  The study of flatness properties of ordered monoids acting on posets was
initiated by S.M. Fakhruddin in the 1980's. Although there exist many papers
which investigate various properties of $S$-posets (posets equipped with a
compatible right action of an ordered monoid $S$) from free to torsion free,
among them generators, there seems to be known very little. In 2008, Laan
characterized generators in the category {\bf Pos}-$S$ of all $S$-posets with
monotone action-preserving maps between them. His characterization is similar
to the case of acts over monoids. We attempt here to collect the knowledge on
generators in the category {\bf Pos}-$S$ and to apply this to proceed on the
questions of homological classification of ordered monoids, that is results of
the type: all generators in the category {\bf Pos}-$S$, satisfy a flatness
property if and only if $S$ has a certain property.
