SOME PROPERTIES OF INTUITIONISTIC FUZZY

SOME PROPERTIES OF INTUITIONISTIC FUZZY EQUIVALENCE RELATIONS AND EQUIVALENCE CLASS TREES ¼ GÖKHAN ÇUVALCIOGLU AND SINEM YILMAZ Abstract. Intuitionistic fuzzy sets, introduced by Atanassov, have gained successful applications in various …elds. Then the intuitionistic fuzzy relations were developed. In this paper, the cover of an intuitionistic fuzzy sets and the equivalence relations with their some property were studied. Finally some properties of intuitionistic fuzzy relations and intuitionistic fuzzy equivalence class w.r.t. IFRs were studied .The class tree that has been attained from the intuitionistic fuzzy equivalence classes was given. 2010 Mathematics Subject Classification. 03E72,47S40 Keywords and phrases. Intuitionistic Fuzzy Equivalence Relations, Intuitionistic Fuzzy Equivalence Classes, Class Trees.

1. Introduction The concept of fuzzy sets was introduced in Zadeh [5] as an extension of crisp sets, the usual two-valued sets in ordinary set theory, by expanding the truth value set to the real unit interval [0; 1]. In crisp set theory, if the membership degree of an element x is 1 then the nonmembership degree is 0. In fuzzy set theory, if the membership degree of an element x is (x) then the nonmembership degree is 1 (x) and thus it is …xed. Intuitionistic fuzzy sets have been introduced by Atanassov in 1983 [1] and form an extension of fuzzy sets. While the nonmembership degree for each element of the universe is …x in fuzzy set theory, in intuitionistic fuzzy set theory, nonmembership degree is a more or less independet degree; the only condition that it is smaller then 1 membership degree. So, If X an universe then there exist two membersip and nonmembership degree for each x 2 X; respectively A (x) and A (x) such that 0 1. A (x) + A (x) Let X and Y be universes. The intuitionistic fuzzy relation between X and Y is de…ned an intuitionistic fuzzy set in X Y in Atanassov [2, 3]. If R is an intuitionistic fuzzy relation between X and Y , x 2 X, y 2 Y then the degree to which x is in relation R with y is denoted by R (x; y). De…nition 1. Let L=[0,1] then L = f(x1 ; x2 ) 2 [0; 1]2 : x1 + x2 1g is a lattice with (x1 ; x2 ) (y1 ; y2 ) :() "x1 y1 and x2 y2 ". The units of this lattice are denoted by 0L = (0; 1) and 1L = (1; 0). The lattice (L ; ) is a complete lattice: for each A supA = (supfx 2 [0; 1] : (y 2 [0; 1])((x; y) 2 A)g; inf fy 2 [0; 1] :( x 2 [0; 1])((x; y) 2 A)g) and 1

L

2

¼ GÖKHAN ÇUVALCIO GLU AND SINEM YILMAZ

inf A = (inf fx 2 [0; 1](y 2 [0; 1])((x; y) 2 A)g; supfy 2 [0; 1] ( x 2 [0; 1])((x; y) 2 A)g): As is well known, every lattice (L ; ) has an equivalent de…nition as an algebraic structure (L; ^; _) where the meet operator ^ and the join operator _ are linked the ordering " " by the following equivalence , for a; b 2 L a b () a _ b = b () a ^ b = a The operators ^ and _ on (L ; ) are def¬ned as follows, for (x1 ; y1 ); (x2 ; y2 ) 2 L : (x1 ; y1 ) ^ (x2 ; y2 ) = (min(x1 ; x2 ); max(y1 ; y2 )) (x1 ; y1 ) _ (x2 ; y2 ) = (max(x1 ; x2 ); min(y1 ; y2 )) De…nition 2. [1]An intuitionistic fuzzy set (shortly IFS) on a universe X is an object of the form A = f< x;

A (x); A (x)

>: x 2 Xg

where A (x)(2 [0; 1]) is called the“degree of membership of x in A ”, A (x)(2 [0; 1])is called the “ degree of nonmembership of x in A ”,and where A and A satisfy the follow¬ng condition: (x 2 X) ( A (x) + A (x) 1): The class of IFSs on a universe X will be denoted IF S(X). An IFS A is said to be contained in an IFS B (notation A B) if and only if, for all x 2 X : A (x) A (x) B (x): B (x) and The intersection (resp.the union) of two IFSs A and B on X is de…nition as the IFSs A \ B = f< x; (resp. A [ B = f< x;

