Class two near-Lie cring

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Class two near-Lie cring, all facts related to Class two near-Lie cring) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

A class two near-Lie cring is an abelian group $L$ (denoted additively) equipped with an additional binary operation $*$ satisfying the following additional conditions:

Condition name	Description
$*$ is a 2-cocycle for trivial group action from $L$ to itself	$(a*(b+c))+(b*c)=((a+b)*c)+(a*b)$ for all $a,b,c\in L$
$*$ is cyclicity-preserving	$a*b=0$ if $\langle a,b\rangle$ is cyclic.
$*$ has class two	$a*(b*c)=(a*b)*c=0$ for all $a,b,c\in L$

Relation with other properties

Stronger properties

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