Quasigroup is not necessarily B-Algebra

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Theorem

Let $\left({S, \circ}\right)$ be a quasigroup.

Then $\left({S, \circ}\right)$ is not necessarily a $B$-algebra.


Proof

As all groups are quasigroups we will use a small group as a counterexample.

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 $(A2)$ does not hold as $1 \circ 1 \ne 0$.

$\blacksquare$


Also see