On Conditions (P′) and (Pw′) for S-posets

Abstract

Partially ordered monoids (or pomonoids) [Formula: see text] acting on a partially ordered set (or poset), briefly [Formula: see text]-posets, appear naturally in the study of mappings between posets, and play an essential role in pomonoid theory. The study of flatness properties of [Formula: see text]-posets was initiated by Fakhruddin in the 1980s, and an extensive theory of flatness properties has been developed in the past several decades. The obtained results have prompted a new progress in the research area of [Formula: see text]-posets. To date, a large number of familiar properties have been generalized from acts to [Formula: see text]-posets (involving free and projective [Formula: see text]-posets, flat [Formula: see text]-posets of various sorts, [Formula: see text]-posets satisfying Conditions [Formula: see text], [Formula: see text] and [Formula: see text], and torsion free [Formula: see text]-posets). Some new properties in [Formula: see text]-posets, such as Conditions [Formula: see text], [Formula: see text] and [Formula: see text], have also been discovered. This paper continues the study of flatness properties of [Formula: see text]-posets. We first introduce Condition [Formula: see text] in the context of [Formula: see text]-posets, and characterize pomonoids by this new property of [Formula: see text]-posets. Unlike the case for acts, pomonoids over which all right [Formula: see text]-posets satisfy Condition [Formula: see text] are stronger than pogroups. Thereby, we introduce Condition [Formula: see text] similar to Condition [Formula: see text]. Furthermore, we describe Conditions [Formula: see text] and [Formula: see text] covers of cyclic [Formula: see text]-posets. Finally, we investigate direct products of [Formula: see text]-posets satisfying Conditions [Formula: see text] and [Formula: see text].