ON HOMOLOGICAL CLASSIFICATION OF MONOIDS BY CONDITION (PWPsc) OF RIGHT ACTS

In this paper, we introduce Condition (PWPsc) as a generalization of Condition (PWPE) of acts over monoids, and we observe that Condition (PWPsc) does not imply Condition (PWPE). In general, we show that Condition (PWPsc) implies the property of being principally weakly flat, and that in left PSF monoids, the converse of this implication is also true. Moreover, we present some general properties and a homological classification of monoids by comparing Condition (PWPsc) with some other properties. Finally, we describe left PSF monoids for which SIS satisfies Condition (PWPsc) for any nonempty set I.