A (x)

A (x)

_

^

B (x); A (x)

B (x); A (x)

^

_

B (x)

B (x)

>: x 2 Xg

>: x 2 Xg)

2. Intuitionistic Fuzzy Equivalence Relation De…nition 3. Let X be a set, R 2 IF R(X): ER1 For every x 2 X,

R (x; x)

=1 R (x; x) = 0 then R is called an intuitionistic fuzzy re‡exsive. ER2 For every x; y 2 X, R (x; y)

R (y; x)

R (x; y) R (y; x) then R is called an intuitionistic fuzzy symmetric. ER3 For every x; y; z 2 X, R (x; y)

R (x; y)

^

R (y; z)

R (x; z)

_ R (y; z) R (x; z) then R is called an intuitionistic fuzzy transitive.

SOME PROPERTIES OF I.F.E.R.S AND EQUIVALENCE CLASS TREES

3

If an intuitionistic fuzzy relation satis…es the previous properties then it is called an intuitionistic fuzzy equivalence relation ( IF ER(X) ). Example 1. Let X = fx1 ; x2 ; x3 ; x4 g, A = f< x; 1; 0 >: x 2 Xg. If we de…ne the intuitionistic relation is de…ned as follows x1 x2 x1 (1:0; 0:0) (0:6; 0:2) R = x2 (0:6; 0:2) (1:0; 0:0) x3 (0:3; 0:4) (0:3; 0:4) x4 (0:6; 0:2) (0:6; 0:2) then R 2 IF ER(X):

x3 (0:3; 0:4) (0:3; 0:4) (1:0; 0:0) (0:3; 0:4)

x4 (0:6; 0:2) (0:6; 0:2) (0:3; 0:4) (1:0; 0:0) S

De…nition 4. Let A 2 IF S(X) and { = fC : C 2 IF S(X)g. If A = then { is called a intuitionistic fuzzy cover for A. We write IF C(A) = f{ : { is a cover for Ag From the above de…nition, A (x) = sup C (x) therefore

C (x)

C

C2{

A (x)

C2{

and

A (x)

= inf C2{

C (x)

therefore

above discussion we have C C 2 { and 0 sup C (x) + inf

A (x)

and C (x) 1: C A

C2{

C2{

C (x)

, for all x 2 X: From the

A.

Therefore, C

A for all

De…nition 5. Let X be a set, A 2 IF S(X), { 2 IF C(A). If we de…ne R{ = f< (x; y); sup(

C (x)

C2{

^

C (x));

then it is clear that R{ 2 IF S(X inf C2{

C (x)

inf (

C2{

C (x)

_

C (x))

>: x; y 2 Xg

X) from the inequality 0

sup

C (x)

+

C2{

1:

Theorem 2.1. Let X be a set, R 2 IF R(X) and { be an intuitionistic fuzzy cover of A. R{ 2 IF ER(X) if and only if for every C1 ; C2 2 { there exists C 2 { such that h1 (C1 ; C2 ) ^

C1 (x)

^

C2 (y)

C (x)

^

C (y)

h2 (C1 ; C2 ) _ C1 (x) _ C2 (y) C (x) _ C (y) where h1 (C1 ; C2 ) = sup C1 (x)^ C2 (x) and h2 (C1 ; C2 ) = inf x2X

x2X

C1 (x)_ C2 (x)

Proof. Let { be a intuitionistic fuzzy cover of A that is satis…es the above properties. For every x 2 X; R{ (x; x)

= sup C2{

therefore R{ (x; x)

C (x) ^

R{ (x; x)

= inf C2{

C (x)

C (x)

C (x)

C2{

sup y2X

_

= sup

C (x) ^

C (y)

=

C2{

sup

R{ (x; y)

inf

C (x)

C (x)

=

R{ (x; y)

R{ (x; y):

C (x)

= inf C2{

C (x)

C2{

therefore R{ (x; x) R{ (x; y): thus R is an intuitionistic fuzzy re‡exive relation.

