Some results on the dot-depth hierarchy

In this paper we pursue the study of the decidability of the dot-depth hierarchy. We give an effective lower bound for the dotdepth of an aperiodic monoid. The main tool for this is the study of a certain operation on varieties of finite monoids in terms of Mal'cev product. We also prove the equality of two decidable varieties which were known to contain all dot-depth two monoids. Finally, we restrict our attention to inverse monoids, and we prove that the class of inverse dot-depth two monoids is locally finite.