Abstract

Lee and Kwon (Sci Math 2:247–251, 1998) defined an ordered semigroup S to be completely regular if $$a \in (a^2Sa^2]$$ for every $$a \in S$$ . 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 $$e \le e^2$$ . 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.

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