Right Regular Representation of 0 is Bijection in B-Algebra

Theorem
Let $\struct {X, \circ}$ be a $B$-algebra.

Then the right regular representation of $\struct {X, \circ}$ $0$ is a bijection.

Proof
Axiom $(\text A 2)$ for $B$-algebras states:


 * $\forall x \in X: x \circ 0 = x$

and so, for all $x \in X$:


 * $\map {\rho_0} x = x$

That is, $\rho_0 = I_X$, the identity mapping on $X$.

The result follows from Identity Mapping is Bijection.