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

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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. 
@article{ZNSL_2007_343_a8,
     author = {N. Kehayopulu and M. Tsingelis},
     title = {$\mathcal{CS}$-indecomposable ordered semigroups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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     publisher = {mathdoc},
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     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_343_a8/}
}
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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/