On generalized inverses and Green's relations

TL;DR

This paper explores generalized inverses within semigroup theory using Green's relations, introducing a new notion of inverse along an element and connecting it to classical inverses like the Moore-Penrose inverse.

Contribution

It defines the inverse along an element in semigroups and demonstrates that classical generalized inverses are special cases of this new concept.

Findings

Introduces the notion of inverse along an element in semigroups

Shows classical generalized inverses are instances of this new class

Provides properties and relations of these inverses

Abstract

We study generalized inverses on semigroups by means of Green's relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse and Moore-Penrose inverse) belong to this class.