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.
$\blacksquare$