On Fuzzy Topological K-Algebras

7/29/2019 On Fuzzy Topological K-Algebras

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International Mathematical Forum, 1, 2006, no. 23, 1113 - 1124

On fuzzy topological K-algebras

M. Akram

University College of Information TechnologyUniversity of the Punjab, Old Campus, Lahore-54000, Pakistan

[email protected]

K. H. Dar

Govt. College University Lahore, Department of MathematicsKatchery Road, Lahore-54000, Pakistan

prof [email protected]

Abstract. In this paper, we introduce the notions of fuzzy topologicalsubalgebras and ideals in K-algebras, and investigate some of their properties.We also discuss the properties of homomorphic image and inverse image offuzzy topological ideals of K-algebras.Keywords: (Fuzzy )continuous maps, (Fuzzy topological )ideals, Hausdorffspaces ,C5-disconnectness, Compactness.Mathematics Subject Classification: 06F35, 94D05.

1 Introduction

K. H. Dar and M. Akram [9] introduced a class of logical algebras: K-algebrason a group G( briefly, K(G)-algebras) using the induced binary operation and have further extended its scope of study in [10] . Fuzzy set was introducedby Zadeh [4]. Since then there have been wide-ranging applications of the fuzzyset theory. Many research workers have fuzzified the various mathematicalstructures, such as topological spaces, functional analysis, loop, group, ring,near ring, vector spaces, automation. In this paper, we introduce the notionsof fuzzy topological subalgebras and ideals in K-algebras, and investigate someof their properties such as fuzzy Hausdorff spaces, fuzzy C5-disconnectness andfuzzy compactness. We also discuss the properties of homomorphic image andinverse image of fuzzy topological ideals of K-algebras.

2 Preliminaries

In this section, we review some definitions and properties that will be used inthe sequel:


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1114 M. Akram and K. H. Dar

Definition 2.1. [9] Let (G, , e) be a group with the identity e such that x2 = efor some x(= e) G. A K-algebra is a structure K = (G, , , e), where isa binary operation on G which is induced from the operation , that satisfiesthe following:

(k1) (a,x,y G) ((a x) (a y) = (a (y1 x

1)) a),

(k2) (a, x G) (a (a x) = (a x1) a),

(k3) (a G) (a a = e),

(k4) (a G) (a e = a),

(k5) (a G) (e a = a1).

If G is abelian , then conditions (k1) and (k2) can be written as follows:

(k1

) (a,x,y G) ((a x) (a y) = y x),(k2) (a, x G) (a (a x) = x),

respectively.

Definition 2.2. [9] A nonempty subset H of a K-algebra K is called a subal-gebra of K if it satisfies:

(i) e H,

(ii) (a, b H) (a b H).

Definition 2.3. Let K1 = (G1, , , e1) and K2 = (G2, , , e2) be K-algebras.A mapping f : K1 K2 of K-algebras is called a homomorphismif f(x y) = f(x) f(y), x, y K1.

Note that if f is a homomorphism, then f(e1) = e2.

Definition 2.4. A nonempty subset A of a K-algebra K is called an ideal ofK if,

(i) e A,

(ii) (y K)(x A) (x y) A and y (y x) A x A).Definition 2.5. [1] Let X be a non-empty set. A fuzzy (sub)set A in X ischaracterized by a membership function A : X [0, 1].

Definition 2.6. [1] Let A and B be two fuzzy sets in X, then following op-erations are valid :


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On fuzzy topologicalK-algebras 1115

(a) (x X)(A B A(x) B(x)),

(b) (x X)(A = B A(x) = B(x)) ,

(c) (x X)(A B = min (A(x), B(x))),

(d) (x X)(A B = max (A(x), B(x))).Definition 2.7. The fuzzy sets X and 1X in X are defined byX = {x X : (x) = 0} and 1X = {x X : (x) = 1} respectively.

Definition 2.8. A fuzzy topology on a set X is a family of fuzzy sets in Xwhich satisfies the following conditions:

(i) X, 1X ,

(ii) If A1, A2 , then A1 A2 ,

(iii) If Ai for all i I, then iIAi .

