Semisimplicity and von Neumann's regularity

The purpose of this paper is to link semigroup theory and category theory by using Green's relations to study regular morphisms and semisimple objects in a category which is filtering in the sense that every morphism admits an epic-monic factorization. For example, the following results are shown for a filtering category C ∶ (1)C is regular if and only if every monic inC is left invertible and every epic inC is right invertible: (2) the monoid of endomorphisms of any semisimple object ofC is regular. Communicated by A.H. Clifford