Antiassociative Structure of Finite Order

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

and:

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.