176

ANNEXURE-I

ON d- ALGEBRA

Introduction: Y.Imai and K.Iseki [1996] introduced two classes of abstract algebras.

BCL-algebra and BCI – algebra. It is known that the notion of BCI – algebra. It is known that

the notion of BCI – algebra is a generalization of BCK-algebras. J. Neggers and H.S.Kim [1909]

introduced the class of d-algebras which is another generalization of BCK-algebras and

investigated relations between d-algebras and BCK-algebras. we investigate some interesting

results on d-algebras.

Section II: Preliminaries

Definition 2.1: A non empty set X with constant 0 and a binary operation * is called a

d-algebra if ,

i. x* x = 0

ii. 0* x = 0

iii. x* y = 0 and y* x = 0 →x = y for all x, y in X

Definition 2.2: Let X be a d-algebra and I be a subset of X. Then I is called d-ideal of X if

i. 0 I

ii. x* y I and Y I → x I

iii. x I and y x → x* y I

Proposition 2.3: The intersection of two d-algebra is a d-algebra with respect to * and Δ .

Proof: Let (X, *) and (Y, Δ) be two d-algebras. Here 0 X and 01 Y.

i. Let x X ∩Y. Thus x X and x Y. So x* x = 0 and x Δ x = 0

ii. Let x X ∩ Y and 0 x. Thus x X and x Y and 0* x = 0 and 0 Δ x = 0.

iii. Let x, y X ∩ Y implies x, y X and x, y Y, and x * y = 0 = y * .

Since X is a d-algebra, we have x = y. x Δ y = 0 and y Δ x = 0 →x = y (Y is d- algebra ).

X ∩ Y forms a d-algebra with respect to * as well as Δ.

177

Proposition 2.4: The union two d-algebra’s (X, *) and (Y, *) form a d-algebra with respect to *

if one contained in other.

Proof: Let X and Y be two d-algebra’s. Here 0 X, 01 Y

i. Let x X Y implies x X or y Y and so x* x = 0 or 0* x=0→ x* x = 0.

ii. Let x X Y → x X or y Y → 0 * x = 0 or 0 * x = 0 →0 * x = 0.

iii. Let x, y X Y → x, y X or x, y Y since x y or y x

(or) x * y = 0 , y * x = 0 → x = y where x, y X and y Y

union of two d-algebras is a d-algebra with respect to *

Proposition 2.5: Kernel of a d-algebra homomorphism forms X into Y is a d-ideal of X.

Proof: f: (X, *) → (Y, Δ) be a homomorphism and define f(x * y) = f (x) Δ f(y).

For all x, y X, kernel of I = {0 x / f(0) = 01) .

i. f(0) = f(0* 0) = f(0) Δ f(0)

= 01 Δ 0

1

= 01

ii. Let x* y I ; y I claim x I

f(x* y) = 01 and f(y) = 0

1

f(x) Δ f(y) = 0

1

f(x) Δ 01 = 0

1

It follows that 01 Δ f(x) = 0

1 by (ii) of d-algebra; f(x) = 0

1 by (iii) of d-algebra, x I

iii. Let x I ; y X→ f(x) = 01 → I x → x X ; y Y → x * y X.

f(x* y) = f(x) Δ f(y) = 01 Δ f(y) = 0

1 by (ii) of d-algebra → x * y I.

I is a d-ideal of d-algebra X.

Definition 2.6 : Let (X,*) be a d-algebra and (I,*) be a d-ideal of X. Then {X / I} =

{a*I ; a X} and a* I = {a * x ; x I} , 0 a * I. It follows that (a* I) * (b* I) = I * (a* b).

Proposition 2.7 : Let (x,*) be a d-algebra and (I,*) be an d-ideal of X then X / I forms

A d-algebra.

178

Proof: Let a* I X / I. (a* I) * (a* I) =I * (a * a) =I * 0 (a* a = 0) = 01.

ii. 01* a = (0* a) * (a* I)

= I*(0* a) (since 0*a=0)

= I* 0

= 0

iii. (a* I) * (b* I) = 01 and (b* I) * (a* I) = 0

Now (a* b) * I = 0 * I=0 and

(b * a) * I = 0 * I=0.

(a * b) * x = 0 for all x I

(a * b) * 0 = 0 (0 I).

It gives that 0* (a * b) = 0 ; a * b X ; 0 Y

a * b = 0 and b * a = 0 → a = b.

