Narrow Results

In this paper, we investigate the notion of dc-po-flat $S$-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 $S$-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.

Over the past four decades an extensive literature covered the properties of $S$-acts. However, only few studies had generalized some known properties of $S$-acts to the $S$-posets. The reversible, and indecomposable properties in $S$-posets have been addressed previously but connectivity has not been defined in $S$-posets yet. Connectivity property was found to be related to those of reversibility and flatness in the category of $S$-acts. The primary objective of this paper is to define connectivity in the category of $S$-posets for both versions: ordered “poconnected” and unordered “connected”. Examples are presented to show the difference between the two versions. The relationship between connectivity with other properties such as reversibility, and indecomposability had also been investigated. We show that the poconnected in $S$-posets is always indecomposable, but the inverse is not true. We also find that the weakly reversible is always connected and indecomposable. These relations among these properties in $S$-posets are different from their corresponding relations in $S$-acts.