuzzy subset of a set X as a function from X into the closed interval 0 and 1 and studied their properties. Fuzzy set theory is a useful tool to describe situations in which the data or imprecise or vague and it is applied to logic , set theory, group theory, ring theory, real analysis, measure theory etc. In 1967, Rosenfeld [13] defined the idea of fuzzy subgroups and gave some of its properties. In 1988, Professor Li Hong Xing [6] proposed the concept of HX ring and derived some of its properties, then Professor Zhong [1,2] gave the structures of HX ring on a class of ring. In this paper, we define the concept of fuzzy normal HX ring and anti-fuzzy normal HX ring of a HX ring by establishing the relation among them and discussed its properties. We also discuss some results based on homomorphism and anti-homomorphism of a fuzzy normal HX ring and anti-fuzzy normal HX ring of a HX ring.

II. PRELIMINARIES In this section, we site the fundamental definitions that will be used in the sequel. Throughout this paper, R = (R, +, ·) is a Ring, e is the additive identity element of R and xy, we mean x.y.

III. FUZZY NORMAL HX RING In this section we define the concept of fuzzy normal HX ring and discuss some related results.

Definition Let R be a ring. Let μ be a fuzzy set defined on R. Let   2R –{} be a HX ring. A fuzzy HX subring  of  is called a fuzzy normal HX ring on  or a fuzzy normal ring induced by μ if the following conditions are satisfied. For all A , B , (AB) =  (BA), where  (A) = max {μ(x) /for all xA  R }.

Example In Example C [10],  is a fuzzy HX ring on .

For all A , B , ( AB) =  (BA).

Hence,  is a fuzzy normal HX ring on .

Theorem If μ is a fuzzy normal subring of a ring R then the fuzzy subset  is a fuzzy normal HX subring of a HX ring .

Proof Let μ be a fuzzy normal subring of R. That is, μ(xy) = μ(yx), for all x,y R.

By Theorem ‘D’ [10],  is a fuzzy HX subring on .

For all A, B ,  (AB) = max{μ(xy) / for all xy AB  R } = max{μ(yx) / for all xy AB  R }, =  (BA)   (AB) =  (BA).

Hence,  is a fuzzy normal HX subring of a HX ring .

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

Theorem Let  and  be any two fuzzy sets defined on R. Let  and  be any two fuzzy normal HX subrings of a HX ring  then the intersection of two fuzzy normal HX subrings,    is also a fuzzy normal HX subring of a HX ring .

Proof Let  and  be any two fuzzy sets defined on R. Let  and  be any two fuzzy normal HX subrings of a HX ring .

By Theorem ‘H’ [10],    is a fuzzy HX subring of a HX ring .

Let A, B . ( )(AB) = min{ (AB),  (AB)} = min{ (BA),  (BA)}, = (  )(BA).   (   )(AB) = (  )(BA).   Hence,    is a fuzzy normal HX subring of a HX ring .

Remark  

The intersection of family of fuzzy normal HX subrings of a HX ring  is also fuzzy normal HX subring of .

Let R be a ring. Let μ and  are fuzzy sets of R and μ   is also a fuzzy set of R then φ   is a fuzzy normal HX subring of  induced by μ   of R. Theorem

If , , φ   are fuzzy normal HX subrings of a HX ring  induced by the fuzzy sets μ,  , μ   of R then φ   =    .

Proof

By Theorem ‘J’ [10], it is clear.

Theorem Let μ and  be fuzzy sets of R. Let   2R – {} be a HX ring. If  and  are any two fuzzy normal HX subrings of  then    is a fuzzy normal HX subring of .

Proof Let μ and  be fuzzy sets of R. Let  and  be any two fuzzy normal HX subrings of .

By Theorem ‘M’[ 10] ,    is a fuzzy HX subring of .

Let A, B  . (  ) (AB) = max{ (AB),  (AB)} = max{ (BA),  (BA)} = (  )(BA).   (   )(AB) = (  )(BA).   Hence,    is a fuzzy normal HX subring of .

Remark  

Union of family of fuzzy normal HX subrings of a HX ring  is also fuzzy normal HX subring of .

Let R be a ring. Let μ and  are fuzzy sets of R then φ   is a fuzzy normal HX subring of  induced by the fuzzy set μ   of R. Theorem

Let R be a ring. Let μ and  be fuzzy sets of R. If ,  , φ   are fuzzy normal HX subrings of a HX ring  induced by μ, , μ   of R then φ   =   .

Proof

By Theorem ‘O’ [10], it is clear.

