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 $\struct {X, \circ}$ with respect to $0$ is a bijection.

Proof

$B$-Algebra Axiom $(\text A 2)$ states:

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

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

$\map {\rho_0} x = x$

That is:

$\rho_0 = I_X$

which is the identity mapping on $X$.

The result follows from Identity Mapping is Bijection.

$\blacksquare$