Geodesic
$\mathcal{CS}$-indecomposable ordered semigroups
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 222-232 
N. Kehayopulu; M. Tsingelis. $\mathcal{CS}$-indecomposable ordered semigroups. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 15, Tome 343 (2007), pp. 222-232. http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a8/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

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