Complete semilattice decompositions of ordered semigroups

An ordered semigroup (po-semigroup) is a partially ordered set S and at the same time a semigroup such that for any a, 6, x Є 5,

For H Ç 5, (#] := {t Є 5Ί(3/ι Є Я) t < h}. In this paper, S stands for an arbitrary ordered semigroup.