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. Mathematics Subject Classification (2010). 20M99, 15A09, 15A24.