The pair (X, ) is called a fuzzy topological space(FTS) and members of arecalled open fuzzy sets( OFSs), and the complement of fuzzy open sets are closed

fuzzy sets(CFSs).

Definition 2.9. [5] Let f be a mapping from a set X into set Y.

(a) Let B be a fuzzy set in Y with membership function B. The inverseimage of B, denoted by f1(B), is the fuzzy set in X with membershipfunction f1(B) defined by f1(B)(x) = B(f(x)) for all x X.

(b) Let A be a fuzzy set in X with membership function A. The image ofA, is denoted by f(A), is the fuzzy set in Y with membership function

f(A) such that

fsup(A)(y) =

supxf1(y) A(x), if f

1(y) = ,

0, otherwise .

Proposition 2.10. Let A (Ai)be a fuzzy set with membership function A(Ai) in X and B be a fuzzy set with membership function B in Y. Letf : X Y be a function. Then

(a) If f is surjective , then f(f1(B)) = B.

(b) f(X) = Y.

(c) f(1X) = 1Y, if f is surjective.

(d) f1(1Y) = 1X.

(e) f1(Y) = X.

(f) f(Ai) = f(Ai).


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3 Fuzzy topological subalgebras

Definition 3.1. A fuzzy topology on a K-algebra K is said to be an indis-crete fuzzy topology if its only elements are empty fuzzy set (K) and wholefuzzy set (1K). A fuzzy topology on a K-algebra K is said to be a discrete

fuzzy topology if it contains all fuzzy subsets of K.

Definition 3.2. [?] A fuzzy set A in a K-algebra K with membership functionA is called a fuzzy subalgebra of K if A(x y) min{A(x), A(y)}, for allx, y K.

Example 3.3. [9] Consider the K-algebra K = (G, , , e), whereG = {e,a,a2, a3, a4} is the cyclic group of order 5 and is given by thefollowing Cayley table:

e a a2 a3 a4

e e a4 a3 a2 a

a a e a4 a3 a2a2 a2 a e a4 e3

a3 a3 a2 a e e4

a4 a4 a3 a2 a e

(a) If we define a fuzzy set as follows:

A =< x, (e

0.5,

a

0.4,

a2

0.3,

a3

0.3,

a4

0.2) >

B =< x, (e

0.6,

a

0.4,

a2

0.5,

a3

0.5,

a4

0.3) >

Then the family {K, 1K, A , B} of fuzzy sets in K is a fuzzy topologyon K because the empty fuzzy set K and the whole fuzzy set 1K are in, and the intersection of any two members of is a member of , andarbitrary union of members of is a member of .

(b) Define a fuzzy set A in K with membership function A defined byA(e) = 0.8 and A(x) = 0.02 for all x = e in K. It is easy to check thatA is a fuzzy subalgebra of K.

Definition 3.4. Let (F1, F1) and (F2, F2

) be fuzzy subspaces of fuzzy topo-

logical spaces (K1, ) and (K2, ) respectively, and let f be a mapping from(K1, ) to (K2, ). Then f is a mapping of (F1, F1) into (F2,

F2

) iff(F1) F2.Furthermore, f is relatively fuzzy continuous if for each open fuzzy set VF2 inF2, the intersection f

1(VF2) F1 is in F1.Moreover, f is relatively fuzzy open if for each open fuzzy set UF1 in F1, theimage f(UF1) is in

.


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Theorem 3.5. Let K1 and K2 be K-algebras and let (F, F) and (G, G) befuzzy subspaces of (K1, ) and(K2, ) respectively. Letf be a fuzzy continuousmapping of K1 into K2 such that f(F1) F2. Then f is relatively fuzzycontinuous mapping of F1 into F2.

Proof. Let VF2 be a fuzzy set in F2 , then there exists V

such thatVF2 = V F2. Since f is fuzzy continuous, it follows that f

1(V) is a fuzzyset in .Hence f1(VF2) F1= f

1(V F2) F1= f1(V) f1(F2) F1=f1(V) F1is a fuzzy set in F1. This completes the proof.

