Connected components of open semigroups in semi-simple Lie groups

This paper studies connected components of open subsemigroups of non-compact semi-simple Lie groups through the control sets in the flag manifolds and their coverings. A method for computing the number of components we call recurrent, which includes the semigroup components, is developed and it is proved that the union of this set of components is a subsemigroup. The idea of mid-reversibility comes up to show that an open semigroup has just one semigroup component if the identity belongs to its closure. A necessary and sufficient condition for mid-reversibility is proved showing that e.g. in a complex group any open semigroup is mid-reversible.