Now a* I = b * I. Therefore a1 = b

1. X / I is a d-algebra.

Proposition 2.8 Every image of a homomorphic d-algebra is a d-algebra.

Proof: Let (X,*) be a d-algebra and f : X → Y be one to one homomorphism’s, f(0) = 0

Claim : f(x) is a d-algebra with respect to Δ

i. Let y f (x), there exists an element x X, such that f(x) = y.

But X is a d-algebra, x* x = 0

f(x* x) = f(0) = 01

f(x) * f(x) = 01

y* y = 0

1 (i) is true for f(x)

ii. Also 01 Δ y = f(0) Δ f(x)

= f(0*x)

= f(0) (X is a d-algebra)

= 01 (ii is true for f(x))

179

iii. Let f(x), f(y) f(X), such that f(x) , f(y) = 01 = f(y) = f(x)

f(x* y) = f(0)

f(y* x) = f(0)

Since f is 1-1, then x * y = 0 and y * x = 0. X is a d-algebra, x=y → f(x) = f(y).

This completes the proof.

MAIN RESULTS:

1. The intersection of d-algebra is a d-algebra.

2. Union of d-algebra is a d-algebra with extra condition.

3. Kernel of a homomorphism on d-algebra is a d-algebra.

4. Image of a homomorphism of a d-algebra into another d-algebra is a d-algebra.

5. If X is a d-algebra and I is a d-ideal then X/I is a d-algebra.

180

ANNEXURE-I1

A FUZZY METRIC TOPOLOGY ON THE REAL LINE STRUCTURES

Introduction: Chang [1988] used the ideal of fuzzy set to introduce fuzzy topological space.

The concept of fuzzy metric introduced by Deng.zi [1982]. George and veeraMani [1994]

modified the concept of fuzzy metric space introduced by Kramosil and Michalek [1975].

Recently it was proved that the topology induced by a fuzzy metric space. we investigate the

properties of fuzzy metric space and we extend this into fuzzy real line topological space with

respect to collection of all open balls in (RL, Md, * )

Definition 1.1: For any two points x, y R, the usual distance d defined by

d (x, y) = │x-y │, is called standard metric on R.

Definition 1.2: A function d : X × X → R is called semi metric on X if the

following conditions are satisfied

(a) d (x, y) ≥ 0; x=y implies d (x, y) = 0

(b) d (x, y) = d (x, y)

(c) d (x , z) ≤ d (x, y) + d (y, z) for all x, y, z R.

Definition 1.3: In a fuzzy real line, a map d : RL→ [0,1] by d (f, g) = min {│f(x) – g(x) │ } ,

for all f, g RL.

Definition 1.4: The 3-tuple ( X, M , * ) is said to be a fuzzy metric space if X is an

arbitrary set, * is a continous t-norm and M is a fuzzy set on X2 ×(0, ∞) satisfying the

following conditions

(i) M (x, y, t) > 0

(ii) M(x, y, t ) = 1 if and only if x =y

(iii) M(x, y, t ) = M (y ,x ,t)

(iv) M(x, y, t ) * M (y, z, s ) ≤ M (x, z, t + s )

(v) M(x, y, ∙ ) : (0, ∞ ) → [0,1], for all x, y, z ε X and t, s M.

181

Definition 1.5: A binary operator * : [0,1]× [0,1] → [0,1] is a continuous t- norm

if * satisfies the following conditions

(i) * is associative and commutative

(ii) * is continuous

(iii) a * 1 = a for every a [0,1]

(iv) a * b ≤ c * d whenever a ≤ c and b ≤ d, for a, b, c, d [0,1].

Definition 1.6: Let (X, M, * ) is said to be a fuzzy metric space. The open ball B (x, r, t ) for t >

0 with centre x X and radius 0 < r < 1 is defined as B(x, r, t) = { y X / M (x, y, t ) ≥

1 - r }.

Definition 1.7: In a metric space ( RL, d ), the 3- tuple (RL, Md * ) where Md (x, y, t)=

t/ [ t+d (x,y)] and a * b = ab is a fuzzy symmetric space. This Md is called standard fuzzy metric

induced by d.

Aim: The purpose is to investigate some properties of fuzzy metric space. The standard fuzzy

metric to fuzzy real topological space is extended with respect to collection of open balls in

(RL, Md, * ).