Definition 3.6. Let 1 and 2 be fuzzy topologies on K-algebras K1 and K2respectively and A be a fuzzy set with membership function A. A functionf : (K1, 1) (K2, 2) is said to be a fuzzy continuous map from (K1, 1) to(K2, 2) if it satisfies following conditions:

(i) For every A 2, f1

(A) 1,

(ii) For every fuzzy subalgebras A(ofK2) in 2, f1(A) is a fuzzy subalgebra

(of K1) in 1.

Theorem 3.7. If1 is a fuzzy topology on the K-algebraK1 and2 is an indis-crete fuzzy topology on the K-algebras K2, then every function f : (K1, 1) (K2, 2) is a fuzzy continuous map.

Proof. Since 2 is an indiscrete fuzzy topology, therefore, 2= {K2, 1K2}.Let f : (K1, 1) (K2, 2) be any mapping of K-algebras. We see that everymember of 2 is a fuzzy subalgebra of K-algebra K2. So it is enough to prove

that for every A 2, f1(A) 1.Let K2 2, then for any x K1,

f1(K2)(x) = K2(f(x))

= 0 [as f(x) K2]

= K1(x) [by definition of empty fuzzy set].

Thus (f1(K2)) = K1 1.On the other hand, if 1K2 2 and x K1, then we have

(f

1

(1K2))(x) = 1K2(f(x))= 1 [as f(x) K2]

= 1K1(x) [by definition of whole fuzzy set].

Thus (f1(1K2)) = 1K1 1. Hence f is a fuzzy continuous map which maps(K1, 1) into (K2, 2).


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Theorem 3.8. Let 1 and 2 be any two discrete fuzzy topologies on the K-algebras K1 and K2 respectively. Then every homomorphism f : (K1, 1) (K2, 2) is a fuzzy continuous map.

Proof. Since 1 and 2 are discrete fuzzy topologies on K-algebras K1 and K2

respectively, for every A 2, f1

(A) 1 [Note that f is not the usual inversehomomorphism from K2 to K1].Let A be a fuzzy subalgebra in 2 with membership function A, thenfor any x, y K1, we have

(f1(A))(x y) = A(f(x y))

= A(f(x) f(y)) [since f is homomorphism]

min{A(f(x)), A(f(y))} [for A is a fuzzy subalgebra of K2]

= min{(f1(A))(x), (f1(A)(y)}.

This shows that f1(A) is a fuzzy subalgebra (of K1) in 1 and hence f is afuzzy continuous map from K1 to K2.

Theorem 3.9. Let1 and2 be two fuzzy topologies defined on the K-algebrasK1 and K2, respectively. Then every homomorphism f : (K1, 1) (K2, 2)need not in general be a fuzzy continuous map.

Proof. To prove this theorem it is sufficient if we prove the result to be falsefor a particular 1 and 2 defined on any K-algebra K as in our definition of afuzzy continuous map we have not assumed K1 and K2 to be distinct.Let K be any K-algebra. Define two fuzzy topologies 1 and 2 on K-algebra

K as 1 = {0K, 1K, } and 2 = {0K, 1K, }, where , : K [0, 1] defined asfollows:

(x) =

1 if x = e

0 x = e

and

(x) =

1 if x = e

0 x = e

where e is identity of a K-algebra. Define f : (K, 1) (K, 2) by f(x) = xfor all x K. Clearly, f is a homomorphism. For x K and 2, we have

(f1())(x) = (f(x)) = (x)

This gives (f1())(x) = (x), for all x K. That is f1() = . Thusf1() / 1, as / 1. Hence f is not a fuzzy continuous map on K.


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Definition 3.10. Let (K1, 1) and (K2, 2) be any two fuzzy topological spaces.A function f : (K1, 1) (K2, 2) is said to be a fuzzy homomorphismif it sat-isfies the following conditions:

f is bijective,

both f and f1 are fuzzy continuous maps.

Definition 3.11. A fuzzy topological space (K, ) is said to be a fuzzy Haus-dorff space if and only if for any two distinct fuzzy points x and y there existopen fuzzy sets F and G such that F G = K.

