On subpullback flat S-posets

Abstract

In 1971, inspired by the work of Lazard and Govorov for R-modules over a ring R, B. Stenström proved that an S-act over a monoid S is isomorphic to a directed colimit of finitely generated free S-acts if and only if it is both pullback flat and equalizer flat. Such acts are now usually called strongly flat. In 1991, S. Bulman-Fleming discovered that every pullback flat S-act is in fact strongly flat by developing a new “interpolation” Condition (PF) for pullback-preservation. In 2005, V. Laan and S. Bulman-Fleming obtained a version of the Lazard-Govorov Theorem for S-posets over a pomonoid S, in which subpullbacks and subequalizers now take on the role previously played by pullbacks and equalizers. However, unlike the situation for acts, S. Bulman-Fleming showed in 2009 that subpullback flat S-posets need not be strongly flat, and also T. Zhao found in 2022 that Condition (PF) does not imply strongly flatness for S-posets. The present paper is devoted to the study of subpullback flatness and Condition (PF) in the context of S-posets over a pomonoid S. First we unexpectedly show that subpullback flatness is equivalent to Condition (PF). Furthermore, we give some classifications of pomonoids by subpullback flatness of (cyclic, Rees factor) S-posets, and give some classes of pomonoids that all (cyclic) S-posets have a PF-cover. Finally, we investigate products of subpullback flat S-posets.

KEYWORDS:

• Homological classification
• pomonoid
• S-poset
• subpullback flatness

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 06F05
• 20M50

Additional information

Funding

The research was supported by the National Natural Science Foundation of China (No. 12001345, 11971383) and the China Postdoctoral Science Foundation (No. 2020M673618XB).