B-Algebra Identity: xy = 0 iff x = y

Theorem

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

Then:

$\forall x, y \in X: x \circ y = 0 \iff x = y$

Proof

Sufficient Condition

Suppose that $x = y$.

Then by $B$-Algebra Axiom $(\text A 1)$:

$x \circ y = x \circ x = 0$

$\Box$

Necessary Condition

Let $x, y \in X$ such that $x \circ y = 0$.

Then:

\(\ds x \circ y\)

\(=\)

\(\ds y \circ y\)

$B$-Algebra Axiom $(\text A 1)$

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds y\)

$B$-algebra operation is right cancellable

Hence the result.

$\blacksquare$