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∈Sgenerated 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 allz ∈ 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 semigroupS such that µ(x, y) = 0 for all x = y ∈ S and µ(z, z) = 0 for somez ∈ 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 inX 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 aG-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. Thenn=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 isn=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 thatis 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 isn=1 µn, where µn = µ ◦ µ ◦ · · · ◦ µ.Proof. Clearly µ1 = 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=1µn](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 suchthat µ is fuzzy left and right compatible.
(1) If µ(x, y) > 0 for some x = y ∈ S, then the G-fuzzy congruencen=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 andθ(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 µ1 = µ ∪ µ−1 ∪ θ. Since θ(z, z) > 0, µ1(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 µ1(t, t) ≥ inf θ(t, t)
≥ max[µ(x, y), µ−1(x, y), θ(x, y)] = µ1(x, y).
That is, µ1 is G-reflexive. By Proposition 3.1, ∪∞
Since θ(x, y) = θ(y, x), θ = θ−1. Thus
µ1(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)]= µ1(y, x).
That is, µ1 is symmetric. By Proposition 2.6, ∪∞
equivalence relation containing µ. Since θ(x, y) ≤ µ(x, y) ≤ µ(zx, zy),
µ1(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)] = µ1(zx, zy)
for all x, y, z ∈ S such that x = y. Since θ(x, x) = θ(zx, zx) for allx, z ∈ S, µ1(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)] = µ1(zx, zx) forall x, z ∈ S. Thus µ1 is fuzzy left compatible. Similarly we mayshow µ1 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), andθ(x, y) ≤ µ(x, y) ≤ ν(x, y). Thus µ1(x, y) ≤ ν(x, y) for all x, y ∈ Ssuch 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 alla ∈ S, max [µ(a, a), µ−1(a, a), θ(a, a)] ≤ ν(a, a) for all a ∈ S. Thusµ1 ⊆ ν. Suppose µk1 ⊆ ν. Then µk+1
(b, c) = (µk1 ◦ µ1)(b, c) =
sup min[µk1(b, d), µ1(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, µn1 ⊆ ν for every natural number n. Thus
n=1 [µ ∪ µ−1 ∪ θ]n = ∪∞
= µ1 ∪ (µ1 ◦ µ1) ∪ (µ1 ◦ µ1 ◦ µ1) · · · ⊆ 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 thatθ(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 ∪ θis 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 µ1 = µ ∪ θ. Then µ1 ⊆ ξ. By the mathematical induction as shown in Theorem3.2 (1), we may show that µn1 ⊆ ξ for every natural number n. Hence
n=1 [µ ∪ θ]n = ∪∞⊆ ξ. Let v = w ∈ S. Then µ1(v, w) =
(µ ∪ θ)(v, w) = θ(v, w) ≤ inf θ(t, t) ≤ µ1(z, z) for every z ∈ S. Suppose
µk1(v, w) ≤ µ1(z, z) for every z ∈ S. Then
(v, w) = sup min [µk1(v, s), µ1(s, w)]
min(µk1(v, s), µ1(s, w)),
min (µk1(v, v), µ1(v, w)), min (µk1(v, w), µ1(w, w))]
≤ max [µ1(z, z), µ1(z, z), µk1(v, w)] = µ1(z, z).
By the mathematical induction, µn1(v, w) ≤ µ1(z, z) for every natural
number n. Clearly µk1(z, z) = µ1(z, z) for k = 1. Suppose µk1(z, z) =µ1(z, z). Since µk1(z, s) ≤ µ1(z, z) for s = z ∈ S, µk+1
sup min [µk1(z, s), µ1(s, z)] = max [ sup
min(µk1(z, s), µ1(s, z)),
min (µk1(z, z), µ1(z, z))] = µ1(z, z). By the mathematical induction,
µn1(z, z) = µ1(z, z) for every natural number n and every z ∈ S. Letp be in S with µ(p, p) = 0. Then µ1(p, p) = θ(p, p) = ξ(p,p) < ξ(p, p).
Since µn1(z, z) = µ1(z, z) for every natural number n and every z ∈ S,
n=1 (µ ∪ θ)n](p, p) = [∪∞n=1 µ1 ](p, p) = µ1(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, KoreaE-mail: [email protected]

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