Theorem 3.12. Let 1 and 2 be the topologies on K-algebras K1 and K2respectively and let f : K1 K2 be a fuzzy homeomorphism. Then K1 is a

fuzzy Hausdorff space if and only if K2 is a fuzzy Hausdorff space.

Proof. Suppose that K1 is a fuzzy Hausdorff space. Let xt and xs be the fuzzypoints in 2 with x = y(x, y K1), then f1(x) = f1(y), as f is one to one.

For z X, we consider

(f1(x1))(z) = x1(f(z)) =

t (0, 1], if f(z)=x;

0, if f(z)= x.

=

t (0, 1], if z = f1(x);

0, if z = f1(x).

= (f1(x))1(z).

That is, (f1(xt))(z) = (f1(x))t(z), for all z K1.

Hence, we have f1(xt) = (f1(x))t.Similarly we can prove that f1(xs) = (f

1(x))s.By definition of a fuzzy Hausdorff space, there exist open fuzzy sets Fx andGy of f

1(xt) and f1(xs) respectively such that Fx Gy = K1. Since f is a

fuzzy continuous map from K1 to K2 and f1 is a fuzzy continuous map fromK2 to K1, there exist open fuzzy sets f(Fx) and f(Gy) of xt and ys respectivelysuch that f(Fx) f(Gy) = f(Fx Gy) = f(K1) = K2. Hence K2 is a fuzzyHausdorff space.Conversely, let (K2, 2) be a fuzzy Hausdorff space. By a similar argument andby also using the fact that both f and f1 are fuzzy continuous maps we can

prove that (K1, 1) is a fuzzy Hausdorff space. The proof is now completed.Definition 3.13. Let be a fuzzy topology on a K-algebra K. A FTS (K, )is said to be a fuzzy C5- disconnected if there exists a fuzzy open and fuzzyclosed set F with membership F such that F = 1K and F = K.A FTS (K, ) is said to be a fuzzy C5- connected if it is not a fuzzy C5-disconnected.


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Theorem 3.14. Let1 and2 be the fuzzy topologies onK(G)-algebrasK1 andK2 respectively. Let f : K1 K2 be a fuzzy continuous surjective mapping. If(K1, 1) is a fuzzy C5-connected, then (K2, 2) is a fuzzy C5-connected.

Proof. Assume that (K2, 2) is a fuzzy C5-disconnected. Then there exist a

fuzzy open and closed set F with membership function F such that F = 1K2and F = K2. Since f is a fuzzy continuous mapping, f1(F) is both OFS

and CFS.Thus f1(F)= 1K1 or f

1(F)= K1 which is impossible.[since F = f(f1(F)) =

f(1K1) = 1K2, and F = f(f1(F)) = f(K1) = K2.]

This is contradiction to our assumption. Hence (K2, 2) is also a fuzzy C5-connected.

Definition 3.15. Let be a fuzzy topology on a K-algebra K and F be afuzzy set in K with membership function F. If a class {< x, Fi} >: i I} ofOFS in K satisfies the condition F {< x, Fi >: i I} , then it is called

a fuzzy open cover of F.A finite subclass of the fuzzy open cover {< x,Fi >: i I} of F, which isalso a fuzzy open cover of F, is called a finite subcover of {< x, Fi >: i I}.A FS F =< x, F > in a FTS (X, ) is called a fuzzy compact, if every fuzzyopen cover of F has a finite subcover.

Theorem 3.16. Let 1 and 2 be the fuzzy topologies on K-algebras K1 andK2 respectively. Let f : K1 K2 be a fuzzy continuous mapping. If F is a

fuzzy compact in (K1, 1), then f(F) is a fuzzy compact in (K2, 2).

Proof. Let A= {Fi : i I}, where Fi =< y, Fi > be a fuzzy open cover off(F). Then B= {f1(Fi) : i I} is a fuzzy open cover of F. Since F is a

fuzzy compact, there exists an finite subcover Fi(i = 1, 2, , n) of F suchthat F ni=1f

1(Fi). Thus

f(F) f(ni=1f1(Fi))

f(F) ni=1f(f1(Fi))

f(F) ni=1Fi

follows. Hence f(F) is a fuzzy compact in (K2, 2).