_


4

For x; y; z 2 X R{ (x; y)

^

R{ (y; z)

=

sup

C (x)

C2{

= = = =

sup ( C1 ;C2 2{

sup (

R{ (x; y)

_

R{ (y; z)

R{ (y; z)

=

inf C2{

= = = =

C (z)

C1 (y)

^

C2 (y)

^

C2 (z))

C1 (y)

^

C2 (y)

^

C1 (x)

^

C2 (z))

C1 (y)

C1 ;C2 2{

^

C (z)

^

C2 (y))

C1 (x)

=

^

^

C (x)

C (y)

inf (

C1 ;C2 2{

_ inf

_

C1 (y)

_

C2 (y)

_

C2 (z))

C1 (y)

_

C2 (y)

_

C1 (x)

_

C2 (z))

C1 ;C2 2{

C (x)

_

C1 (y)

C (z)

=

_

C2 (y))

C1 (x)

_

_

C1 (x)

sup (

C1 (x)

^

C1 (y)

^

C2 (y)

^

C2 (z))

_

C2 (z))

C2 (z))

R{ (x; z)

i.e. R{ (x; y) _ R{ (y; z) R{ (x; z) for every x; y; z 2 X: Therefore R{ 2 IF ER(X) Conversely, let R{ 2 IF ER(X) then R{ (x; y) ^ R{ (y; z) R{ (x; z) and R{ (y; z) R{ (x; z) for every x; y; z 2 X. Hence

C1 ;C2 2{

C2 (z))

C (z)

_

inf (h1 (C1 ; C2 ) _

C1 ;C2 2{

C (y)

C2{

C1 (x)

inf (( inf

C1 ;C2 2{

^

R{ (x; z)

for every x; y; z 2 X;

_

C1 (x)

C2 (z))

R{ (x; z)

inf (

inf

^

^

C (x)

C1 ;C2 2{

C2{

C (y)

C1 (x)

sup (h1 (C1 ; C2 ) ^

sup

^

^ sup

C2{

sup (( sup

C1 ;C2 2{

C2{ R{ (x; y)

C (y)

C1 ;C2 2{

C1 ;C2 2{

i.e. and

^

sup(

C (x)

C2{

therefore there exist C 2 { for C1 ; C2 2 { such that sup( C (x) ^ C (z)) then we get C2 (y) ^ C2 (z)

^

C1 (x)

R{ (x; y) _

C (z))

^

C1 (y)

^

C2{

h1 (C1 ; C2 )^

C1 (x)^ C2 (z)

= sup ( y2X

C1 (x)^ C1 (y)^ C2 (y)^ C2 (z))

and inf ( C1 ;C2 2{

C1 (x)

_

C1 (y)

_

C2 (y)

_

C2 (z))

inf ( C2{

C (x)

_

C (z))

C (x)^ C (z)


therefore there exist C 2 { for C1 ; C2 2 { such that inf ( C (x) _ C (z)), then we get C2 (z)

5

C1 (x)_ C1 (y)_ C2 (y)_

C2{

h2 (C1 ; C2 )_

C1 (x)_ C2 (z)

= inf ( y2X

C1 (x)_ C1 (y)_ C2 (y)_ C2 (z))

C (x)_ C (z)

De…nition 6. Let R 2 IF R(X); C 2 IF S(X). If C (x)

^ C (x) _

C (y)

R (x; y)

C (y)

R (x; y)

