Lazard set

A subset of the free magma , i.e. the free non-associative structure over (cf. also Associative rings and algebras). The elements of correspond to completely bracketed words over (or rooted planar binary trees with leaves labelled by generators ; cf. also Binary tree). These are defined recursively as brackets where are bracketed words of lower weight; bracketed words of weight one correspond to the generators . A subset is said to be closed, if for each element one has . Given two elements , one writes to denote the element

Consider trees and subsets defined as follows:

(a1)

A Lazard set is a subset such that for any finite, non-empty and closed subset one has:

for some , (a1) holds and, moreover, .

Lazard sets may be shown to coincide with Hall sets (cf. Hall set). Thus, they give bases of the free Lie algebra over ; that is, one may associate a Lie polynomial to each element of a Lazard set such that the free Lie algebra (over ; cf. Lie algebra, free) is freely generated (as a module over a commutative ring ) by the Lie polynomials . Lazard's elimination process may then be phrased as follows: One has the direct sum decomposition (as a module over a commutative ring ):

where is the Lie subalgebra freely generated by .

Lazard sets were introduced by X. Viennot [a1] in order to unify combinatorial constructions of bases of the free Lie algebra. The Lyndon basis (see Lyndon word) was thought to be of a different nature from the one considered by M. Hall [a2], and generalizations of it were proposed by many authors. Viennot gave a unifying framework for all these constructions. One may present Lazard sets in terms of words, rather than trees in . It can then be shown that a unique tree structure is attached to every word of a Lazard set. Moreover, a Lazard set of words is totally ordered, as is a Lazard set of trees, and it is a complete factorization of the free monoid. That is, every word is a unique non-increasing product of Lazard words. This result makes explicit the link between bases of free Lie algebras and complete factorizations of free monoids.

See also Hall word.

References

[a1]	X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Lecture Notes in Mathematics , 691 , Springer (1978)
[a2]	M. Hall, "A basis for free Lie rings and higher commutators in free groups" Proc. Amer. Math. Soc. , 1 (1950) pp. 57–581
[a3]	C. Reutenauer, "Free Lie algebras" , London Math. Soc. Monographs New Ser. , 7 , Oxford Univ. Press (1993)