Strongly Flat and PO-Flat S-Posets

Abstract

ABSTRACT For a monoid S, a (left) S-act is a nonempty set B together with a mapping S × B → B sending (s, b) to sb such that s(tb) = (st)b and 1b = b for all s, t ∈ S and b ∈ B. Over the past three decades, an extensive theory of flatness properties has been developed (involving free acts, projective acts, strongly flat acts, Condition (P), flat acts, weakly flat acts, principally weakly flat acts, and torsion free acts). A recent and complete discussion of this area is contained in the monograph Monoids, Acts and Categories by Kilp et al. (2000). Partially ordered acts over a partially ordered monoid S, or S-posets appear naturally in the study of mappings between posets. Preliminary work on flatness properties of S-poset, was done by Fakhruddin in the 1980s (see Fakhruddin, 19861988), and continued in recent (Bulman-Fleming and Laan, 2005; Shi et al., 2005). In Bulman-Fleming and Laan (2005), the Stenström-Govorov-Lazard theorem was shown in the context of S-posets. Tensor products of S-posets, free, projective and flat S-posets, as well as an analogue of Condition (P) and Condition (E) were introduced in these articles, but strongly flat, flat, weakly flat, principally weakly flat acts and torsion free S-posets were not considered. The present article addresses these matters.