3-additive Lie cring

Definition

A 3-additive Lie cring is a Lie cring $L$ (where the abelian group structure is denoted additively and the cring operation is denoted $*$) satisfying the following condition:

The maps $L\times L\times L\to L$ given by $(x,y,z)\mapsto (x*y)*z$ and $(x,y,z)\mapsto x*(y*z)$ are both additive in each of their three inputs.

As such, this would give six conditions, but because of skew symmetry, this is equivalent to both the following two conditions holding:

Condition name	Description
Inner 3-additivity	$((x_{1}+x_{2})*y)*z=((x_{1}*y)*z)+((x_{2}*y)*z)$ for all $x_{1},x_{2},y,z\in L$
Outer 3-additivity	$(x*y)*(z_{1}+z_{2})=((x*y)*z_{1})+((x*y)*z_{2})$

Note also that because $x*0=0*x=0$ for all $x$, we can deduce multiplication by integral scalars, including negative ones, can be pulled out of any triple product.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
class two Lie cring
Lie ring