The Non-symmetric Operad Pre-lie Is Free

Operads are a specific tool for encoding type of algebras. For instance there are operads encoding associative algebras, commutative and associative algebras, Lie algebras, pre-Lie algebras, dendriform algebras, Poisson algebras and so on. A usual way of describing a type of algebras is by giving the generating operations and the relations among them. For instance a Lie algebra L is a vector space together with a bilinear product, the bracket (the generating operation) satisfying the relations [x, y] = −[y, x] and [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z ∈ L. The vector space of all operations one can perform on n distinct variables in a Lie algebra is Lie(n), the building block of the symmetric operad Lie. Composition in the operad corresponds to composition of operations. The vector space Lie(n) has a natural action of the symmetric group, so it is a symmetric operad. The case of associative algebras can be considered in two different ways. An associative algebra A is a vector space together with a product satisfying the relation (xy)z = x(yz). The vector space of all operations one can perform on n distinct variables in an associative algebra is As(n), the building block of the symmetric operad As. The vector space As(n) has for basis the symmetric group Sn. But, in view of the relation, one can look also at the vector space of all order-preserving operations one can perform on n distinct ordered variables in an associative algebra: this is a vector space of dimension 1 generated by the only operation x1 · · ·xn. So the non-symmetric operad Ãs describing associative algebras is 1-dimensional for each n: this is the terminal object in the category of non-symmetric operads.