Lazard-divided Lie ring

Definition
A Lazard-divided Lie ring is a Lie ring $$L$$ equipped with additional multilinear operations, one for each prime number $$p$$, of the form:

$$t_p: L \times L \times \dots \times L \to L$$

where there are $$p$$ copies of $$L$$

such that the following holds for all $$x_1,x_2,\dots,x_p \in L$$:

$$pt_p(x_1,x_2,\dots,x_p) = [[ \dots [x_1,x_2],\dots,x_p]$$

and further, such that every identity for which some multiple is an identity in Lie ring theory must hold.

The operations $$t_p$$ are called the Lazard division operations.

More abstractly, a Lazard-divided Lie ring is a representation of the defining ingredient::Lazard-divided Lie operad.

Related notions

 * Lazard-divided Lie operad
 * Lazard-divided Lie subring
 * Lazard-divided Lie ideal