Theorem Let  and  be fuzzy normal HX subrings of HX rings 1 and 2 respectively then  ×  is also a fuzzy normal HX subring of a HX ring 1 × 2.

Proof Let μ and  be fuzzy sets of R. Let  and  be any two fuzzy normal HX subrings of 1 and 2 respectively.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

By Theorem ‘Q’ [10],  ×  is a fuzzy HX subring of 1 × 2.

Let (A, B), (C, D)  1 × 2. ( ×  ) ((A, B)·(C ,D)) = ( ×  ) (AC, BD) = min { (AC), (BD)} = min{  (CA),  (DB)} = ( ×  ) (CA , DB) = ( ×  ) ((C, D) · (A , B))   ( ×  ) ((A, B) · (C,D)) = ( ×  ) ((C, D) · (A , B)).   Hence,  ×  is a fuzzy normal HX subring of a HX ring 1 × 2.

Theorem Let  and  be fuzzy subsets of the HX rings 1 and 2 respectively, such that  (A ) ≤ ( Q1 ) for all A  1 , Q1 being the identity element of 2 . If ( × ) is a fuzzy normal HX subring of 1 × 2 then  is a fuzzy normal HX subring of 1.

Proof Let  ×  be a fuzzy normal HX subring of 1 × 2.

By Theorem ‘S’[10],  is a fuzzy HX subring of 1.

Let A, B  1.

Given,  (A ) ≤ ( Q1 ) for all A  1.  (AB) = min { (AB),  (Q1Q1)} = ( × )((A, Q1 ) · (B , Q1)) = ( × )( (B , Q1) · (A, Q1 )) = min { (BA),  (Q1Q1)} = (BA).   (AB) =  (BA).  Hence,  is a fuzzy normal HX subring of 1.

Theorem Let  and  be fuzzy subsets of the HX rings 1 and 2 respectively , such that  (A ) ≤  (Q ) for all A  2 , Q being the identity element of 1. If  ×  is a fuzzy normal HX sub ring of 1 × 2 then  is a fuzzy normal HX subring of 2.

Proof Let  ×  be a fuzzy normal HX subring of 1 × 2.

By Theorem ‘T’ [10],  is a fuzzy HX subring of 2.

Let A, B  2.

Given,  (A) ≤  ( Q ) for all A  2.  (AB) = min {(QQ),  (AB)} = (  × )((Q , A) · (Q , B)) = (  × )((Q , B) · (Q , A)) = min {(QQ), (BA)} = (BA)   (AB) = (BA)  Hence,  is a fuzzy normal HX subring of 2.

IV. HOMOMORPHISM AND ANTI-HOMOMORPHISM OF A FUZZY NORMAL HX SUBRING OF A HX RING In this section, we introduce the concept of an image, pre-image of fuzzy subset of a HX ring and discuss the properties of homomorphic and anti homomorphic images and pre images of fuzzy normal HX subring of a HX ring  .

Theorem Let 1 and 2 be any two HX rings on the rings R 1 and R2 respectively. Let f : 1  2 be a homomorphism onto HX rings. Let λ be a fuzzy normal HX subring of 1 then f ( λ ) is a fuzzy normal HX subring of 2 , if λ has a supremum property and λ is f-invariant.

Proof Let  be a fuzzy subset of R1 and λ is a fuzzy normal HX subring of 1.

By Theorem 3.4[12], f ( λ ) is a fuzzy HX subring of 2.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

There exist X,Y 1 such that f(X), f(Y)  2, (f (λ)) (f(X) f(Y)) = (f (λ)) (f(XY)) , = λ (XY) = λ (YX) = (f (λ)) (f(YX)) = (f (λ)) (f(Y)f(X))  (f (λ )) (f(X)f(Y)) = (f (λ)) (f(Y)f(X)).  Hence, f (λ ) is a fuzzy normal HX subring of 2.

Theorem Let 1 and 2 be any two HX rings on R1 and R2 respectively. Let f: 1  2 be a homomorphism on HX rings. Let  be a fuzzy normal HX subring of 2 then f –1 () is a fuzzy normal HX subring of 1.

Proof Let  be a fuzzy subset of R2 and  be a fuzzy normal HX subring of 2 .

By Theorem 3.5 [12], f –1 () is a fuzzy HX subring of 1.

For any X,Y  1, f(X) , f(Y)  2, (f –1()) (XY) =  (f(XY)) =  (f(X) f(Y)) =  (f(Y) f(X)) =  (f(YX)) = (f –1()) (YX) –1  (f ( )) (XY) = (f –1()) (YX) –1  Hence, f ( ) is a fuzzy normal HX subring of 1.

