Antiassociative Structure of Finite Order

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

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

Proof
Let $S = \left\{{a_0, \ldots, a_{n - 1}}\right\}$.

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


 * $\forall x \in S: x \circ a_i = a_{\left({i + 1}\right) \pmod n}$

Then $\forall j, k, m \in \left[{0 \,.\,.\, n-1}\right]$:

and:

As $n \ge 2$ it follows that:
 * $m + 1 \ne m + 2 \pmod n$

and so $\forall j, k, m \in \left[{0 \,.\,.\, n - 1}\right]$:


 * $\left({a_j \circ a_k}\right) \circ a_m \ne a_j \circ \left({a_k \circ a_m}\right)$

Hence the result.