The Dot-Depth of a Generating Class of Aperiodic Monoids is Computable

Given a finite alphabet A and a sequence of positive integers congruences on A*, denoted by and related to a version of the Ehrenfeucht-Fraisse game, have been defined by Thomas in order to give a new proof that the Brzozowski’s dot-depth hierarchy of star-free languages is infinite. A natural extension of some of the results of Thomas states that the monoid variety corresponding to level k of the Straubing hierarchy (the Straubing hierarchy is closely related to the Brzozowski’s dot-depth hierarchy) can be characterized in terms of the monoids . In this paper, it is shown that the dot-depth of the ’s is computable.