Now we obtain a Theorem of relative fuzzy homomorphism inspired by [5].Theorem 3.17. Let K be a K-algebra and let be a fuzzy topology on K-algebra K. Let A be a fuzzy topological algebra in K. Then the inversion mapf : A A defined by f(x) = x1 and the inner automorphism h : A Adefined by h(g) = aga1 are all relative fuzzy homomorphisms, where a {x :A(x) = A(e)}.


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Proof. Clearly f is one-to-one. Since

f(A)(y) = supzf1(y)

A(z) = A(y1) = A(y) y A.

That is, f(A) = A. Since f1(x) = x1 is relatively fuzzy continuous, f is

relatively fuzzy open. Thus f is a relative fuzzy homomorphism. Let ra :A A be a right translation defined by ra(x) = xa and la : A A be lefttranslation defined by la(x) = ax. Then

(ra(A))(x) = supzr1a (x)

A(z) = A(xa1)

min(A(x), A(a1)) = min(A(x), A(e))

= A(x) = A(xa1a)

min(A(xa1), A(a))

= A(xa1) = (ra(A))(x).

Thus ra(A) = A. Let : A A A be a map defined by (x) = (x, a)and : A A A be a map defined by (x, y) = xy. Then ra = .Since and are fuzzy continuous, ra is fuzzy continuous. Since r

1a = ra1,

ra is a fuzzy homomorphism. Similarly, la is a fuzzy homomorphism. Sinceh is a composition of ra1 and la, h is relatively fuzzy homomorphism. Thiscompletes the proof.

4 Fuzzy topological ideals

Definition 4.1.A fuzzy set A in a K-algebra K with membership functionA is called a fuzzy ideals of K if it satisfies:

(i) (x G) (A(e) A(x)),

(ii) (x, y G) (A(x) min{A(x y, A(y (y x))}).

Example 4.2. [9] Consider the K(S3)-algebra K = (S3, , , e) on the sym-metric group S3 = {e,a,b,x,y,z} where e = (1), a = (123), b = (132),x = (12), y = (13), z = (23), and is given by the following Cayley ta-ble:

e x y z a b

e e x y z b ax x e a b z yy y b e a x z z z a b e y xa a z x y e bb b y z x a e


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Define a fuzzy set A in K with membership function A by A(e) = t1, A(p) =t2, for all p = e, where t1 > t2 in [0, 1]. It is easy to check that A is a fuzzyideal of K.

Definition 4.3. Let K be a K-algebra and a fuzzy topology on K. Let A

be a fuzzy K-algebra with induced fuzzy topology A. Then A is called a fuzzytopological K-algebra if for each a K the self mapping ar : (A, A) (A, A)defined by

ra(g) = g a g K, (1)

is relatively fuzzy continuous.

Theorem 4.4. Letf : K1 K2 be a homomorphism ofK-algebras. Let and be the fuzzy topologies on K1 and K2 respectively such that = f1().If B is a fuzzy topological ideal of K2 with membership function B. Thenf1(B) is a fuzzy topological ideal of K1 with membership function f1(B).

Proof. It is easy to show that f1(B)(e) f1(B)(x), for all x K1.For any x, y K1, we have

f1(B)(x) = B(f(x))

min{B(f(x y)), B(f(y (y x)))}

min{B(f(x) f(y)), B(f(y) f(y x)))}

= min{f1(B)(x y), f1(B)(y (y x))}.

Hence f1(B) is a fuzzy ideal of K1.Let F1 be an open fuzzy set in f1(B) on f

1(B). Since f is a fuzzy continuous

mapping of (K1, ) into (K2, ), f is a relatively fuzzy continuous mapping of(f1(B), f1(B)) into (B,

B). If there exists open fuzzy set F2

B such that

f1(F2) = F1 (2)

Then

r1a (F1)(g) = F1(ra(g))) = F1(g a)

= f1(F2)(g a) = F2(f(g a))

= F2(f(g) f(a))

r1a (F1)(g) = F2(f(g) f(a)) (3)

Since B is a fuzzy topological in K2, the mapping rb : (B, B) (B, B)defined by

rb(y) = y b b K2 (4)


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is relatively fuzzy continues.

r1a (F1)(g) = F2(f(g) f(a)) = F2(rf(a)(f(g)))

= r1f(a)(F2)(f(g)) = 1f (r

1f(a)(F2))(g)

This implies that r

1

a (F1) = f

1

(r

1

f(a)(F2)) andso r1a (F1) f1(B) = f1(r1

f(a)(F2)) f1(B) is open in the induced fuzzy

topology on f1(B).

