Q- Fuzzy Soft Ring

Q- Fuzzy Soft Ring

Dr. N. Sarala , B. Suganya

Department of Mathematics, Department of Mathematics,

1. college for women(Autonomous), Bharathidasan University constituent College of Nagapattinam, TamilNadu,(India) Arts and science,

   Nagapattinam, Tamilnadu(India)

   Abstract: – In this paper, the study of Q- fuzzy soft ring by combining soft set theory. The notions of Q- fuzzy soft ring as defined and several related properties and structural characteristics are investigated some related properties., then the definition of Q- fuzzy soft ring and the theorem of homomorphic image and homomorphic pre-image are given.

   Keywords: Soft set, Fuzzy soft set, soft ring, Fuzzy soft ring, soft homomorphism, Fuzzy Soft isomorphism,Q-fuzzy set , Q- Fuzzy soft ring

   1. INTRODUCTION

      The concept of soft sets was introduced by Molodtsov [3] in 1999, soft sets theory has been extensively studied by many authors. It is well known that the concept of fuzzy sets, introduced by Zadeh [9], has been extensively applied to many scientific fields.

      In 1971, Rosenfeld [4] applied the concept to the theory of groupoids and groups. Inan et al. have already introduced the definition of fuzzy soft rings and studied some of their basic properties. A.Solairaju and R.Nagarajan[7] have introduced and defined a new algebraic structure called Q-fuzzy subgroups. (Dr.N.sarala and B.Suganya, 2014) presented some properties of fuzzy soft groups . further (Dr.N.sarala and B.Suganya, 2014) introduced on normal fuzzy soft groups.

      In this paper, we study Q-fuzzy soft ring theory by using fuzzy soft sets and studied some of algebraic properties.

   2. PRELIMINARIES

      In this section, we first recall the basic definitions related to fuzzy soft sets which would be used in the sequel.

      Definition2.1

      Suppose that U is an initial universe set and E is a set of parameters, let P(U) denotes the power set of U .A pair (F,E) is called a soft set over U where F is a mapping given by F: E P(U) .

      Clearly, a soft set is a mapping from parameters to P(U), and it is not a set, but a parameterized family of subsets of the Universe.

      Definition 2.2.

      Let U be an initial Universe set and E be the set of parameters. Let A E. A pair (F, A) is called fuzzy soft set over U where F is a mapping given by F: AIU, where IU denotes the collection of all fuzzy subsets of U.

      Definition 2.3

      Let X be a group and (F,A) be a soft set over X. Then (F,A) is said to be a soft group over X iff F(a) < X, for each a A.

      Definition 2.4

      Let X be a group and (f, A) be a fuzzy soft set over X. Then (f, A) is said to be a fuzzy soft group over X iff for each a A and x, y X,

      1. fa(x . y) T (fa(x), fa(y)

      2. fa(x-1) fa(x)

      That is, for each a A, fa is a fuzzy subgroup in Rosenfeld's sense [4]

      Definition 2.5

      Let (f,A) be a soft set over a ring R. Then (f, A) is said to be a soft ring over R if and only if f(a) is sub ring of R for each a A.

      Definition 2.6

      Let R be a soft ring. A fuzzy set in R is called

      fuzzy soft ring in R if

      (i) ((x+y)) T{(x), (y)}

      1. (-x) (x) and

      2. ((xy)) T{(x), (y)} , for all x,y R.

      Definition 2.7

      Let (,): X Y is a fuzzy soft function, if is a homomorphism from x y then (,) is said to be fuzzy soft homomorphism. if is a isomorphism from X Y and is 1-1 mapping from A on to B then (,) is said to be fuzzy soft isomorphism.

   3. Q-FUZZY SOFT RINGS

Definition 3.1:

Let R be a soft ring. A fuzzy set in R is called

Q- fuzzy soft ring in R if

(i) ((x+y) , q) T{(x,q), (y,q)}

1. (-x, q) (x,q) and

2. ((xy) , q) T{(x,q), (y,q)}, for all x,y R.

& qQ

Proposition 3.1:

Every imaginable Q- fuzzy soft ring is a Q- fuzzy soft ring of R.

Proof:

Assume that is imaginable Q- fuzzy soft ring of R, then we have

((x+y) , q) T{(x,q), (y,q)} (-x, q) (x,q) and

((xy) , q) T{(x,q), (y,q)} , for all x,y R.

