Combinatorics of Hall trees and Hall words

AbstractWe present combinatorial properties of Hall trees and Hall words. We give new proofs of the unique factorisation of words into decreasing products of Hall words. An order < H on the free monoid is then constructed upon the unique factorisation of words. When the Hall set is the set of Lyndon words, the order < H coincides with the lexicographical order (with which Lyndon words are defined). Motivated by this fact, we give combinatorial properties of Hall words, related to the order < H, which generalize known properties of Lyndon words to Hall words