On the inverse braid and reflection monoids of type $B$

TL;DR

This paper explores the relationship between inverse braid monoids and inverse reflection monoids of type B, providing a presentation of the latter by extending known relations from braid and Coxeter groups.

Contribution

It establishes a new connection between inverse braid monoids and inverse reflection monoids of type B, offering a presentation of the inverse reflection monoid of this type.

Findings

Established a relation between inverse braid monoid of type B and inverse reflection monoid of type B.

Provided a presentation for the inverse reflection monoid of type B.

Extended known relations from braid and Coxeter groups to inverse monoids.

Abstract

There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.