(b, c)-inverse, inverse along an element, and the Schützenberger category of a semigroup

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 Schützenberger category of a semigroup. Applications to the Reverse Order Law are given. C Green’s relations and the Schützenberger category of a semigroup In this first section, we provide the reader with the necessary definitions and results regarding semigroups and categories. In particular, we recall the definition of the Schützenberger category of a semigroup and the interpretation of Green’s relations in this setting. Section 2 then presents the main result of the article (Theorem C.7), that (b, c)-inverses (and inverses along an element) are genuine inverses when considered as morphisms in the corresponding Schützenberger category. Finally, applications to the Reverse Order Law are given in Section 3.