Topology of the Semigroup of Singular Endomorphisms

The topological interpretations of some of the algebraic properties of the semigroup Sn of singular endomorphisms of an n-dimensional vector space over K are discussed here. Since Sn is known to be an idempotent generated regular semigroup, we pay more attention to the topological properties of the set En of idempotents in Sn. The local structure of En is shown to be that of a C^infinity-manifold and of a finite-dimensional vector bundle over the Grassmann manifolds. The topology of the biorder relations and sandwich sets are also discussed.