Equivalence of definitions of inverse property loop

This article gives a proof/explanation of the equivalence of multiple definitions for the term inverse property loop
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a loop $(L,*)$.

1. Existence of left and right inverses: There exist bijective maps $\lambda,\rho:L \to L$ such that $\lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L$.
2. Existence of two-sided inverses: There exists a bijective map ${}^{-1}: L \to L$ such that $a^{-1} * (a * b) = (b * a) * a^{-1} = b$ for all $a,b \in L$.

Proof

(1) implies (2)

It suffices to show that $\lambda = \rho$; then we can take the inverse map to equal that.

Given: A loop $(L,*)$ with a map $\lambda$ bijective maps $\lambda,\rho:L \to L$ such that $\lambda(a) * (a * b) = (b * a) * \rho(a) = b \ \forall \ a, b \in L$.

To prove: For any $x \in L$, $\lambda(x) = \rho(x)$.

Proof: Consider the product $(\lambda(x) * x) * \rho(x)$.

Putting $a = b = x$ and using the property of $\rho$, this simplifies to $\lambda(x)$. On the other hand, we know that $\lambda(x) * x = \lambda(x) * (x * e) = e$ where $e$ is the identity element, so the expression $(\lambda(x) * x) * \rho(x)$ simplifies to $\rho(x)$. Thus, $\lambda(x) = \rho(x)$.

(2) implies (1)

We can take both $\lambda$ and $\rho$ as the inverse map.