& qQ

Since is imaginable, we have min{(x,q) ,(y,q)} = T{min{(x,q), (y,q)},

min{x,q) , (y,q) }}

T {(x,q) , (y,q)}

min {(x,q) , (y,q)}

Proposition 3.3:

Let R and R' be two rings and : R R' be a soft homomorphism. If and fa is a Q-fuzzy soft ring of R then the pre-image 1 (fa) Q-fuzzy soft ring of R.

Proof:-

Assume that fa is a Q-fuzzy soft ring of R'. Let x, y R & q Q

(FSR1)

(i) ((x+y),q)) = ((x+y),q))

-1[fa] fa

= ((x,q ), (y,q)) fa

T { ((x,q)), ((y,q))} fa fa

T { (x,q), (y,q)}

-1[fa] -1[fa]

(FSR2)

(ii) (-x,q) = ((-x,q))

-1[fa] fa

((x,q) fa

and so

T{(x,q) ,(y,q)} = min { (x,q) , (y,q) }

It follows that

((x+y), q ) T{(x,q) , (y,q)}

(FSR3)

(x,q)

-1[fa]

x,y R, qQ

= min {(x,q) , (y,q)} for all

(iii) ((xy),q)) = ((xy),q))

-1[fa] fa

Hence is a Q-fuzzy soft ring of R.

Proposition 3.2:

If is Q-fuzzy soft ring R and is an endomorphism of R, then [] is a Q- Fuzzy soft ring of R

Proof:

For any x,y R, we have

(FSR1)

(i) []((x+y),q)) = (((x+y),q))

= ((x,q ),(y,q))

T { ((x,q)),((y,q))}

T { [] (x,q),[](y,q)}

(FSR2)

(ii) [](-x,q) = ((-x,q))

((x,q))

[](x,q)

(FSR3)

(iii) []((xy),q)) = (((xy),q))

= ((x,q ), (y,q))

T {(x,q) , (y,q )}

T { (x,q) , (y,q )}

T { [] (x,q),[](y,q)} Hence [] is a Q-fuzzy soft ring of R.

= ((x,q ), (y,q)) fa

T { ((x,q)), ((y,q))} fa fa

T { (x,q), (y,q) }

-1[fa] -1[fa] Hence 1 (fa) is a Q-fuzzy soft ring of R.

Proposition 3. 4

Let: R R' be an epimorphism and fa be fuzzy soft set in R' . If [fa] is q-fuzzy soft ring of R then fa is q-fuzzy soft ring of R.

Proof :-

Let x,y R, Then there exist a,b R such that

(a) = x , (b) =y. It follows that

(FSR1)

(i) ((x+y),q)) = ((x+y),q))

[fa] fa

(FSR2)

= ((x,q ), (y,q)) fa

T { ((x,q)), ((y,q))} fa fa

T { (x,q), (y,q)}

[fa] [fa]

(FSR1)

(i) f((x1+y1), q) = sup (z,q)

(z,q) f-1((x1+y1),q)

T{(x0,q), (y0,q)

= T{sup (h,q), sup (h,q)} (h,q) f-1(x1,q) , (h,q) f-1(y1,q)

= min {f(x1,q) , f (y1,q)}

(ii) (-x,q) = ((-x,q))

[fa] fa

((x,q)) fa

(x,q)

[fa]

(FSR3)

(iii) ((xy),q)) = ((xy),q))

[fa] fa

= ((x,q ), (y,q)) fa

T { ((x,q)), ((y,q))} fa fa

T { (x,q), (y,q)}

[fa] [fa] Hence [fa] is a Q-fuzzy soft ring of R.

Proposition 3.5:

Onto homomorphic image of a Q-fuzzy soft ring with the sup property is Q-fuzzy soft ring of R.

Proof:

Let f: RR' be an onto homomorphism of Q- fuzzy soft rings and let be a sup property of Q-fuzzy soft ring of R.