for all x; y 2 X then C is called an intuitionistic fuzzy pre-class of R. It is clear that the family of the intuitionistic fuzzy pre-class of R is partially ordered set with respect to including relation " ". The maximal intuitionistic fuzzy pre-class is called class of R. We write the family of the intuitionistic fuzzy pre-class of R with {(R). Proposition 2.2. Let X be a set, A 2 IF S(X), { = f{ : { is a cover for Ag. The relation ' : { ! IF R(X) de…ned by { 7! R{ is a well-de…ned relation. Corollary 2.3. Let R 2 IF T RA (X) then there exist an intuitionistic fuzzy cover { of A such that R = R{ . Corollary 2.4. If { = {(R) is a family of intuitionistic fuzzy class of R then R{(R) = R. Example 2. Let X = fx1 ; x2 ; x3 ; x4 g, A = f< x; 1; 0 >: x 2 Xg. If we de…ne {1 = fC1 ; C2 ; C3 ; C4 g, {2 = fC1 ; C2 ; C3 ; C5 g as the following C1 C2 C3 C4 (1:0; 0:0) (0:1; 0:7) (0:3; 0:4) (0:6; 0:2) (0:1; 0:7) (1:0; 0:0) (0:5; 0:1) (0:7; 0:2) (0:3; 0:4) (0:5; 0:1) (1:0; 0:0) (0:3; 0:6) (0:6; 0:2) (0:7; 0:2) (0:3; 0:6) (1:0; 0:0) C1 C2 C3 C5 x1 (1:0; 0:0) (0:1; 0:7) (0:3; 0:4) (0:6; 0:2) {2 = x2 (0:1; 0:7) (1:0; 0:0) (0:5; 0:1) (0:7; 0:2) x3 (0:3; 0:4) (0:5; 0:1) (1:0; 0:0) (0:1; 0:8) x4 (0:6; 0:2) (0:7; 0:2) (0:3; 0:6) (1:0; 0:0) Then the result of the intuitionistic fuzzy relation is as follows; x1 x2 x3 x4 x1 (1:0; 0:0) (0:6; 0:2) (0:3; 0:4) (0:6; 0:2) R{1 = R{2 = x2 (0:6; 0:2) (1:0; 0:0) (0:5; 0:1) (0:7; 0:2) x3 (0:3; 0:4) (0:5; 0:1) (1:0; 0:0) (0:5; 0:2) x4 (0:6; 0:2) (0:7; 0:2) (0:5; 0:2) (1:0; 0:0) in addition, if {3 = {1 [ {1 then the result is C5 C1 C2 C3 C4 x1 (1:0; 0:0) (0:1; 0:7) (0:3; 0:4) (0:6; 0:2) (0:6; 0:2) {3 = x2 (0:1; 0:7) (1:0; 0:0) (0:5; 0:1) (0:7; 0:2) (0:7; 0:2) x3 (0:3; 0:4) (0:5; 0:1) (1:0; 0:0) (0:3; 0:6) (0:1; 0:8) x4 (0:6; 0:2) (0:7; 0:2) (0:3; 0:6) (1:0; 0:0) (1:0; 0:0)

x1 {1 = x2 x3 x4


6

and R{1 = R{2 = R{3 therefore, it can be said that the function ' should not be always a monomorphism. 3. Class Trees with Respect to Intuitionistic Fuzzy Equivalence Relations De…nition 7. Let X be a set, A 2 IF S(X); R 2 IF ER(X) and a 2 X. [a]R = f< x;

[a]R ; [a]R

where [a]R = R (a; x), [a]R = equivalence class of a w.r.t R.

R (a; x)

>: x 2 Xg

is called an intuitionistic fuzzy

From the de…nition of [a]R it is clear that 0 [a]R 2 IF R(X).

[a]R

+

[a]R

1 i.e.

Example 3. If the intuitionistic fuzzy relation will be used like the previous example the result of the intuitionistic fuzzy equivalence classes will be as follows, [x1 ]R [x2 ]R [x3 ]R [x4 ]R

= f(x1 ; 1; 0); (x2 ; 0:6; 0:2); (x3 ; 0:3; 0:4); (x4 ; 0:6; 0:2)g = f(x1 ; 0:6; 0:2); (x2 ; 1; 0); (x3 ; 0:3; 0:4); (x4 ; 0:6; 0:2)g = f(x1 ; 0:3; 0:4); (x2 ; 0:3; 0:4); (x3 ; 1; 0); (x4 ; 0:3; 0:4)g = f(x1 ; 0:6; 0:2); (x2 ; 0:6; 0:2); (x3 ; 0:3; 0:4); (x4 ; 1; 0)g

Theorem 3.1. Let X be a set, R 2 IF R(X). C is an intuitionistic fuzzy equivalence class of R if and only if C is an intuitionistic fuzzy class of R. Proof. First we show that an intuitionistic fuzzy equivalence class of R is an intuitionistic fuzzy pre-class of R. Let x; y 2 X; [a]R (x)

and [a]R (x)

^

[a]R (y)

=

R (a; x)

^

R (a; y)

=

R (x; a)

^

R (a; y)

R (x; y)

_

[a]R (y)

=

R (a; x)

_

R (a; y)

=

R (x; a)

_

R (a; y)

