Principally weakly and weakly coherent monoids

We shall call a monoid S principally weakly (weakly) left coherent if direct products of non-empty families of principally weakly (weakly) flat right S-acts are principally weakly (weakly) flat. Such monoids have not been studied in general. However, Bulman-Fleming and McDowell proved that a commutative monoid S is (weakly) coherent if and only if the act S^I is weakly flat for each non-empty set I . In this paper we introduce the notion of finite (principal) weak flatness for characterizing (principally) weakly left coherent monoids. Also we investigate monoids over which direct products of acts transfer an arbitrary flatness property to their components.