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Kangweon-Kyungki Math. Jour. 14 (2006), No. 2, pp. 241–248
Abstract. We define a G-fuzzy congruence, which is a generalizedfuzzy congruence, and characterize the G-fuzzy congruence gener-ated by a left and right compatible fuzzy relation on a semigroup.

The concept of a fuzzy relation was first proposed by Zadeh [9]. Sub-
sequently, Goguen [2] and Sanchez [7] studied fuzzy relations in vari-ous contexts. In [5] Nemitz discussed fuzzy equivalence relations, fuzzyfunctions as fuzzy relations, and fuzzy partitions. Murali [4] developedsome properties of fuzzy equivalence relations and certain lattice the-oretic properties of fuzzy equivalence relations. Samhan [6] discussedthe fuzzy congruence generated by a fuzzy relation on a semigroupand studied the lattice of fuzzy congruences on a semigroup. Guptaet al. [3] proposed a generalized definition of a fuzzy equivalence rela-tion on a set, which we call G-fuzzy equivalence relation in this paper,and developed some properties of that relation. In [8] Tan developedsome properties of fuzzy congruences on a regular semigroup. Chon[1] characterized the G-fuzzy congruence generated by a fuzzy relationon a semigroup and gave some lattice theoretic properties of G-fuzzycongruences on semigroups. The present work has been started as acontinuation of these studies.

In section 2 we define a G-fuzzy congruence and review some ba-
sic definitions and properties of fuzzy relations and G-fuzzy congru-ences. In section 3 we find the G-fuzzy congruence generated by a
Received November 13, 2006.

2000 Mathematics Subject Classification: 03E72.

Key words and phrases: G-fuzzy equivalence relation, G-fuzzy congruence.

This work was supported by a research grant from Seoul Women’s University
left and right compatible fuzzy relation

*µ *on a semigroup

*S *such that
sup

*µ*(

*x, y*)

*> *0 for some

*x *=

*y ∈ S*, find the G-fuzzy congruence

*x*=

*y∈S*generated by a left and right compatible fuzzy relation

*µ *on a semi-group

*S *such that

*µ*(

*x, y*) = 0 for all

*x *=

*y ∈ S *and

*µ*(

*z, z*)

*> *0 for all

*z ∈ S*, and show that there does not exist the G-fuzzy congruence gen-erated by a left and right compatible fuzzy relation

*µ *on a semigroup

*S *such that

*µ*(

*x, y*) = 0 for all

*x *=

*y ∈ S *and

*µ*(

*z, z*) = 0 for some

*z ∈ S*.

We recall some basic definitions and properties of fuzzy relations
and G-fuzzy congruences which will be used in the next section.

Definition 2.1. A function

*B *from a set

*X *to the closed unit
interval [0, 1] in R is called a

*fuzzy set *in

*X*. For every

*x ∈ B*,

*B*(

*x*) iscalled a

*membership grade *of

*x *in

*B*.

The standard definition of a fuzzy reflexive relation

*µ *in a set

*X*
demands

*µ*(

*x, x*) = 1. Gupta et al. ([3]) weakened this definition asfollows.

Definition 2.2. A

*fuzzy relation µ *in a set

*X *is a fuzzy subset of

*X × X*.

*µ *is

*G-reflexive *in

*X *if

*µ*(

*x, x*)

*> *0 and

*µ*(

*x, y*)

*≤ *inf

*µ*(

*t, t*) for
all

*x, y ∈ X *such that

*x *=

*y*.

*µ *is

*symmetric *in

*X *if

*µ*(

*x, y*) =

*µ*(

*y, x*)for all

*x, y *in

*X*. The composition

*λ ◦ µ *of two fuzzy relations

*λ, µ *in

*X *is the fuzzy subset of

*X × X *defined by
(

*λ ◦ µ*)(

*x, y*) = sup min(

*λ*(

*x, z*)

*, µ*(

*z, y*))

*.*
A fuzzy relation

*µ *in

*X *is

*transitive *in

*X *if

*µ ◦ µ ⊆ µ*. A fuzzyrelation

*µ *in

*X *is called

*G-fuzzy equivalence relation *if

*µ *is G-reflexive,symmetric, and transitive.

