Self-Distributive Quasigroup with at least Two Elements is not Associative

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Theorem

Let $\struct {S, \odot}$ be a self-distributive quasigroup.

Let $S$ have at least $2$ elements.


Then $\odot$ is not an associative operation.


Proof

Aiming for a contradiction, suppose $\odot$ is associative operation.

Let $a, b \in S$ such that $a \ne b$.

Then:

\(\ds \forall a, b \in S: \, \) \(\ds \paren {a \odot a} \odot \paren {a \odot b}\) \(=\) \(\ds a \odot \paren {a \odot b}\) Definition of Self-Distributive Structure
\(\ds \) \(=\) \(\ds \paren {a \odot a} \odot b\) Definition of Associative Operation
\(\ds \) \(=\) \(\ds \paren {a \odot b} \odot \paren {a \odot b}\) Definition of Self-Distributive Structure
\(\ds \leadsto \ \ \) \(\ds a \odot a\) \(=\) \(\ds a \odot b\) Definition of Quasigroup
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds b\) Definition of Quasigroup

which contradicts our supposition that $a \ne b$.

$\blacksquare$


Sources

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