On weakly pullback flat S-posets

Abstract

In 2005, Bulman-Fleming and Laan established an analog of the Lazard–Govorov–Stenström theorem in the convex of [Formula: see text]-posets, which shows that an [Formula: see text]-poset [Formula: see text] is strongly flat if and only if [Formula: see text]-preserves subpullbacks and subequalizers if and only if [Formula: see text] satisfies condition (P) and condition (E). Using subpullback diagrams, the equivalent descriptions of flatness and properties (P), (PWP) and (WP) were presented by Golchin and Rezaei in 2009. In this paper, we first give the definition of weak pullback flatness by using subpullback diagram. We then introduce condition (E[Formula: see text]) and prove that a right [Formula: see text]-poset [Formula: see text] is weakly pullback flat if and only if [Formula: see text] satisfies condition (P) and condition (E[Formula: see text]). At the same time, we correct an error in the paper “Strongly flat and po-flat [Formula: see text]-posets” by Shi, Comm. Algebra 33 (2005) 4515–4531. After that we investigate homological classification problems for cyclic [Formula: see text]-posets and Rees factor [Formula: see text]-posets using weak pullback flatness. Finally, we characterize the pomonoids over which every right [Formula: see text]-poset has a weakly pullback flat cover.