Flatness properties of S-posets with an approach to down-closed subposets

In this paper, we investigate the notion of dc-po-flat [Formula: see text]-posets as the ones for which the associated tensor functors transfer merely down-closed embeddings (embeddings with down-closed images in codomains) to embeddings. We investigate derived flatness notions in regard to dc-po-flatness in parallel with po-flatness notions and give examples to clarify new notions and their implications. As the characterization of flat acts by means of embeddings into cyclic acts, stated by Fleischer, is not valid for [Formula: see text]-posets, it eventuates in introducing the new notion of cyclical po-flatness, situated strictly between weak po-flatness and po-flatness, though, we express a counterpart characterization for dc-po-flatness. At the end, we expose relationships between some po-flatness properties and regular injectivity.