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 dot-depth 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. In this paper we pursue the study of the dot-depth hierarchy, a hierarchy of star-free (i.e. recognizable aperiodic) languages with connections to finite monoid theory, formal logic (Thomas [30]) and computational complexity (Barrington and Thérien [1]). The dot-depth hierarchy was first introduced by Brzozowski and Cohen [3] in 1971 and was studied by numerous authors since. It consists in a strictly increasing sequence of classes of star-free languages whose union is the class of all star-free languages. Precise definitions are given in Section 2. The dot-depth of a star-free language L is the order of the least level of this hierarchy containing L . The main open problem regarding the dot-depth hierarchy is its decidability: for a given star-free language, can we decide at which level of the hierarchy it lies, that is, can we decide its dot-depth? This question was answered positively only for the first level of the hierarchy by Simon [25], and partial answers were given for its second level (Straubing [28], Weil [34] and Straubing and Weil [29]). It is known that the dot-depth of a star-free language depends only on its syntactic monoid, so that the problem translates to an equivalent problem on finite aperiodic monoids. As was done in most previous papers on the subject, we rely mostly on this algebraic approach. The main results which we wish to present here are the following. First, in Section 3, we consider an operation on M-varieties (or pseudo-varieties of * Partial support form the following is gratefully acknowledged: PRC “Mathématiques et Informatique”, ESPRIT-BRA project 3166 “ASMICS”, NSF grant DMS8702019.