Theorem 4.5. Letf : K1 K2 be an isomorphism of K-algebras. Let and be the fuzzy topologies on K1 and K2 respectively such that f() = . If Ais a fuzzy topological ideal of K1 with membership function A. Then f(A) isa fuzzy topological ideal of K2 with membership function f(A).

Proof. It is easy to show that f(A)(e) f(A)(x), for all x K2.Given x, y K2, let x0 f1(x), y0 f1(y) such that

A(x0)= suptf1(x)(t), A(y0)= suptf1(y)(t). Then, we have

f(A)(x) = suptf1(x)

A(t)

A(x0)

min{A(x0 y0), A(y0 (y0 x0))}

= min{ suptf1(xy)

A(t), suptf1(y(yx))

A(t)}

= min{f(A)(x y), f(A)(y (y x))}.

Hence f(A) is a fuzzy ideal of K2.Now we show that the mapping rb : (f(A),

f(A)

) (f(A), f(A)

) defined by

rb(y) = y b (5)

is relatively fuzzy continuous for each b K2. Let UA be a fuzzy set in A.Then there exists a fuzzy set U in such that UA = U A. Since f is one-one,it follows that f(UA) = f(U A) = f(U) f(A) which is a fuzzy set in

f(A).

This shows that f is relatively fuzzy open.Let Vf(A) be a fuzzy set in

f(A). The onto mapping of f implies that for each

b K2, there exists a K1 such that

b = f(a) (6)

Thusf1(r1b (Vf(A))

(x) = f1(r1f(a)

(Vf(A))(x) = r1

f(a)(Vf(A))

(f(x))

= Vf(A)(rf(a)(f(x))) = Vf(A)(f(x) f(a))

= f1(Vf(A))(x a) = f1(Vf(A))(ra(x))

= r1a (f1(Vf(A))(x).


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This implies that f1(r1a (Vf(A))) = r1(f1(Vf(A))). The mapping ra : (A, A)

(A, A) defined by ra(x) = x a is relatively fuzzy continuous and f is rela-tively fuzzy continuous map (A, A) (f(A), f(A)).

Hence f1(r1b (Vf(A))) A = r1a (f

1(Vf(A))) A is a fuzzy set in A.

References

[1] L. A. Zadeh, Fuzzy sets, Information Control, 8(1965) 338-353.

[2] C. Chang, Fuzzy topological space, J. Math. Anal. Appl., 24(1968), 182-190.

[3] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512517.

[4] R. Luwen, Fuzzy topological spaces and fuzzy compactness, J. Math. andAnal. Appl., 56(1976), 621-633.

[5] D. H. Foster, Fuzzy topological groups, J. Math. Anal. Appl. 67 (1979),549564.

[6] D. M. Ali and A. K. Srivastava , On fuzzy connectedness , Fuzzy Sets andSystems, 28(1988), 203-208.

[7] A. K. Chaudhuri and P. Das, Fuzzy connected sets in fuzzy topologicalspaces, Fuzzy Sets and Systems, 49(1992), 223-229.

[8] Y. B. Jun, Fuzzy topological BCK-algebras, Math. Japon. 38(6)(1993),

1059-1063.[9] K. H. Dar and M. Akram, On a K-algebra built on a group, SEA Bull.

Math. 29(1) (2005), 41-49.

[10] K. H. Dar and M. Akram, Characterization of a K(G)-algebra by selfmaps, SEA Bull. Math. 28(4) (2004), 601-610.

Received: October 3, 2005

On Fuzzy Topological K-Algebras

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