Right Regular Representation of 0 is Bijection in B-Algebra
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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.
$\blacksquare$