Thai Journal of Mathematics Volume 4(2006) Number 2 : 403–415 www.math.science.cmu.ac.th/thaijournal On the Least (Ordered) Semilattice Congruence in Ordered Γ-Semigroups M. Siripitukdet and A. Iampan Abstract : In this paper, we firstly characterize the relationship between the (ordered) filters, (ordered) s-prime ideals and (ordered) semilattice congruences in ordered Γ-semigroups. Finally, we give some characterizations of semilattice congruences and ordered semilattice congruences on ordered Γ-semigroups and prove that 1. n is the least semilattice congruence, 2. N is the least ordered semilattice congruence, 3. N is not the least semilattice congruence in general. Keywords : Ordered Γ-semigroup; (ordered) filter; (ordered) s-prime ideal; Least (ordered) semilattice congruence. 2000 Mathematics Subject Classification : 20M99, 06F99, 06B10. 1 Preliminaries In 1998, Gao [8] gives some characterizations of semilattice congruences and ordered semilattice congruences on ordered semigroups. Now we also character- ize the semilattice congruences and ordered semilattice congruences on ordered Γ-semigroups and give some characterizations of semilattice congruences and or- dered semilattice congruences on ordered Γ-semigroups analogous to the character- izations of semilattice congruences and ordered semilattice congruences on ordered semigroups. Let M and Γ be any two nonempty sets. M is called a Γ-semigroup [3, 4] if there exists a mapping M × Γ × M −→ M , written as (a,γ,b) −→ aγb, satisfying the following identity (aαb)βc = aα(bβc) for all a, b, c ∈ M and α, β ∈ Γ. A Γ- semigroup M is called a commutative Γ-semigroup if aγb = bγa for all a, b ∈ M and γ ∈ Γ. A nonempty subset K of a Γ-semigroup M is called a sub-Γ-semigroup of M if aγb ∈ K for all a, b ∈ K and γ ∈ Γ. For examples of Γ-semigroups, see [1, 3, 4]. A partially ordered Γ-semigroup M is called an ordered Γ-semigroup (po-Γ- semigroup) if for any a, b, c ∈ M and γ ∈ Γ,a ≤ b implies aγc ≤ bγc and cγa ≤ cγb.