On generalized inverses and Green’s relations

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. There exist many specific generalized inverses in the literature, such as the group inverse, the Moore-Penrose inverse [1] or the Drazin inverse ([2], [1]). Necessary and sufficient conditions for the existence of such inverses are known ([4], [2], [5], [6], [7], [8], [14], [9]), as are their properties. If one looks carefully at these results, it appears that these existence criteria all involve Green’relations [4], and that all inverses have double commuting properties. So one may wonder whether we could unify these different notions of invertible. We propose here to define a new type of generalized inverse, the inverse along an element, that is based on Green’s relation’s L, R and H [4], and ∗email: [email protected]