(b, c)-INVERSE, INVERSE ALONG AN ELEMENT, AND THE SCHUTZENBERGER CATEGORY OF A SEMIGROUP
Xavier Mary · HAL (Le Centre pour la Communication Scientifique Directe) · 2021
In this short note, we study two specific generalized inverses, the (b, c)-inverse of Drazin and the inverse along an element of the present author, in a categorical way. Precisely, we prove that the (b, c)-inverse and the inverse along an element in a semigroup are actually genuine inverse when considered as morphisms in the Schutzenberger category of the semigroup. This category was first defined and studied by Costa and Steinberg, and its definition is based on Green's preoders and relations. In the last section, applications to the Reverse Order Law are given and some new formulas are provided. Examples are also given.