The least regular order with respect to a regular congruence on ordered .-semigroups

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 .-semigroups. We can see that any semigroup can be considered as a .-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered .-semigroups and prove the following statements on an ordered .-semigroup M: (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the .-semigroup M/. with respect to a regular congruence . on M. (3) The regular congruences are not ordered semilattice congruences in general.