Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

Abstract

We provide existence criteria and characterizations for outer inverses in a semigroup belonging to the prescribed Green's R-, L -and H -classes. These results generalize the well-known problem of finding outer inverses of a matrix over a field with the prescribed range or/and null space. Outer inverses belonging to the prescribed Green's R-and L -classes also represent extensions of Drazin's (b, c)-inverses and Mary's inverses along an element. We provide an overview of other such extensions that have emerged recently and compare them with the extensions introduced in this paper.