# Basic commutator

regular commutator

An object inductively constructed from the elements of a given set and from brackets, in the following manner. The elements of are considered by definition to be basic commutators of length 1, and they are given an arbitrary total order. The basic commutators of length , where where is an integer, are defined and ordered as follows. If are basic commutators of lengths smaller than , then is considered to be a basic commutator of length if and only if the following conditions are met: 1) are basic commutators of lengths and , respectively, and ; 2) ; and 3) if , then . The basic commutators of length not exceeding thus obtained are arbitrarily ordered, subject to the condition that , while preserving the order of the basic commutators of lengths less than . Any set of basic commutators constructed in this way is a base of the free Lie algebra with as set of free generators [1].

#### References

 [1] A.I. Shirshov, "On bases of free Lie algebras" Algebra i Logika , 1 : 1 (1962) pp. 14–19 (In Russian)