Braids and partial permutations

We study the inverse braid monoid IBn introduced by Easdown and Lavers in 2004. We completely describe the factorizable structure of IBn and use this to give a new proof of the Easdown-Lavers presentation; we also derive several new presentations, each of which gives rise to a new presentation of the symmetric inverse monoid. We then define and study the pure inverse braid monoid IPn which is related to IBn in the same way that the pure braid group is related to the braid group.