Theorem Let 1 and 2 be any two HX rings on R1 and R2 respectively. Let f: 1  2 be an anti homomorphism onto HX rings. Let λ be a fuzzy normal HX subring of 1, then f(λ) is a fuzzy normal HX subring of 2 , if λ has a supremum property and λ is f-invariant.

Proof Let  be a fuzzy subset of R1 and λ is a fuzzy normal HX subring of 1.

By Theorem 3.6[12], f (λ ) is a fuzzy HX subring of 2.

There exist X, Y 1 such that f(X), f(Y)  2, (f (λ)) (f(X) f(Y)) = (f (λ)) (f(YX)) , = λ (YX) = λ (XY) = (f (λ)) (f(XY)) = (f (λ)) (f(Y)f(X))  (f (λ )) (f(X)f(Y)) = (f (λ)) (f(Y)f(X)).  Hence, f (λ ) is a fuzzy normal HX subring of 2.

Theorem Let 1 and 2 be any two HX rings on R1 and R2 respectively. Let f: 1  2 be an anti homomorphism on HX rings. Let  be a fuzzy normal HX subring of 2 then f –1() is a fuzzy normal HX subring of 1.

Proof Let  be a fuzzy subset of R2 and  be a fuzzy normal HX subring of 2 .

By Theorem 3.7 [12], f –1 () is a fuzzy HX subring of 1.

For any X,Y  1, f(X) , f(Y)  2, (f –1()) (XY) =  (f(XY)) =  (f(Y) f(X)) =  (f(X) f(Y)) =  (f(YX)) = (f –1()) (YX) –1  (f ( )) (XY) = (f –1()) (YX) –1  Hence, f ( ) is a fuzzy normal HX subring of 1.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

V. ANTI-FUZZY NORMAL HX RING In this section we define the concept of anti-fuzzy normal HX ring and discuss some related results.

Definition Let R be a ring. Let μ be a fuzzy set defined on R. Let   2R – {} be a HX ring. A fuzzy HX subring  of  is called an antifuzzy normal HX ring on  or an anti-fuzzy ring induced by μ if the following conditions are satisfied. For all A , B ,  (AB) =  (BA), where  (A) = min { μ(x)/for all xA  R }.

Theorem If μ is an anti-fuzzy normal subring of a ring R then the fuzzy subset  is an anti-fuzzy normal HX subring of a HX ring .

Proof Let μ be an anti-fuzzy normal subring of R. That is, μ(xy) = μ(yx), for all x,y R.

By Theorem 3.6 [11] ,  is an anti-fuzzy normal HX subring of a HX ring .

For all A, B ,  (AB) = min{μ(xy) / for all xy AB  R } = min{μ(yx) / for all xy AB  R } =  (BA)  (AB) =  (BA).

Hence,  is an anti-fuzzy normal HX subring of a HX ring .

Theorem Let μ and  be any two fuzzy sets of R. Let   2R – {} be a HX ring. Let  and  be any two fuzzy normal HX subrings of a HX ring  then the intersection of two anti-fuzzy normal HX subrings,    is also an anti-fuzzy normal HX subring of a HX ring .

Proof It is clear.

Remark  

The intersection of family of anti-fuzzy normal HX subrings of a HX ring  is also an anti-fuzzy HX subring of .

Let R be a ring. Let μ and  are fuzzy sets of R and μ   is also a fuzzy subset of R then φ   is an anti-fuzzy normal HX subring of  induced by μ   of R. Theorem

Let R be a ring. Let μ and  are fuzzy sets of R. If  ,  , φ   are anti-fuzzy normal HX subrings of a HX ring  induced by μ ,  , μ   of R then φ   =    .

Proof It is clear.

Theorem Let μ and  are any two fuzzy sets of R. Let   2R – {} be a HX ring. If  and  be any two anti-fuzzy normal HX subrings of  then    is also an anti-fuzzy normal HX subring of .

Proof It is clear.

Remark  

Union of family of anti-fuzzy normal HX subrings of a HX ring  is also an anti-fuzzy normal HX subring of .

Let R be a ring. Let μ and  be fuzzy sets of R then φ   is an anti-fuzzy normal HX subring of  induced by μ   of R. Theorem

Let R be a ring. Let μ and  be fuzzy sets of R. If  ,  , φ   are anti-fuzzy normal HX subrings of a HX ring  induced by μ ,  , μ   of R then φ   =   .

Proof It is clear.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

Theorem Let  and  be anti-fuzzy normal HX subrings of HX rings 1 and 2 respectively then the cartesian anti-product of  and  denoted as  ×  is also an anti-fuzzy normal HX subring of 1 × 2.

