Taming the pseudovariety DAb

Schützenberger started in the mid 1970's the investigation of pseudovarieties of finite semigroups whose regular D-classes are subsemigroups, motivated by their relevance in formal language theory according to Eilenberg's correspondence. In our present work, we focus on the special case DAb, where such subsemigroups are Abelian groups. Steinberg and the speaker introduced a strong form of decidability for pseudovarieties called tameness. Besides standard algorithmic properties such as the word problem for relatively free algebras over a suitable signature, it involves a purely topological property: for the finite systems of equations of interest (which depend on the sought applications), the solutions modulo the pseudovariety in the chosen signature are dense in the set of all solutions in the free profinite semigroup. The purpose of this talk is to establish that the pseudovariety DAb is tame. The main ingredient is a normal form for pseudowords over DAb.

(Joint work with M. Kufleitner and J. Ph. Wächter)