Commutator collecting process

In group theory, a branch of mathematics, the commutator collecting process is a method for writing an element of a group as a product of generators and their higher commutators arranged in a certain order. The commutator collecting process was introduced by Philip Hall in 1934 and articulated by Wilhelm Magnus in 1937. The process is sometimes called a "collection process".

The process can be generalized to define a totally ordered subset of a free non-associative algebra, that is, a free magma; this subset is called the Hall set. Members of the Hall set are binary trees; these can be placed in one-to-one correspondence with words, these being called the Hall words; the Lyndon words are a special case. Hall sets are used to construct a basis for a free Lie algebra, entirely analogously to the commutator collecting process. Hall words also provide a unique factorization of monoids.