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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∈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. 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 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 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 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.
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Department of MathematicsSeoul Women’s UniversitySeoul 139–774, KoreaE-mail: [email protected]

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