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 semigroups. 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: (...

We first introduce the notion of a completion. Completions are inductive properties which may be expressed in a declarative way and which may be combined. We show that completions may be used for describing structures or transformations which appear in combinatorial topology. We present two completions, 〈CUP〉 and 〈CAP〉, in order to define, in an axiomatic way, a remarkable collection of acyclic...

We consider smooth completion of algebraic manifolds. Having some information about its singular completions or about completions of its images we prove purity of cohohomology of the set at infinity. We deduce also some topological properties. The work is based on the study of perverse direct images for algebraic maps.

The second statement of Corollary 3 was first proved by Almeida and Volkov using techniques coming from symbolic dynamics [2]. Our next result generalizes a result of Rhodes and the author showing that all elements of finite order in F̂V(A) are group elements, which played a key role in proving that such elements are in fact idempotent [7]. Let N̂ denote the profinite completion of the monoid of ...