On a Class of Factorizable Inverse Monoids Associated with Braid Groups

Abstract

We define and study the permeable braid monoid 𝔓 ℬ _n . This monoid is closely related to the factorizable braid monoid 𝔉 ℬ _n introduced by Easdown et al. (2004), and is obtained from Artin's braid group ℬ _n by modifying the notion of braid equivalence. We show that 𝔓 ℬ _n is a factorizable inverse monoid with group of units ℬ _n and semilattice of idempotents (isomorphic to) 𝔈 𝔮 _n , the join semilattice of equivalence relations on {1,…, n}. We give several presentations of 𝔓 ℬ _n each of which extend Artin's presentation of ℬ _n . We then introduce the pure permeable braid monoid 𝔓 𝒫 _n which is related to 𝔓 ℬ _n in the same way that the pure braid group 𝒫 _n is related to ℬ _n . We show that 𝔓 𝒫 _n is the union of its maximal subgroups, each of which is (isomorphic to) a quotient of 𝒫 _n . We obtain semidirect product decompositions for these quotients, analogous to Artin's decomposition of 𝒫 _n . This structure leads to a solution to the word problem in 𝔓 ℬ _n . We conclude by giving a presentation of 𝔓 𝒫 _n which extends Artin's presentation of 𝒫 _n .

Key Words:

• Braids
• Factorizable inverse monoids
• Presentations

2000 Mathematics Subject Classification:

• Primary 20F36
• Secondary 20F05

Acknowledgments

This work was completed while the author was a postgraduate student at the University of Sydney.

Notes

Communicated by D. Easdown.