Definition 2.3. Let

*µ *be a fuzzy relation in a set

*X*.

*µ *is called

*fuzzy left (right) compatible *if

*µ*(

*x, y*)

*≤ µ*(

*zx, zy*) (

*µ*(

*x, y*)

*≤ µ*(

*xz, yz*))for all

*x, y, z ∈ X*. A G-fuzzy equivalence relation on

*X *is called a

*G-fuzzy left congruence (right congruence) *if it is fuzzy left compatible
G-Fuzzy congruences generated by compatible fuzzy relations
(right compatible). A G-fuzzy equivalence relation on

*X *is a

*G-fuzzycongruence *if it is a G-fuzzy left and right congruence.

Definition 2.4. Let

*µ *be a fuzzy relation in a set

*X*.

*µ−*1 is defined
as a fuzzy relation in

*X *by

*µ−*1(

*x, y*) =

*µ*(

*y, x*).

It is easy to see that (

*µ ◦ ν*)

*−*1 =

*ν−*1

*◦ µ−*1 for fuzzy relations

*µ *and
Proposition 2.5.

*Let µ be a fuzzy relation on a set X. Then*
*n*=1

*µn is the smallest transitive fuzzy relation on X containing µ,*
*where µn *=

*µ ◦ µ ◦ · · · ◦ µ.*
*Proof. *See Proposition 2.3 of [6].

Proposition 2.6.

*Let µ be a fuzzy relation on a set X. If µ is*
*n*=1

*µn, where µn *=

*µ ◦ µ ◦ · · · ◦ µ.*
*Proof. *See Proposition 2.4 of [6].

Proposition 2.7.

*If µ is a fuzzy relation on a semigroup S that*
*is fuzzy left and right compatible, then so is ∪∞*
*Proof. *See Proposition 3.6 of [6].

3. G-fuzzy congruences generated by fuzzy relations
In this section we characterize the G-fuzzy congruence generated by
a left and right compatible fuzzy relation on a semigroup.

Proposition 3.1.

*Let µ be a fuzzy relation on a set S. If µ is*
*n*=1

*µn, where µn *=

*µ ◦ µ ◦ · · · ◦ µ.*
*Proof. *Clearly

*µ*1 =

*µ *is G-reflexive. Suppose

*µk *is G-reflexive.

*µk*+1(

*x, x*) = (

*µk ◦ µ*)(

*x, x*) = sup min[

*µk*(

*x, z*)

*, µ*(

*z, x*)]

*≥ *min[

*µk*(

*x, x*)

*, µ*(

*x, x*)]

*> *0
for all

*x ∈ S*. Let

*x, y ∈ S *with

*x *=

*y*. Then
inf

*µk*+1(

*t, t*) = inf (

*µk ◦ µ*)(

*t, t*)
= inf sup min[

*µk*(

*t, z*)

*, µ*(

*z, t*)]

*≥ *inf min[

*µk*(

*t, t*)

*, µ*(

*t, t*)]

*≥ *min [ inf

*µk*(

*t, t*)

*, *inf

*µ*(

*t, t*)]

*≥ *min[

*µk*(

*x, z*)

*, µ*(

*z, y*)]
for all

*z ∈ S *such that

*z *=

*x *and

*z *=

*y*. That is,
min[

*µk*(

*x, z*)

*, µ*(

*z, y*)]

*.*
inf

*µ*(

*t, t*)

*≥ *min [

*µk*(

*x, x*)

*, µ*(

*x, y*)]
inf

*µk*(

*t, t*)

*≥ *min [

*µk*(

*x, y*)

*, µ*(

*y, y*)]

*.*
Since

*µk*+1(

*t, t*)

*≥ µk*(

*t, t*)

*≥ µ*(

*t, t*) for

*k ≥ *1,
inf

*µk*+1(

*t, t*)

*≥ *min [

*µk*(

*x, x*)

*, µ*(

*x, y*)]
inf

*µk*+1(

*t, t*)

*≥ *min [

*µk*(

*x, y*)

*, µ*(

*y, y*)]

*.*
inf

*µk*+1(

*t, t*)

*≥ *max [
min(

*µk*(

*x, z*)

