Inclusion Relations Between Some Congruences Related to the Dot-depth Hierarchy

Abstract A complete class of generators for Straubing's dot-depth k monoids has been characterized as a class of quotients of the form A∗/∼ m where A∗ denotes the free monoid over a finite alphabet A, m denotes a k-tuple of positive integers, and ∼ m denotes a congruence related to an Ehrenfeucht-Fraisse 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 ∼ m . 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.

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