on homological classification of monoids by condition (psc) and new classification on condition (pe)

In 1997, golchin and renshaw introduced condition (pe) and showed that this condition implies weak flatness, although the converse is not generally valid. in this paper, we present condition (psc) as a generalization of condition (pe). we also see that condition (psc) implies weak flatness, but the converse is not necessarily true. however, for left p sf monoids the converse is holds. moreover, we discuss some general properties and provide a homological classification of monoids by comparing condition (psc) with some other properties. furthermore, a new homological classification of monoids is presented by comparing condition (pe) with other properties.