Some Topological Properties of Rational Sets
Pierre‐Cyrille Héam · Universitätsbibliothek Gießen · 2001
In this paper we give some topological properties of rational sets. The profinite topology was first introduced for the free group by M. Hall, Jr. and by Reutenauer for the free monoid. It is the initial topology for the monoid morphisms from the free monoid into a finite discrete group. For a variety of finite groups $\V$, the pro-$\V$ topology is defined in the same way by replacing "group" by "group in $\V$" in the definition. We prove in this paper that the set of accumulation points for the pro-Vtopology of a rational language of the free monoid is a rational language too (if $\V$ is extension closed) and we give an algorithm to compute it for the profinite and the pro-$p$ topologies. We also give a polynomial time algorithm to test whether a rational set of the free group is the profinite closure of a rational language of the free monoid. Finally we prove that there exist languages which are closed and open and which are not group languages.