*, µ*(

*z, y*))

*,*
min (

*µk*(

*x, x*)

*, µ*(

*x, y*))

*, *min (

*µk*(

*x, y*)

*, µ*(

*y, y*))]
= sup min[

*µk*(

*x, z*)

*, µ*(

*z, y*)]
= (

*µk ◦ µ*)(

*x, y*) =

*µk*+1(

*x, y*)

*.*
That is,

*µk*+1 is G-reflexive. By the mathematical induction,

*µn *is G-reflexive for

*n *= 1

*, *2

*, . . . . *Thus inf [

*∪∞*
*n*=1

*µn*](

*t, t*) = inf sup[

*µ*(

*t, t*)

*, *(

*µ ◦*
*µ*)(

*t, t*)

*, . . . *]

*≥ *sup [ inf

*µ*(

*t, t*)

*, *inf (

*µ ◦ µ*)(

*t, t*)

*, . . . *]

*≥ *sup[

*µ*(

*x, y*)

*,*
(

*µ ◦ µ*)(

*x, y*)

*, . . . *] = [

*∪∞*
*n*=1

*µn*](

*x, y*). Clearly [

*∪∞*
*n*=1

*µn*](

*x, x*)

*> *0. Hence

*n*=1

*µn *is G-reflexive.

G-Fuzzy congruences generated by compatible fuzzy relations
Theorem 3.2.

*Let µ be a fuzzy relation on a semigroup S such*
*that µ is fuzzy left and right compatible.*
(1)

*If µ*(

*x, y*)

*> *0

*for some x *=

*y ∈ S, then the G-fuzzy congruence*
*n*=1 [

*µ ∪ µ−*1

*∪ θ*]

*n, where θ is a fuzzy rela-*
*tion on S such that θ*(

*z, z*) =

*sup µ*(

*x, y*)

*for all z ∈ S and*
*θ*(

*x, y*) =

*θ*(

*y, x*)

*≤ min *[

*µ*(

*x, y*)

*, µ*(

*y, x*)]

*for all x, y ∈ S withx *=

*y.*
(2)

*If µ*(

*x, y*) = 0

*for all x *=

*y ∈ S and µ*(

*z, z*)

*> *0

*for all z ∈ S,*
*then the G-fuzzy congruence generated by µ is ∪∞*
(3)

*If µ*(

*x, y*) = 0

*for all x *=

*y ∈ S and µ*(

*z, z*) = 0

*for some z ∈ S,*
*then there does not exist the G-fuzzy congruence generated byµ.*
*Proof. *(1) Let

*µ*1 =

*µ ∪ µ−*1

*∪ θ*. Since

*θ*(

*z, z*)

*> *0,

*µ*1(

*z, z*)

*> *0
for all

*z ∈ S*. Let

*x, y ∈ S *with

*x *=

*y*. Then

*θ*(

*x, y*)

*≤ µ*(

*x, y*)

*≤*
sup

*µ*(

*x, y*) =

*θ*(

*t, t*) for all

*t ∈ S*. Thus
inf

*µ*1(

*t, t*)

*≥ *inf

*θ*(

*t, t*)

*≥ *max[

*µ*(

*x, y*)

*, µ−*1(

*x, y*)

*, θ*(

*x, y*)] =

*µ*1(

*x, y*)

*.*
That is,

*µ*1 is G-reflexive. By Proposition 3.1,

*∪∞*
Since

*θ*(

*x, y*) =

*θ*(

*y, x*),

*θ *=

*θ−*1. Thus

*µ*1(

*x, y*) = max [

*µ*(

*x, y*)

*, µ−*1(

*x, y*)

*, θ*(

*x, y*)]
= max [

*µ−*1(

*y, x*)

*, µ*(

*y, x*)

*, θ−*1(

*x, y*)]
= max[

*µ−*1(

*y, x*)

*, µ*(

*y, x*)

*, θ*(

*y, x*)]=

*µ*1(

*y, x*)

*.*
That is,

*µ*1 is symmetric. By Proposition 2.6,

*∪∞*
equivalence relation containing

*µ*. Since

*θ*(

*x, y*)

*≤ µ*(

*x, y*)