Proof Let μ and  be fuzzy sets of R. Let  and  be anti-fuzzy normal HX subrings of HX rings 1 and 2 respectively.

By Theorem 4.2[11],  ×  is an anti-fuzzy HX subring of 1 × 2.

Let (A, B), (C, D)  1 × 2. ( ×  ) ((A, B)·(C ,D)) = (×  ) (AC, BD) = max{  (AC),  (BD)} = max{ (CA),  (DB)} = ( ×  ) (CA, DB) = ( ×  ) ((C, D) · (A , B)) ( ×  ) ((A, B)·(C ,D)) = ( ×  ) ((C, D) · (A , B)).

Hence,  ×  is also an anti-fuzzy normal HX subring of a HX ring .

Theorem Let  and  be fuzzy subsets of the HX rings 1 and 2 respectively, such that  (A )  ( Q1) for all A  1, Q1 being the identity element of 2 .If ( ×  ) is an anti-fuzzy normal HX subring of 1 × 2 then  is an anti-fuzzy normal HX subring of 1.

Proof Let ( ×  ) is an anti-fuzzy normal HX subring of 1 × 2.

By Theorem 4.3[11],  is an anti-fuzzy HX subring of 1.

Given,  (A)  (Q1) for all A  1.  (AB) = max { (AB),  (Q1Q1)} = ( × ) ((A, Q1 ) · (B , Q1)) = ( × ) ( (B , Q1) · (A, Q1 )) = max { (BA),  (Q1Q1)} =  (BA)  (AB) =  (BA) Hence,  is an anti-fuzzy normal HX subring of 1.

Theorem Let  and  be fuzzy subsets of the HX rings 1 and 2 respectively , such that  (A )   (Q ) for all A  2 , Q being the identity element of 1. If  ×  is an anti-fuzzy normal HX sub ring of 1 × 2 then  is an anti-fuzzy normal HX subring of 2.

Proof Let  ×  be an anti-fuzzy normal HX subring of 1 × 2.

By Theorem 4.4[11],  is an anti-fuzzy HX subring of 2.

Let A, B  2.

Given  (A)   (Q) for all A  2  (AB) = max {(QQ),  (AB)} = (×) ((Q, A) · (Q, B)) = (×) ((Q, B) · (Q, A)) = max {(QQ), (BA)}  (AB) =  (BA)}.

Hence,  is an anti-fuzzy normal HX subring of 2.

Theorem Let  be a fuzzy set defined on R. Let λμ be a fuzzy normal HX subring of  if and only if (λμ)c is an anti-fuzzy normal HX subring of .

Proof It is clear.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

VI. HOMOMORPHISM AND ANTI-HOMOMORPHISM OF AN ANTI-FUZZY NORMAL HX SUBRING OF A HX RING In this section, we introduce the concept of an anti-image, anti-pre-image of an anti-fuzzy normal HX subring of a HX ring and discussed its properties under homomorphism and anti-homomorphism.

Theorem Let 1 and 2 be any two HX rings on the rings R1 and R2 respectively. Let f: 1  2 be a homomorphism onto HX rings. Let λ be an anti-fuzzy normal HX subring of 1 then f ( λ ) is an anti-fuzzy normal HX subring of 2, if λ has an infimum property and λ is f-invariant.

Proof Let  be a fuzzy subset of R1 and λ is an anti-fuzzy HX subring of 1.

There exist X, Y 1 such that f(X) , f(Y)  2, (f (λ)) (f(X) – f(Y)) = (f (λ)) (f(X–Y)), = λ (X–Y)  max {λ (X), λ (Y)} = max {(f (λ)) (f(X)), (f (λ)) (f(Y))} (f (λ)) (f(X) – f(Y))  max {(f (λ)) (f(X)), (f (λ)) (f(Y))}. ((f (λ)) (f(X) f(Y)) = (f (λ)) (f(XY)) , = λ (XY)  max {λ (X), λ (Y)} = max {(f (λ)) (f(X)), (f (λ)) (f(Y))} (f (λ)) (f(X)f(Y))  max {(f (λ)) (f(X) , (f (λ)) (f(Y))}.

Hence, f (λ) is an anti-fuzzy HX subring of 2.

Also (f (λ)) (f(X) f(Y)) = (f (λ)) (f(XY)) , = λ (XY) = λ (YX) = (f (λ)) (f(YX)) (f (λ)) (f(X)f(Y)) = (f (λ)) (f(Y)f(X)).

