Antiassociative Structure of Finite Order

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Theorem

Let $n \in \N$ such that $n > 2$.


Then there exists an algebraic structure $\struct {S, \circ}$ of order $n$ such that $\circ$ is antiassociative on $S$.


Proof

Let $S = \set {a_0, \ldots, a_{n - 1} }$.

Let $\circ$ be a binary operation defined on $S$ such that:

$\forall x \in S: x \circ a_i = a_{\paren {i + 1} \pmod n}$


Then $\forall j, k, m \in \closedint 0 {n - 1}$:

\(\ds \paren {a_j \circ a_k} \circ a_m\) \(=\) \(\ds a_{\paren {m + 1} \pmod n}\)

and:

\(\ds a_j \circ \paren {a_k \circ a_m}\) \(=\) \(\ds a_j \circ a_{\paren {m + 1} \pmod n}\)
\(\ds \) \(=\) \(\ds a_{\paren {m + 2} \pmod n}\)

As $n \ge 2$ it follows that:

$m + 1 \ne m + 2 \pmod n$

and so $\forall j, k, m \in \closedint 0 {n - 1}$:

$\paren {a_j \circ a_k} \circ a_m \ne a_j \circ \paren {a_k \circ a_m}$


Hence the result.

$\blacksquare$