Equivalence of definitions of gyrogroup

The definitions that we have to prove are equivalent
We want to prove that the following two definitions are equivalent.

Minimal 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:

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

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 $$\operatorname{gyr}[a,b]c \in G$$ such that:

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


 * Gyroautomorphism: $$\operatorname{gyr}[a,b]$$ (i.e., the map that sends $$c$$ to $$\operatorname{gyr}[a,b]c$$) is a magma automorphism of $$G$$. This is called the Thomas gyration, or gyroautomorphism, of $$G$$.


 * Left loop property: The following are equal as automorphisms of $$G$$:

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

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


 * Two-sided identity and two-sided inverse: There is a unique element $$e \in G$$ such that $$e$$ is a two-sided neutral element and every element has a unique two-sided inverse element with respect to $$e$$. In other words:

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

and for all $$a \in G$$, there exists a unique two-sided inverse $$b \in G$$ such that:

$$b * a = a * b = e$$

The element $$b$$ is denoted $$a^{-1}$$.


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

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


 * Gyroautomorphism: $$\operatorname{gyr}[a,b]$$ (i.e., the map that sends $$c$$ to $$\operatorname{gyr}[a,b]c$$) is a magma automorphism of $$G$$. This is called the Thomas gyration, or gyroautomorphism, of $$G$$.


 * Left loop property: The following are equal as automorphisms of $$G$$:

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

Related facts

 * Semigroup with left neutral element where every element is left-invertible equals group
 * Monoid where every element is left-invertible equals group

Facts used

 * 1) uses::Binary operation on magma determines neutral element, which follows from uses::equality of left and right neutral element

Proof
We will show that the minimal definition implies the maximal definition, i.e., the existence of a left neutral element and left inverses, along with the other axioms, forces the left neutral element to be the unique two-sided neutral element and forces the left inverse to be the unique two-sided inverse.

Proof of two-sided neutral element and two-sided inverse
Given: A gyrogroup $$(G,*)$$ (as per the minimal definition) with left neutral element $$e$$.

To prove: $$e$$ is a two-sided neutral element and every element has a two-sided inverse. Explicitly, for all $$a \in G$$, we have $$e * a = a * e = a$$, and there exists $$b \in G$$ such that $$b * a = a * b = e$$.

Proof: We fix $$a \in G$$.

Proof of uniqueness of neutral element
This is direct from Fact (1) -- a two-sided neutral element, if it exists, must be unique.

Proof of uniqueness of two-sided inverse
We will show something stronger, any left inverse must equal any right inverse.

Given: A gyrogroup $$(G,*)$$ with two-sided neutral element $$e$$. An element $$a \in G$$ with left inverse $$b$$ and right inverse $$d$$. In other words, $$b * a = a * d = e$$.

To prove: $$b = d$$

Proof: