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.We show that Mary’s inverse along an element, Drazin’s (b, c)-inverse, and Bott-Duffin (e,f)-inverse of a given element are just three different ways of representing the same notion – the outer inverse of this element belonging to the prescribed Green’s ℋ-class. Hence, outer inverses belonging to the prescribed Green’s ℛ- and ℒ-classes represent extensions of (b,c)-inverses and 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.