Let x1, y1 R1, and x0 f-1(x1), y0 f-1(y1) be such that (x 0,q) = sup (h,q),

(h,q) f-1(x1) and

(y 0,q) = sup (h,q)

(h,q) f-1(y1) Respectively, then we can deduce that

(FSR2)

1. f(-x1,q) = sup (z,q)

   (z,q) f-1( -x1,q)

   (x0 , q)

   sup (,q) (h,q) f-1(x1,q)

   = f(x1,q)

   (FSR3)

2. f((x1y1), q) = sup (z,q)

(z,q) f-1((x1y1),q)

T{(x0,q), (y0,q)}

= T { sup (h,q) , sup (h,q)} (h,q) f-1(x1,q) , (h,q) f-1(y1,q)

= min {f(x1,q) , f (y1,q)} Hence f is a Q-fuzzy soft ring of R1.

Proposition 3. 6:

Let T be a continuous t-norm and Let f be a soft homomorphism on R. If is Q-fuzzy soft of R, then f is Q-fuzzy soft ring of f(R).

Proof:

Let A1 = f -1(y1,q) , A2 = f -1(y2,q) and A12 = f -1((y1+y2), q) where y1,y2 f(R), q Q

Consider the set

A1+ A2 = { x R / (x,q) = (a1,q) + (a2,q) } for some (a1,q)A1 and (a2,q) A2.

If (x,q) A1+A2 , then (x,q) = (x1,q) + (x2,q) for some (x1,q) A1 and (x2,q) A2 so that we have

f (x,q) = f(x1,q)+ f(x2,q)

= y1+ y2

Since (x,q) f-1((y1,q)+ (y2,q)) = A12 .Thus A1+A2

A12

It follows that

(FSR1)

(i) f ((y1+y2), q ) = sup{(x,q)/(x,q)

f-1(y1+ y2,q)}

= sup {(x,q) / (x,q) A12 }

sup {(x,q)/ (x,q) A1+A2}

sup {((x1,q)+(x2,q))/(x1,q) A1 and

(x2,q) A2}

sup {S((x1,q) , (x2,q))/ (x1,q) A1

and (x2,q) A2}

Since T is continuous. For every > 0 , we see that if

sup {(x1,q) / (x1,q)A1} + (x1*, q) } and

sup {(x2,q) / (x2,q) A2} + (x2*,q)}

T{sup{(x1,q)/ (x1,q)A1} ,

sup { (x2,q) / (x2,q)A2 } +T((x1*,q), (x2*,q)

(FSR2)

(FSR3)

*(-x,q) = (-x,q) +1 (0,q)

(x,q) +1- (0,q)

= (x,q)

*((xy),q) = ((xy),q) + 1 (0,q)

T((x,q), (y,q)) +1- (0,q))

T((x,q)+1- (0,q),

((y,q)+1- (0,q))

= T (*(mx,q) , *(my,q)).

Choose (a ,q) A and (a ,q) A

such that

3. CONCLUSION:

1 1 2 2

sup{(x1,q) / (x1,q) A1} + (a1,q) and

sup{(x2,q) / (x2,q) A2} +(a2,q) .

Then we have

T{sup{(x1,q)/(x1,q)A1}, sup{(x2,q)/(x2,q)

A2}+T((a1,q),(ma2,q)

Consequently, we have

f((y1+ y2), q ) sup{ T((x1,q), (x2,q))

/ (x1,q) A1 ,(x2,q) A2}

T(sup{(x1,q) / (x1,q) A1},

sup{(x2,q) / (x2,q) A2}

T {(f(y1,q) , f(y2,q)}

Similarly we can show f(-x,q ) f(x,q) and f(xy,q) T {(f(x,q) , f(y,q)}

Hence f is Q-fuzzy soft ring of f(R).

Proposition3.7:

Let be a Q-fuzzy soft ring R and let * be a Q- fuzzy set in N defined by *(x,q) = (x,q) +1-(0,q) for all x N. Then * is a normal Q-fuzzy subgroup of R

Proof :

In this paper we investigate the notion of Q-fuzzy soft ring. This work focused on Q-fuzzy soft rings of fuzzy soft rings. To extend this work one could study the properties of fuzzy soft sets in other algebraic structure.

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For any x, yR and q Q we have

(FSR1)

*((x+y),q) = ((x+y),q) + 1 (0,q)

T((x,q), (y,q)) +1- (0,q))

T((x,q)+1-(0,q),

((y,q)+1-(0,q))

= T (*(mx,q) , *(my,q)).