Algebra Seminar talk
Relatively free profinite semigroups
For their intimate connections with the theory of rational languages, relatively free profinite semigroups have become a well-established tool in many aspects of the theory and its applications, particularly in the framework of Eilenberg-type algebraic classification schemes. They serve as a powerful descriptive tool and, despite their rather general tendency to be uncountable, they often provide a route to decidability results. By focusing our attention on rich structural entities, they also fit well into the solid ground of classical Mathematics, opening at the same time doors to toolboxes and drawing analogies with other areas, such as group theory, topology, and dynamical systems.
The purpose of the talk is to introduce and put into (the author's) perspective recent results concerning relatively free profinite semigroups. They are all concerned with various structural aspects of free profinite semigroups over `large' pseudovarieties. Until recently, not much was known in such cases.