*≤ µ*(

*zx, zy*),

*µ*1(

*x, y*) = max [

*µ*(

*x, y*)

*, µ−*1(

*x, y*)

*, θ*(

*x, y*)]
= max [

*µ*(

*x, y*)

*, µ*(

*y, x*)

*, θ*(

*x, y*)] = max [

*µ*(

*x, y*)

*, µ*(

*y, x*)]

*≤ *max [

*µ*(

*zx, zy*)

*, µ*(

*zy, zx*)]

*≤ *max [

*µ*(

*zx, zy*)

*, µ*(

*zy, zx*)

*, θ*(

*zx, zy*)]
= max [

*µ*(

*zx, zy*)

*, µ−*1(

*zx, zy*)

*, θ*(

*zx, zy*)] =

*µ*1(

*zx, zy*)
for all

*x, y, z ∈ S *such that

*x *=

*y*. Since

*θ*(

*x, x*) =

*θ*(

*zx, zx*) for all

*x, z ∈ S*,

*µ*1(

*x, x*) = max [

*µ*(

*x, x*)

*, µ−*1(

*x, x*)

*, θ*(

*x, x*)]

*≤ *max [

*µ*(

*zx, zx*)

*,θ*(

*zx, zx*)] = max [

*µ*(

*zx, zx*)

*, µ−*1(

*zx, zx*)

*, θ*(

*zx, zx*)] =

*µ*1(

*zx, zx*) forall

*x, z ∈ S*. Thus

*µ*1 is fuzzy left compatible. Similarly we mayshow

*µ*1 is fuzzy right compatible. By Proposition 2.7,

*∪∞*
fuzzy left and right compatible. Thus

*∪∞*
gruence containing

*µ*. Let

*ν *be a G-fuzzy congruence containing

*µ*.

Then

*µ*(

*x, y*)

*≤ ν*(

*x, y*),

*µ−*1(

*x, y*) =

*µ*(

*y, x*)

*≤ ν*(

*y, x*) =

*ν*(

*x, y*), and

*θ*(

*x, y*)

*≤ µ*(

*x, y*)

*≤ ν*(

*x, y*). Thus

*µ*1(

*x, y*)

*≤ ν*(

*x, y*) for all

*x, y ∈ S*such that

*x *=

*y*. Since

*ν*(

*a, a*)

*≥ ν*(

*x, y*)

*≥ µ*(

*x, y*) for all

*a, x, y ∈S *such that

*x *=

*y*,

*θ*(

*a, a*) =
sup

*µ*(

*x, y*)

*≤ ν*(

*a, a*) for all

*a ∈*
*S*. Since

*ν*(

*a, a*)

*≥ µ*(

*a, a*) =

*µ−*1(

*a, a*) and

*ν*(

*a, a*)

*≥ θ*(

*a, a*) for all

*a ∈ S*, max [

*µ*(

*a, a*)

*, µ−*1(

*a, a*)

*, θ*(

*a, a*)]

*≤ ν*(

*a, a*) for all

*a ∈ S*. Thus

*µ*1

*⊆ ν*. Suppose

*µk*1

*⊆ ν*. Then

*µk*+1
(

*b, c*) = (

*µk*1

*◦ µ*1)(

*b, c*) =
sup min[

*µk*1(

*b, d*)

*, µ*1(

*d, c*)]

*≤ *sup min [

*ν*(

*b, d*)

*, ν*(

*d, c*)] = (

*ν ◦ ν*)(

*b, c*) for
all

*b, c ∈ S*. That is,

*µk*+1

*⊆ *(

*ν ◦ν*). Since

*ν *is transitive,

*µk*+1
the mathematical induction,

*µn*1

*⊆ ν *for every natural number

*n*. Thus

*n*=1 [

*µ ∪ µ−*1

*∪ θ*]

*n *=

*∪∞*
=

*µ*1

*∪ *(

*µ*1

*◦ µ*1)

*∪ *(

*µ*1

*◦ µ*1

*◦ µ*1)

*· · · ⊆ ν*.

(2) Let

*x, y ∈ S *with

*x *=

*y*. Since

*µ*(

*x, y*) = 0, inf

*µ*(

*t, t*)

*≥ µ*(

*x, y*).

