On a complete set of generators for dot-depth two

A complete set of generators for Straubing’s dot-depth-two monoids has been characterized as a set of quotients of the form A*/ wCn, , ,,, where n and M denote positive integers, A* denotes the free monoid generated by a finite alphabet A, and mCn,,,) denote congruences related to a version of the Ehrenfeucht-Frays& game. This paper studies combinatorial properties of the N~,,~)‘s and in particular the inclusion relations between them. Several decidability and inclusion consequences are discussed.