B-Algebra is Quasigroup

Theorem
Let $\left({X, \circ}\right)$ be a $B$-algebra.

Then $\left({X, \circ}\right)$ is a quasigroup.

Proof
By the definition of a quasigroup it must be shown that $\forall x \in X$ the left and right regular representations $\lambda_x$ and $\rho_x$ are permutations on $X$.

As $\left({X, \circ}\right)$ is a magma:


 * $\forall x \in X$ the codomain of $\lambda_x$ and $\rho_x$ is $X$.

Hence it is sufficient to prove that $\lambda_x$ and $\rho_x$ are bijections.

We have that:
 * $B$-Algebras are left cancellable
 * $B$-Algebras are right cancellable
 * Cancellable elements have injective regular representations.

Therefore:


 * $\forall x \in X$: $\lambda_x$ and $\rho_x$ are injective mappings.

We also have that regular representations of $B$-algebras are surjective.

Therefore:


 * $\forall x \in X$: $\lambda_x$ and $\rho_x$ are both injective and surjective mappings.

Hence by definition are $\lambda_x$ and $\rho_x$ bijections for all $x \in X$.

Note
The converse does not necessarily hold as demonstrated here.