The work developed focused on the theory of pseudovarieties of finite semigroups and ordered semigroups. In general, one is given a certain class of such structures and one seeks algorithmic characterizations of the pseudovarieties they generate. A fruitful method involves studying relatively free profinite semigroups and some of their relatively free algebras with respect to enriched algebraic structures, as well as the induced topology. The connections of such structures with symbolic dynamics have also proved to shed light on their understanding and were again exploited.