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 Lazard-divided Lie operad.