Characterization of Pomonoids by Properties of Generators

Setareh Irannezhad, Ali Madanshekaf

Published 2015-03-17Version 1

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.