Simple semigroups in finite categories

TL;DR

This paper classifies finite categories with two objects where one endomorphism monoid is a group, showing the other monoid contains a simple semigroup, and constructs categories from Rees matrix semigroups.

Contribution

It establishes a structural link between groups and simple semigroups within finite categories with two objects, providing a classification and construction method.

Findings

Endomorphism monoid with a group influences the other monoid to contain a simple semigroup.

Categories with a group endomorphism can be constructed from Rees matrix semigroups.

Structural characterization of finite categories with two objects and specific endomorphism properties.

Abstract

In this paper, we classify finite categories with two objects such that one of the endomorphism monoids is a group. We prove that having a group on one side affects the structure of the other endomorphism monoid, and we prove that it is going to contain a simple semigroup. We also prove the other direction, that if we have a Rees matrix semigroup we can construct a category with two objects such that one of the objects is a group.