Lyndon–Shirshov basis and anti-commutative algebras

Chen, Fox, Lyndon (1958) [10], and Shirshov (1958) [29], introduced non-associative Lyndon–Shirshov words and proved that they form a linear basis of a free Lie algebra, independently. In this paper we give another approach to definition of Lyndon–Shirshov basis, i.e., we find an anti-commutative Gröbner–Shirshov basis S of a free Lie algebra such that Irr(S) is the set of all non-associative Lyndon–Shirshov words, where Irr(S) is the set of all monomials of N(X), a basis of the free anti-commutative algebra on X, not containing maximal monomials of polynomials from S. Following from Shirshovʼs anti-commutative Gröbner–Shirshov bases theory (Shirshov, 1962 [32]), the set Irr(S) is a linear basis of a free Lie algebra.