On completely regular and Clifford ordered semigroups

Lee and Kwon (Sci Math 2:247-251, 1998) defined an ordered semigroup S to be completely regular if a is an element of(a(2)Sa(2)] for every a is an element of S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e <= e(2). Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed.