# Remarks on Enveloping Semigroups

## TL;DR

This paper investigates the structure of enveloping semigroups of simple groups, characterizing certain submonoids, introducing the concept of a navel, and analyzing orbit counts in type A.

## Contribution

It provides a detailed structural analysis of enveloping semigroups, introduces the navel concept, and derives orbit generating series for type A groups.

## Key findings

- All J-coirreducible connected stabilizer submonoids are classified.
- The navel of a reductive monoid is defined and studied.
- The generating series for G×G-orbits in type A is derived.

## Abstract

The local structures of enveloping semigroups of simple groups are investigated. All J-coirreducible connected stabilizer submonoids are determined. The notion of a navel of a reductive monoid is introduced. The cross-section lattice of the enveloping monoid is shown to be atomic. In type A, the generating series for the number of $G\times G$-orbits is found.