R (x; y)

therefore [a]R is an intuitionistic fuzzy pre-class of R. Let C be an intuitionistic fuzzy classs of R then there exist an a 2 X such that C (x) C (a) therefore C (a) and C (x) C (x)

=

C (a)

^

C (x)

R (a; x)

=

[a]R (x)

C (x)

=

C (a)

_

C (x)

R (a; x)

=

[a]R (x)

and

Conversely, let [a]R is an intuitionistic fuzzy equivalence class of R and let [a]R then C (a) sup R (a; x) [a]R (a) = R (a; a) x2X

and

C (a)

[a]R (x)

=

R (a; a)

inf

x2X

R (a; x)

if we use this result, [a]R (x)

and

C (x)

=

C (x)

^

C (a)

R (a; x)

=

[a]R (x)

C


[a]R (x)

C (x)

=

C (x)

therefore we get [a]R = C.

_

C (a)

R (a; x)

=

7

[a]R (x)

Proposition 3.2. Let X be a set, R 2 IF ER(X) and let a; b 2 X: [a]R = [b]R if and only if R (a; b) = R (a; a) and R (a; b) = R (a; a). Proof. Let [a]R = [b]R then R (a; b) = [a]R (b) = [b]R (b) = R (b; b) = R (a; a) and R (a; b) = [a]R (b) = [b]R (b) = R (b; b) = R (a; a) Conversely, let R (a; b) = R (a; a) and R (a; b) = R (a; a) then [a]R (x) = R (a; x) R (a; b) ^ R (b; x) = R (b; x) = [b]R (x) therefore [a]R (x) [b]R (x): From the same way we get [b]R (x) [a]R (x) then [b]R (x) = [a]R (x) and R (a; b) ^ R (b; x) = R (b; x) = [b]R (x) therefore [a]R (x) = R (a; x) (x) (x): [a]R [b]R From the same way we get [b]R (x) [a]R (x) then [b]R (x) = [a]R (x) Proposition 3.3. Let X be a set, R 2 IF ER(X) and let a; b 2 X, [a]R 6= [a]R then h1 ([a]R ; [b]R ) < R (x; x) and h2 ([a]R ; [b]R ) > R (x; x) ,for every x 2 X: Proof. We assume that h1 ([a]R ; [b]R ) = R (x; x) and h2 ([a]R ; [b]R ) = ,for every x 2 X: Then sup ( [a]R (x) ^ [b]R (x)) = R (x; x) therefore

R (x; x)

x2X

sup ( x2X

R (a; x)

^

R (x; b))

sup x2X

R (a; b)

=

R (a; b)then R (x; x)

R (a; b)

from the same way we get R (a; b) R (x; x) hence R (x; x) = R (a; b) i.e. [a]R = [b]R If we use the above method we get [a]R = [b]R : This is conradict to our assumption. So; h1 ([a]R ; [b]R ) < R (x; x) and h2 ([a]R ; [b]R ) > R (x; x), for every x 2 X: De…nition 8. Let X be a set,A 2 IF S (X) ,R 2 IF R (X) : Let a 2 X; ; 2 [0; 1] and 0 + 1; n o (1) [a]R = x : [a]R (x) = R (a; x) n o (2) [a]R = x : [a]R (x) = R (a; x) n o (3) [a]R = x : [a]R (x) = R (a; x) ; [a]R (x) = R (a; x) Theorem 3.4. Let X be a set, A 2 IF S (X) ,R 2 IF R (X), and 0 + 1 [a]R =

[a]R \

[a]R

;

2 [0; 1]


8

Proof. Let X be a set,A 2 IF S (X) ,R 2 IF R (X) . ; + 1 n x: n = x: n = x:

[a]R =

=

[a]R (x)

;

[a]R (x) [a]R (x)

[a]R \

[a]R

2 [0; 1] and 0

o

[a]R (x)

o ^ [a]R (x) o n \ x : [a]R (x)

o

Proposition 3.5. Let X be a set,A 2 IF S (X) ,R 2 IF R (X) and a 2 X: ; 2 [0; 1] and 0 + 1; R (a; x) + 1 [a]R =

[a]R =

[a]R

Proof. 8x 2 [a]R : R (a; x) 1 =) R (a; x) 1 and R (a; x) + 0 (a; x) + (a; x) 1 =) (a; x) 1 R R R so 1 1 ) R (a; x) R (a; x) we get [a]R [a]R : 8x 2 [a]R : R (a; x) R (a; x)

