Inclusion relations between some congruences related to the dot-depth hierarchy Original Research Article

A complete class of generators for Straubingʹs dot-depth k monoids has been characterized as a class of quotients of the form A∗/∼m̄ where A∗ denotes the free monoid over a finite alphabet A, m̄ denotes a k-tuple of positive integers, and ∼m̄ denotes a congruence related to an Ehrenfeucht-Fraïssé game. In this paper, we first reduce the complete class of generators for dot-depth k to a complete class whose members are of dot-depth exactly k. We then study all the inclusion relations between the resulting congruences ∼m̄. Several applications of these relations are discussed. For instance, a conjecture of Pin (which was shown by the author to be false in general) is shown to be true in an important special case.