Antiassociative Structure of Finite Order

Theorem
Let $n \in \N$ and $n > 2$.

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

Proof
Let $S = \{a_1,...,a_n\}$

And let:


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

Then $\forall j, k, m \in [1,..,n]$:

And

So $\forall j, k, m \in [1,..,n]$:


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

Hence the result.