On a class of factorizable inverse monoids associated with braid groups

We define and study the permeable braid monoid ℬBn. This monoid is closely related to the factorizable braid monoid F-fraktur sign Bn introduced by Easdown et al. (2004), and is obtained from Artin's braid group Bn by modifying the notion of braid equivalence. We show that ℬBn is a factorizable inverse monoid with group of units Bn and semilattice of idempotents (isomorphic to) Gn, the join semilattice of equivalence relations on {1,..., n}. We give several presentations of ℬBn each of which extend Artin's presentation of Bn. We then introduce the pure permeable braid monoid ℬPn which is related to ℬBn in the same way that the pure braid group Pn is related to Bn. We show that ℬPn is the union of its maximal subgroups, each of which is (isomorphic to) a quotient of Pn. We obtain semidirect product decompositions for these quotients, analogous to Artin's decomposition of Pn. This structure leads to a solution to the word problem in ℬBn. We conclude by giving a presentation of ℬPn which extends Artin's presentation of Pn. Note: Some of the scientific symbols cannot be represented correctly in the abstract. Please read with caution and refer to the original publication.