Quasigroup is not necessarily B-Algebra
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Theorem
Let $\struct {S, \circ}$ be a quasigroup.
Then $\struct {S, \circ}$ is not necessarily a $B$-algebra.
Proof
From Group is Quasigroup we take an arbitrary small group.
Consider the Cayley table of the group of order $3$:
- $\begin{array}{c|cccccc}
& 0 & 1 & 2 \\
\hline 0 & 0 & 1 & 2 \\ 1 & 1 & 2 & 0 \\ 2 & 2 & 0 & 1 \\ \end{array}$
By inspection we see that $B$-Algebra Axiom $(\text A 2)$ does not hold as $1 \circ 1 \ne 0$.
$\blacksquare$