Characterization of Pomonoids by Properties of Generators

TL;DR

This paper explores the properties of generators in the category of S-posets, aiming to classify ordered monoids based on the flatness properties of their generators, extending previous characterizations.

Contribution

It consolidates knowledge on generators in Pos-S and applies this to homological classification of ordered monoids based on flatness properties.

Findings

Characterization of generators in Pos-S category.

Connection between flatness properties of generators and monoid properties.

Extension of known results to homological classification.

Abstract

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.