$\mathcal{CS}$-indecomposable ordered semigroups

An ordered semigroup $S$ is called $\mathcal{CS}$-indecomposable if the set $S\times S$ is the only complete semilattice congruence on $S$. In this paper we prove that each ordered semigroup is, uniquely, complete semilattice of $\mathcal{CS}$-indecomposable semigroups, which means that it can be decomposed into $CS$-indecomposable components in a unique way. Furthermore, the $\mathcal{CS}$-indecomposable ordered semigroups are exactly the ordered semigroups which do not contain proper filters.