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

There are well-known relations between braid and symmetric groups as well as Artin–Brieskorn braid groups and Coxeter groups: the latter are the factor-groups of the Artin–Brieskorn braid groups. The inverse braid monoid is related to the inverse symmetric monoid in the same way. 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 latter monoid.