Thus

*µ *is G-reflexive. Since

*µ*(

*x, y*) = 0,

*µ *is symmetric. By Propo-sition 2.5, Proposition 2.6, and Proposition 3.1,

*∪∞*
equivalence relation containing

*µ*. Since

*µ *is fuzzy left and right com-patible from the hypothesis,

*∪∞*
*n*=1

*µn *is a G-fuzzy congruence con-
taining

*µ *by Proposition 2.7. Let

*ν *be a G-fuzzy congruence con-taining

*µ*. By the mathematical induction as shown in Theorem 3.2(1), we may show that

*µn ⊆ ν *for every natural number

*n*. Hence

*∪∞*
*n*=1

*µn *=

*µ ∪ *(

*µ ◦ µ*)

*∪ *(

*µ ◦ µ ◦ µ*)

*· · · ⊆ ν*.

(3) Suppose

*ξ *is the G-fuzzy congruence generated by

*µ*. Then

*ξ*(

*z, z*)

*> *0 for every

*z ∈ S*. Let

*θ *be a fuzzy relation such that

*θ*(

*a, b*) =

*ξ*(

*a,b*) for all

*a, b ∈ S*. Then

*θ*(

*z, z*)

*> *0 for all

*z ∈ S*. Let

*x, y ∈ S *with

*x *=

*y*. Since

*ξ *is G-reflexive, inf

*ξ*(

*t, t*)

*≥ ξ*(

*x, y*). Since

*θ*(

*a, b*) =

*ξ*(

*a,b*) for all

*a, b ∈ S*, inf

*θ*(

*t, t*)

*≥ θ*(

*x, y*). Since

*µ*(

*x, y*) = 0,
inf (

*µ ∪ θ*)(

*t, t*)

*≥ *inf

*θ*(

*t, t*)

*≥ *(

*µ ∪ θ*)(

*x, y*). That is,

*µ ∪ θ *is G-
reflexive. Since

*ξ *is symmetric,

*θ *is symmetric. Since

*θ *is symmetric
G-Fuzzy congruences generated by compatible fuzzy relations
and

*µ*(

*x, y*) = 0,

*µ ∪ θ *= (

*µ ∪ θ*)

*−*1. That is,

*µ ∪ θ *is symmetric. ByProposition 2.5, Proposition 2.6, and Proposition 3.1,

*∪∞*
is a G-fuzzy equivalence relation containing

*µ*. Since

*θ*(

*a, b*) =

*ξ*(

*a,b*)
for all

*a, b ∈ S *and

*ξ *is fuzzy left and right compatible,

*θ *is fuzzy leftand right compatible. Since

*µ *is fuzzy left and right compatible,

*µ ∪ θ*is fuzzy left and right compatible. By Proposition 2.7,

*∪∞*
is a G-fuzzy congruence containing

*µ*. Since

*θ*(

*a, b*) =

*ξ*(

*a,b*)

*≤ ξ*(

*a, b*)
and

*µ*(

*a, b*)

*≤ ξ*(

*a, b*) for all

*a, b ∈ S*,

*µ ∪ θ ⊆ ξ*. Let

*µ*1 =

*µ ∪ θ*.

Then

*µ*1

*⊆ ξ*. By the mathematical induction as shown in Theorem3.2 (1), we may show that

*µn*1

*⊆ ξ *for every natural number

*n*. Hence

*n*=1 [

*µ ∪ θ*]

*n *=

*∪∞*
*⊆ ξ*. Let

*v *=

*w ∈ S*. Then

*µ*1(

*v, w*) =
(

*µ ∪ θ*)(

*v, w*) =

*θ*(

*v, w*)

*≤ *inf

*θ*(

*t, t*)

*≤ µ*1(

*z, z*) for every

*z ∈ S*. Suppose

*µk*1(

*v, w*)

*≤ µ*1(

*z, z*) for every

*z ∈ S*. Then
(

*v, w*) = sup min [

*µk*1(

*v, s*)

*, µ*1(

*s, w*)]
min(

*µk*1(

*v, s*)

*, µ*1(

*s, w*))

*,*
min (

*µk*1(

*v, v*)

*, µ*1(

*v, w*))