Hence, f (λ) is an anti-fuzzy normal HX subring of 2.

Theorem Let 1 and 2 be any two HX rings on R1 and R2 respectively. Let f: 1  2 be a homomorphism on HX rings. Let  be an anti-fuzzy normal HX subring of 2 then f –1 () is an anti-fuzzy normal HX subring of 1.

Proof Let  be a fuzzy subset of R2 and  be an anti-fuzzy HX subring of 2 .

For any X,Y  1, f(X) , f(Y)  2, (f –1()) (X–Y) =  (f(X–Y)) =  (f(X) – f(Y)),  max { (f(X)) ,  (f(Y))} = max {(f –1()) (X) , (f –1()) (Y)} –1 (f ()) (X–Y)  max{( f –1()) (X) , (f –1()) (Y)}. (f –1()) (XY) =  (f(XY)) =  (f(X) f(Y))  max { (f(X)) ,  (f(Y))} = max {(f –1()) (X), (f –1()) (Y)} (f –1()) (XY)  max{(f –1()) (X) , (f –1()) (Y)}.

Hence, f –1() is an anti-fuzzy HX subring of 1.

Also, (f –1()) (XY) =  (f(XY)) =  (f(X) f(Y)) =  (f(Y) f(X)) =  (f(YX)) = f –1()) (YX) (f –1()) (XY) = f –1()) (YX).

–1 Hence, f () is an anti-fuzzy normal HX subring of 1.

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Fuzzy and Anti-Fuzzy Normal HX Subring of A HX Ring (IJIRST/ Volume 3 / Issue 04/ 057)

Theorem Let 1 and 2 be any two HX rings on the rings R1 and R2 respectively. Let f: 1  2 be an anti-homomorphism onto HX rings.

Let λ be an anti-fuzzy normal HX subring of 1 then f ( λ ) is an anti-fuzzy normal HX subring of 2 , if λ has an infimum property and λ is f-invariant.

Proof Let  be a fuzzy subset of R1 and λ is an anti-fuzzy HX subring of 1, then There exist X, Y  1 such that f(X), f (Y)  2 (f (λ)) (f(X) – f(Y)) = (f (λ)) (f(Y–X)) , = λ (Y–X)  max {λ (Y), λ (X)} = max {(f (λ)) (f(Y)), (f (λ)) (f(X))} (f (λ)) (f(X) – f(Y))  max {(f (λ)) (f(X)), (f (λ)) (f(Y))}. (f (λ)) (f(X) f(Y)) = (f (λ)) (f(YX)), = λ (YX)  max {λ (Y), λ (X)} = max {(f (λ)) (f(Y)), (f (λ)) (f(X))} (f (λ)) (f(X)f(Y))  max {(f (λ)) (f(X)) , (f (λ)) (f(Y))}.

Hence, f (λ) is an anti-fuzzy HX subring of 2.

Also (f (λ)) (f(X) f(Y)) = (f (λ)) (f(YX)) , = λ (YX) = λ (XY) = (f (λ)) (f(XY)) (f (λ)) (f(X)f(Y)) = (f (λ)) (f(Y)f(X)).

Hence, f (λ) is an anti-fuzzy normal HX subring of 2.

Theorem Let 1 and 2 be any two HX rings on R1 and R2 respectively. Let f: 1  2 be an anti-homomorphism on HX rings. Let  be an anti-fuzzy normal HX subring of 2 then f–1 () is an anti-fuzzy normal HX subring of 1.

Proof Let  be a fuzzy subset of R2 and  be an anti-fuzzy HX subring of 2 .

For any X,Y  1, then f (X ) , f ( Y )  2 (f –1()) (X–Y) =  (f(Y–X)) =  (f(Y) – f(X)),  max { (f(Y)),  (f(X))}  max {(f –1()) (Y), (f –1()) (X)} –1 (f ()) (X–Y)  max{( f –1()) (X), (f –1()) (Y)}. (f –1()) (XY) =  (f(YX)) =  (f(Y) f(X))  max { (f(Y)),  (f(X))}  min {(f –1()) (Y), (f –1()) (X)} –1 (f ()) (XY)  max{( f –1()) (Y), (f –1() )(X)}.

–1 Therefore, f () is an anti-fuzzy HX subring of 1.

Also, (f –1()) (XY) =  (f(XY)) =  (f(Y) f(X)) =  (f(X) f(Y)) =  (f(YX)) = f –1()) (YX) (f –1()) (XY) = f –1()) (YX).

–1 Hence, f () is an anti-fuzzy normal HX subring of 1.

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