Section II: SOME PROPERTIES OF FUZZY METRIC SPACE

In this section, the properties of fuzzy metric space are discussed.

Proposition 2.1: ( RL, d ) is a metric space.

Proof: Let f, g R. It gives that

(i) d (f, g ) = min { │f(x) – g(x) │ } x ε R

≥ min { any other value }

≥ 0

(ii) d (f, g ) = 0 implies min { │f(x)- g(x) │} = 0

x ε R

182

│f(x)- g(x) │ = 0 for all x in R, and so f(x) - g(x) = 0 for all x in R.

Therefore f(x) = g(x) for all x in R implies f = g.

(iii). d ( f, g ) = min { │f(x) – g(x) │ }

x R

= min { │g(x) – f(x) │ }

x R

= d (g, f ).

(iv) d (f, g ) = min { │f(x)- g(x) │ }

x R

= min { │f(x)-h(x) + h(x) – g(x) │ }

x R

≤ min { │f(x)- h(x) │ + │h(x) – g(x) │ }

x R

≤ min { │ f(x ) – h(x) │ } + min { │ h(x) – g(x) │}

x R

≤ d (f, h) + d (h, g).

(RL, d ) forms a metric space.

Proposition 2.2: Union of open balls in ( RL, md , * ) is also open ball in it.

Proof: B ( f α, rα, tα ) is open ball for all index α.

Claim: B (fα , rα , tα ) is open ball.

α

183

f = min { fα (x) }

α

r = max { rα }

α

t = max { tα }. Let g ε U B (fα, rα, t α ) implies g B (fδ, rδ, tδ ) for some ‘δ’.

α

g B (f δ,rδ ,tδ ) implies g RL ; Md ( fδ, g , tδ ) > 1- rδ. … ( 1 )

So f < f α , t α < t for all α and 1/tα ≥ 1/t for all α.

f - g ≤ fα – g; d (f, g ) ≤ d ( f α, g ); (d (f, g)) / t ≤ (d (f α, g ) ) / t for all α.

(1 + d (f, g )) /t ≤ (1 + d (fα , g ) ) / t for all α.

(t + d(f, g)) /t ≤ (t α + d(fα, g )) / tα for all α

t / t + d (f, g) ≥ tα / t + d ( fα , g ) for all α.

In particular t / t + d(f, g) ≥ t δ / (t δ + d (fδ, g )) ≥ 1-r i

Md (f, g, t ) ≥ Md (fδ, g , tδ ) ≥ 1-r by ( 1 ) .

g is an interior point of R implies g B (f, r, t ).

Proposition 2.3: The intersection of all open balls is also open ball in it.

Proof: B(fi,ri,ti) are open ball for all ‘i’.

Claim: ∩ B(fi, ri ,ti ) is open ball.

i

Let h ∩ B (fi, ri , ti ) therefore h B (f i , r i , ti) for all i = 1,2,…, .n.

f = min { f1,f2,…, fn}

r = max { r1 ,r2 ,…, rn}

184

t = max { t1, t2 ,…, .tn}

h R Md(fi, h ,ti ) > 1-ri for all i.

Md (fi, h, ti) > 1-r for all i.

We know that f < f i for all i.

f – h ≤ fi – h

d (f, h) ≤ d (fi ,h ) for all i. ti ≤ t for all i and 1/ti ≥ 1/t.

(d (f ,h)) /t ≤ (d (fi, h)) / ti

(1+ d (f, h) ) / t ≤ (1 + d(fi , h )) / ti

(t + d (f, h )) / t ≤ (ti + d (fi, h)) / ti

Implies t / t+ d(f, h ≥ ti / ti+ d(fi, h )

Md (f, h, t) ≥ Md ( fi , h, ti ) for all i.

Md(f ,h, t) ≥ 1-r . therefore g B (f, r, t ). g is an interior point of RL.

∩ B (fi, ri, ti ) is an open ball in (RL, Md, * ).

τ L = the collection of all open balls in fuzzy metric space (RL, Md, * ) , where ‘Md’ is the

standard fuzzy metric induced by d.

So τ L = { B(f, r, t ) ; g ε RL and Md (f, g, t ) > 1-r }. Then (RL, τ L ) is a topological space ,

called standard fuzzy real topological space.

Corallory: The metric space ( RL, d ) is complete if and only if the standard fuzzy metric space

(RL, Md, * ) is complete.

