Lazard set

A subset $ T $ of the free magma $ M ( A ) $, i.e. the free non-associative structure over $ A $ (cf. also Associative rings and algebras). The elements of $ M ( A ) $ correspond to completely bracketed words over $ A $ (or rooted planar binary trees with leaves labelled by generators $ a _ {1} , a _ {2} , \dots $; cf. also Binary tree). These are defined recursively as brackets $ t = [ t ^ \prime , t ^ {\prime \prime } ] $ where $ t ^ \prime , t ^ {\prime \prime } $ are bracketed words of lower weight; bracketed words of weight one correspond to the generators $ a _ {1} , a _ {2} , \dots $. A subset $ E \subset M ( A ) $ is said to be closed, if for each element $ t = [ t ^ \prime , t ^ {\prime \prime } ] \in E $ one has $ t ^ \prime , t ^ {\prime \prime } \in E $. Given two elements $ s, t \in M ( A ) $, one writes $ [ st ^ {p} ] $ to denote the element

$$ [ \dots [ [ s, t ] , t ] \dots t ] \qquad (p \textrm{ closing brackets} ) . $$

Consider trees $ t _ {0} \dots t _ {n} $ and subsets $ T _ {0} \dots T _ {n + 1 } \subset M ( A ) $ defined as follows:

$$ \tag{a1 } \left . \begin{array}{l} t _ {0} \in T _ {0} = A, \\ t _ {1} \in T _ {1} = \left \{ {[ tt _ {0} ^ {p} ] } : {p \geq 0, t \in T _ {0} \setminus t _ {0} } \right \} , \\ \dots \dots \\ t _ {n} \in T _ {n} = \left \{ {[ tt _ {n - 1 } ^ {p} ] } : {p \geq 0, t \in T _ {n - 1 } \setminus t _ {n - 1 } } \right \} , \\ t _ {1} \in T _ {1} = \left \{ {[ tt _ {0} ^ {p} ] } : {p \geq 0, t \in T _ {0} \setminus t _ {0} } \right \} , \\ \dots \dots \\ t _ {n} \in T _ {n} = \left \{ {[ tt _ {n - 1 } ^ {p} ] } : {p \geq 0, t \in T _ {n - 1 } \setminus t _ {n - 1 } } \right \} , \\ \end{array} \right \} $$

A Lazard set is a subset $ L \subset M ( A ) $ such that for any finite, non-empty and closed subset $ E \subset M ( A ) $ one has:

$$ L \cap E = \{ t _ {0} > \dots > t _ {n} \} $$

for some $ n \geq 0 $, (a1) holds and, moreover, $ T _ {n + 1 } \cap E = \emptyset $.

Lazard sets may be shown to coincide with Hall sets (cf. Hall set). Thus, they give bases of the free Lie algebra over $ A $; that is, one may associate a Lie polynomial $ \psi ( t ) $ to each element $ t \in L $ of a Lazard set such that the free Lie algebra $ {\mathcal L} ( A ) $ (over $ A $; cf. Lie algebra, free) is freely generated (as a module over a commutative ring $ K $) by the Lie polynomials $ \{ {\psi ( t ) } : {t \in L } \} $. Lazard's elimination process may then be phrased as follows: One has the direct sum decomposition (as a module over a commutative ring $ K $):

$$ {\mathcal L} ( A ) = K \psi ( t _ {0} ) \oplus \dots \oplus K \psi ( t _ {n} ) \oplus {\mathcal L} _ {n + 1 } , $$

where $ {\mathcal L} _ {n + 1 } $ is the Lie subalgebra freely generated by $ T _ {n + 1 } $.

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 $ M ( A ) $. 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)