Partial symmetry, reflection monoids and Coxeter groups

TL;DR

This paper introduces the theory of reflection monoids, connecting inverse semigroups with reflection groups, and identifies key examples, their structures, and orders, laying groundwork for further research.

Contribution

It initiates the theory of reflection monoids, linking inverse semigroups to reflection groups, and provides new examples and order calculations.

Findings

Identified important inverse semigroups as reflection monoids

Introduced new examples of reflection monoids

Determined the orders of these monoids

Abstract

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.