Acta Mathematica Sinica, English Series May, 2012, Vol. 28, No. 5, pp. 975–982 Published online: November 25, 2011 DOI: 10.1007/s10114-011-9702-x Http://www.ActaMath.com Acta Mathematica Sinica, English Series © Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2012 The Least Regular Order with Respect to a Regular Congruence on Ordered Γ-Semigroups Manoj SIRIPITUKDET Aiyared IAMPAN Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand E-mail : manojs@nu.ac.th aiyared.ia@up.ac.th Abstract The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M: (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congru- ence ρ on M. (3) The regular congruences are not ordered semilattice congruences in general. Keywords Ordered semigroup, (ordered) Γ-semigroup, regular congruence, ordered semilattice con- gruence and (least) regular order MR(2000) Subject Classification 20N99, 06F99, 06B10 1 Preliminaries Recall that a semilattice congruence N on an ordered semigroup S is an ordered semilattice congruence on S. Thus the S/N endowed with the multiplication (x) N · (y) N =(xy) N is still a semigroup. If we define (x) N  (y) N if and only if (x) N =(xy) N , then it can be easily seen that (S/N ; ·, ) is an ordered semigroup. We now call this semigroup (S/N ; ·, ) the natural ordered semigroup induced by the ordered semilattice congruence N on S. In 1997, Xie and Wu [1] proved that the semilattice congruence N on an ordered semigroup S is the least regular semilattice congruence. In 2004, Dutta and Adhikari [2] introduced the concepts of ordered Γ-semigroups. In 2004, Xu and Ma [3] showed that there exists an order- preserving bijection between the set of all prime ideals of the ordered semigroup (S; ·, ≤) and the set of all prime ideals of (S/N ; ·, ). Moreover, they gave some necessary and sufficient conditions for the natural ordered semigroup (S/N ; ·, ) to be a chain. In 2006, Siripitukdet and Iampan [4] characterized the relationship among the (ordered) filters, (ordered) s-prime ideals and (ordered) semilattice congruences in ordered Γ-semigroups and gave some characterizations of semilattice congruence and ordered semilattice congruence on ordered Γ-semigroups. They proved that Received November 27, 2009, revised April 15, 2010, accepted October 22, 2010