Definition:Associative Operation

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Let $S$ be a set.

Let $\circ : S \times S \to S$ be a binary operation.

Then $\circ$ is associative if and only if:

$\forall x, y, z \in S: \paren {x \circ y} \circ z = x \circ \paren {y \circ z}$


Arbitrary Non-Associative Order 3 Structure

Consider the algebraic structure of order $3$ defined by the Cayley table:

$\begin{array}{c|cccc} \circ & a & b & c \\ \hline a & b & c & b \\ b & b & a & c \\ c & a & c & c \\ \end{array}$

\(\displaystyle \paren {a \circ a} \circ b\) \(=\) \(\displaystyle b \circ b\)
\(\displaystyle \) \(=\) \(\displaystyle a\)
\(\displaystyle a \circ \paren {a \circ b}\) \(=\) \(\displaystyle a \circ c\)
\(\displaystyle \) \(=\) \(\displaystyle b\)

demonstrating non-associativity.

Also note that $a \circ b \ne b \circ a$, so $\circ$ is non-commutative as well.

Also see

Historical Note

The term associative was coined by William Hamilton in about $1844$ while thinking about octonions, which aren't.