+

1 =)

R (a; x)

so we get [a]R [a]R : Hence we get [a]R = [a]R = [a]R :

1

and

[a]R =

R (a; x) R (a; x)

1

)

R (a; x)

[a]R and from above theorem

Proposition 3.6. Let X be a set,A 2 IF S (X) ,R 2 IF R (X) and every x 2 X; values of R (a; x) are 1; 2 ; :::; n and values of R (a; x) are 1 ; 2 ; ::: n . Let these values are put in order like that; ::: ::: 1 2 n; 1 2 n .If there exists , like that; < k 1 then [a]R = kk [a]R k 1 < k; k Proof. Let X be a set,A 2 IF S (X) and R 2 IF R (X) : Let ; 2 [0; 1] and 0 + 1: < k 1 k 1 < k and k 8x 2 k [a]R =) R (a; x) k =) R (a; x) =) x 2 [a]R 8x 2 [a]R =) R (a; x) and k > k 1 =) R (a; x) or (a; x) < k k R if R (a; x) =) x 2 [a] : k k R if R (a; x) < k then 9i 2 f1; 2; :::; ng ; R (a; x) = i : But we know that the value of i must be k > i > immpossible. So we get [a]R = k [a]R :

k 1

and it is


With the same way, we can show that [a] =

[a]R =

k

9

[a]R : Thus;

[a]R \ [a]R

=

k

[a]R \

=

k k

[a]R

k

[a]R

Example 4. Let X = fx1 ; x2 ; x3 ; x4 g and R 2 IF ER(X) as following; x1 x2 x3 x4 x1 (1:0; 0:0) (0:6; 0:2) (0:3; 0:4) (0:6; 0:2) R = x2 (0:6; 0:2) (1:0; 0:0) (0:5; 0:1) (0:7; 0:2) x3 (0:3; 0:4) (0:5; 0:1) (1:0; 0:0) (0:5; 0:2) x4 (0:6; 0:2) (0:7; 0:2) (0:5; 0:2) (1:0; 0:0) for 0 < 1 < 0:1 < 2 < 0:2 < 3 < 0:3 < 4 < 0:4 0:4 < 5 < 0:5 < 6 < 0:6 < 7 < 0:7 < 8 < 1 1 [x1 ]R = 1 [x2 ]R = 1 [x3 ]R = 1 [x4 ]R = X 2 [x1 ]R = 2 [x2 ]R = 2 [x3 ]R = 2 [x4 ]R = X [x ] = [x ] = [x ] = X 2 3 4 R R R 3 3 3 4 [x2 ]R = 4 [x4 ]R = X 5 [x2 ]R = 5 [x4 ]R = X 3 [x1 ]R = 4 [x1 ]R = 5 [x1 ]R = 6 [x1 ]R = 6 [x2 ]R = 6 [x4 ]R = fx1 ; x2 ; x4 g 4 [x3 ]R = 5 [x3 ]R = fx2 ; x3 ; x4 g [x ] = fx 2 R 2 ; x3 g 7 [x ] = fx 4 R 2 ; x4 g 7 7 [x1 ]R = 8 [x1 ]R = fx1 g [x ] = fx 2 2g R 8 [x ] = [x 3 R 3 ]R = 8 [x3 ]R = fx3 g 6 7 [x ] = fx g 4 R 4 8 The class tree that has been attained from the intuitionistic fuzzy equivalence classes of the elements of X w.r.t R as following:

10


References [1] KT Atanassov, Intuitionistic Fuzzy Sets, VII ITKR’S Session, So…a, 1983 [2] KT Atanassov, Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 20, (1986), 87- -96. [3] KT Atanassov, More on Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 33, (1989), 37- -45. [4] KT Atanassov, Intuitionistic Fuzzy Sets, Phiysica-Verlag, Heidelberg, 1999. [5] LA Zadeh, Fuzzy Sets, Information and Control, 8, (1965), 338- -353. The University of Mersin Faculity of Art and Sciences, Department of Mathematics Mersin TURKEY E-mail address: [email protected] The University of Mersin Faculity of Art and Sciences, Department of Mathematics Mersin TURKEY E-mail address: [email protected]