Profinite semigroups provide powerful tools to understand properties of classes of regular languages. Until very recently however, little was known on the structure of "large" relatively free profinite semi- groups. In this paper, we present new results obtained for the class of all finite aperiodic (that is, group-free) semigroups. Given a finite alphabet X, we focus on the following problems: (1) the word problem for ω-terms on X evaluated on the free pro-aperiodic semigroup, and (2) the computation of closures of regular languages in the ω-subsemigroup of the free pro-aperiodic semigroup generated by X.FCT through the Centro de Matemática da Universidade do Minho and the Centro de Matemática da Universidade do PortoEuropean Community Fund FEDERESF programme “Automata: from Mathematics to Applications (AutoMathA)”Pessoa Portuguese-French project Egide-Grices 11113Y
