Ordered semigroups having the $P$-property

The main results of the paper are the following. The ordered semigroups which have the $P$-property are decomposable into archimedean semigroups. Moreover, the ordered semigroups which have the $P$-property are decomposable into semigroups having the $P$-property. Conversely, if an ordered semigroup $S$ is a complete semilattice of semigroups which have the $P$-property, then $S$ itself has the $P$-property as well. An ordered semigroup is $CS$-indecomposable and has the $P$-property if and only if it is archimedean. If $S$ is an ordered semigroup, then the relation $N:=\{(a,b)\mid N(a)=N(b)\}$ (where $N(a)$ is the filter of $S$ generated by $a$ $(a\in S)$) is the least complete semilattice congruence on $S$ and the class $(a)_{N}$ is $CS$-indecomposable subsemigroup of $S$ for every $a\in S$. The concept of the $P_m$-property is introduced and a characterization of the $P_m$-property in terms of the $P$-property is given. Our methodology simplifies the proofs of the corresponding results of semigroups (without order)