Left gyrogroup

Definition
A magma with underlying set $$G$$ and binary operation $$*$$ is termed a gyrogroup if the following hold:


 * Left identity and left inverse: There is an element $$e \in G$$ such that $$e$$ is a left neutral element and every element has a left inverse with respect to $$e$$. In other words:

$$e * x = x \forall x \in G$$

and for all $$a \in G$$, there exists $$b \in G$$ such that:

$$b * a = e$$


 * Gyroassociativity: For any $$a,b,c \in G$$, there is a unique element $$gyr[a,b]c \in G$$ such that:

$$a * (b * c) = (a * b) * (\operatorname{gyr}[a,b]c)$$


 * Gyroautomorphism: $$\operatorname{gyr}[a,b]$$ is a magma automorphism of $$G$$ for all $$a,b \in G$$. This is called the Thomas gyration, or gyroautomorphism, of $$G$$.


 * Weak loop property: If $$b * a = e$$, then

$$gyr[a,b] = \operatorname{id}$$

Thus, the definition of left gyrogroup differs from that of gyrogroup, only in that the left loop property is relaced by the weak loop property.

Stronger structures

 * Group
 * Gyrogroup

Representations of left gyrogroups
By a representation of a left gyrogroup, we mean an expression of it as a left gyrotransversal of a subgroup.