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

Xavier MARY

doi:10.52547/cgasa.15.1.255

Abstract

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.

Highlights

In this first section, we provide the reader with the necessary definitions and results regarding semigroups and categories
Nambooripad in the case of regular semigroups [12] (Other kind of categories are studied, notably in the case of inverse semigroups, see for instance [7] and references therein). It was remarked by Costa and Steinberg [2, Theorem 3.3] that the subcategory of left S-acts with principal left ideals as objects and inner equivariant maps as morphisms is equivalent to a category constructed directly from S, which they call the Schützenberger category of the semigroup
The Schützenberger category D(S) of a semigroup S has for objects the elements of S, and morphisms are triples f = (a, x, b) with (b, c)-inverse and the Schützenberger category x ∈ aS1∩S1b

Summary

Introduction

We provide the reader with the necessary definitions and results regarding semigroups and categories. (If part) Assume that a −a→b b is invertible and let b −→e a be its inverse.

Results

Conclusion