1.FUZZY GROUPS

Example 1: Let G = {e,b,c,d} be a klein’s four group and let A be s fuzzy subset of G defined by

A(e) = 1, A(b) = t1, A(c) = t2 and A(d) = t3 such that t3 = t1 t2 and 1 ≥ t1 ≥ t2 ≥ t3. Then it is easily

verified that A is fuzzy group of G under product. But Gt2

A is not a group of G; since bc = d

€Gt2

A = {a, b, c}.

Example -3: Let G be a Klein’s four group. G = {e, f, g, fg}, where f2 = e = g

2 and fg = gf for

0 ≤ i≤ 5.let ti έ [0,1] such that 1 =t0 > t1 > …. . t5. Define fuzzy subset µ and v : G→ [0.1] as

follows. µ(x) = t1, µ(f) = t2 and µ(g) = µ(fg) = t4 and v(x) = t0, v(f) = t1 v(g) = t3 and v(fg) = t5. It

can be seen that µ and v are fuzzy groups of G. Furthermore (µỤv)(0) = t0 , (µỤv)(fg) = t4,

(µỤv)(g) = t2 but µỤv is not a fuzzy group of G. In fact that the union of two equivalent fuzzy

groups is a fuzzy group.

6.FUZZY LEFT IDEAL

Example-1; On a four element hemi ring (R,+,•) defined by the following two tables:

+ 0 1 2 3 • 0 1 2 3

0 0 1 2 3 0 0 0 0 0

1 1 1 2 3 1 0 1 1 1

2 2 2 2 4 2 0 1 1 1

3 3 3 3 2 3 0 1 1 1

Consider a fuzzy set A where µA(0) = 0.4, µA(0) = 0.2 and µA(x) = 0.2, µA(x) = 0.2 for all

x ≠ 0. It is very difficult to verify that A is in Fuzzy left ideal

Example-2: Let N be the set of all non negative integers an let

µ(x) = 1 if x έ {4}

½ if x έ {2}- {4},

0 otherwise, where {n} denotes the set of all non-negative integers divided by n.

Then (N,+,•) is a hemi ring and A is its left h- ideal.

7. FUZZY LEFT H- IDEAL

Example-1: Let S= {a, b, c, d, e} be a semi group with the following cayley table:

• a b c d e

a a a a a a

b a a a a a

c a a c c e

d a a c d e

e a a c c e

Define a fuzzy set A in S by µ(a, q) = 0.6, µ(b, q) = 0.5, µ(c, q) = 0,4, µ(d, q) = µ(e, q) = 0.3. by

routine calculation, we can check that A Q- fuzzy left h- ideals.

Example-3: Let S be the set of natural numbers including 0 and s is a hemi ring with usual

addition and multiplication. Define a fuzzy set µ : S→ [0,1] by

µ(x)= 1 if x is even or 0

0 otherwise. By routine calculations, we know that µ is fuzzy left h- ideals of S.

14. FUZZY MIDDLE COSET

15.PSEUDO FUZZY COSSETS

Example: Let G be the klein four group. Then G = {e, a, b, ab} where a2 = e = b

2, ab = ba and e

the identity element of G. Define µ ; G→ [0,1] as follows

µ(x) = ½ if x =a

1 if x = e

¼ if x = b, ab . Now we calculate the pseudo fuzzy cossets of µ. For the identity

element e of the group G we have (eµ)P = µ.

(aµ)P(x) = ½ if x =a

¼ if x =e

1/8 if x = b, ab

(bµ)P(x)= 1/3 if x = e

1/6 if x =a

1/12 if x = b, ab

((ab)µ)P(x) = ¼ if x = e

1/8 if x =a

1/16 if x = b, ab. It is easy to check that all the above pseudo

fuzzy cossets of µ are fuzzy groups of µ.

16. PSEUDO FUZZY DOUBLE COSSETS

Example-1; Let R = {a,b,c,d} be a set with two binary operations as follows.

+ a b c d • a b c d

A a B c d a a a a a

B b A d e b a a a a

C c D b a c a a a a

d d C a b d a a b b

Then (R.+,•) is a near ring. We define fuzzy set A in R as follows; for every q έ Q µA(a, q) = 1; µA(b, q) =

1/3, µA(c, q) = 0 = µA(d, ) by routine calculation, we can check that A is both Q fuzzy sub near ring and Q-

fuzzy R- subgroup of R