*, *min (

*µk*1(

*v, w*)

*, µ*1(

*w, w*))]

*≤ *max [

*µ*1(

*z, z*)

*, µ*1(

*z, z*)

*, µk*1(

*v, w*)] =

*µ*1(

*z, z*)

*.*
By the mathematical induction,

*µn*1(

*v, w*)

*≤ µ*1(

*z, z*) for every natural
number

*n*. Clearly

*µk*1(

*z, z*) =

*µ*1(

*z, z*) for

*k *= 1. Suppose

*µk*1(

*z, z*) =

*µ*1(

*z, z*). Since

*µk*1(

*z, s*)

*≤ µ*1(

*z, z*) for

*s *=

*z ∈ S*,

*µk*+1
sup min [

*µk*1(

*z, s*)

*, µ*1(

*s, z*)] = max [ sup
min(

*µk*1(

*z, s*)

*, µ*1(

*s, z*))

*,*
min (

*µk*1(

*z, z*)

*, µ*1(

*z, z*))] =

*µ*1(

*z, z*). By the mathematical induction,

*µn*1(

*z, z*) =

*µ*1(

*z, z*) for every natural number

*n *and every

*z ∈ S*. Let

*p *be in

*S *with

*µ*(

*p, p*) = 0. Then

*µ*1(

*p, p*) =

*θ*(

*p, p*) =

*ξ*(

*p,p*)

*< ξ*(

*p, p*).

Since

*µn*1(

*z, z*) =

*µ*1(

*z, z*) for every natural number

*n *and every

*z ∈ S*,

*n*=1 (

*µ ∪ θ*)

*n*](

*p, p*) = [

*∪∞*
*n*=1

*µ*1 ](

*p, p*) =

*µ*1(

*p, p*)

*< ξ*(

*p, p*) for some

*p ∈ S *such that

*µ*(

*p, p*) = 0. Hence

*∪∞*
*n*=1 (

*µ ∪ θ*)

*n*, which is a G-fuzzy
congruence containing

*µ*, is contained in

*ξ*. This contradicts that

*ξ *isthe G-fuzzy congruence generated by

*µ*.

1. I. Chon,

*Generalized fuzzy congruences on semigroups, (submitted)*.

2. J. A. Goguen,

*L-fuzzy sets*, J. Math. Anal. Appl. 18 (1967), 145–174.

3. K. C. Gupta and R. K. Gupta,

*Fuzzy equivalence relation redefined*, Fuzzy Sets
4. V. Murali,

*Fuzzy equivalence relation*, Fuzzy Sets and Systems 30 (1989), 155–
5. C. Nemitz,

*Fuzzy relations and fuzzy function*, Fuzzy Sets and Systems 19
6. M. Samhan,

*Fuzzy congruences on semigroups*, Inform. Sci. 74 (1993), 165–175.

7. E. Sanchez,

*Resolution of composite fuzzy relation equation*, Inform. and Con-
8. V. Tan,

*Fuzzy congruences on a regular semigroup*, Fuzzy Sets and Systems
9. L. A. Zadeh,

*Fuzzy sets*, Inform. and Control 8 (1965), 338–353.

Department of MathematicsSeoul Women’s UniversitySeoul 139–774, Korea

*E-mail*:

[email protected]
Source: http://www.kkms.org/kkms/vol14_2/14209.pdf

DURHAM DALES CLASSIC RELIABILITY TRIAL REGULATIONS Promoted by: Durham Automobile Club Limited Held under the General Regulations of the Motor Sports Association, (incorporating the provisions of the International Sporting code of the FIA), and Durham Automobile Club Limited is a motor sport club recognised by the R.A.C. Motor Sports Association, and the Auto Cyc

DETERMINACIÓN DE NIVELES RESIDUALES DE TETRACICLINA EN CANALES BOVINAS POR LA TÉCNICA DE ELISA EN EL FRIGORÍFICO FRIOGAN (LA DORADA) DETERMINATION OF RESIDUAL TETRACYCLINE LEVELS IN En las políticas sanitarias colombianas se BOVINE CARCASSES BY MEANS estableció la importancia del control de residuos OF ELISA IN THE FRIOGAN de medicamentos veterinarios, consid