On deleted diagonal acts of completely (0-) simple semigroups

If [Formula: see text] is a semigroup without identity, then the deleted diagonal act of [Formula: see text] is the right [Formula: see text]-act [Formula: see text]. In [S. Bulman-Fleming and A. Gilmour, Flatness properties of diagonal acts over monoids, Semigroup Forum 79 (2009) 298–314] the authors answered the question of when [Formula: see text] is flat, satisfies Condition [Formula: see text] or [Formula: see text] for a completely [Formula: see text] simple semigroup (always represented here in regular Rees matrix form). In this paper we answer similar question for some other properties. There are also some results that can arise.