LAUSR

We generalize the Lyndon–Shirshov words to the Lyndon–Shirshov Ω-words on a set X and prove that the set of all the nonassociative Lyndon–Shirshov Ω-words forms a linear basis of the free Lie Ω-algebra on the set X. From this, we establish Grobner–Shirshov bases theory for Lie Ω-algebras. As applications, we give Grobner–Shirshov bases of a free λ-Rota–Baxter Lie algebra, of a free modified λ-Rota–Baxter Lie algebra, and of a free Nijenhuis Lie algebra and, then linear bases of these three algebras are obtained.