Is a pre-Lie algebra homomorphism multiplicative?

Question

A pre-Lie algebra $A$ is a vector space with a binary operation $(x, y) \mapsto xy$ satisfying $(xy)z − x(yz) = (yx)z − y(xz)$ for all $x, y, z$.

I assume that a homomorphism of pre-Lie algebras is defined as a linear map that preserves the above relation. Am I right in assuming this definition?

Further, does this imply that the homomorphism is multiplicative? I could use help in showing this.

Answer 1

A homomorphism of pre-Lie algebra is a linear map $f\colon A\rightarrow B$ with $f(a\cdot b)=f(a)\circ f(b)$, where $(A,\cdot)$ and $(B,\circ)$ are the pre-Lie products. This definition is for all "nonassociative" algebras ("not necessarily associative" algebras).

Answer 2

It does not really mean anything that the map "preserves the relation"; what is preserves is the product.

Consider the corresponding property for a morphism of associative algebras: a morphism is a linear map that preserves the product. Asking it to "preserve the associativity" is not really meaningful.