Associative Operation/Examples/Non-Associative/Arbitrary Order 3 Structure

Example of Non-Associative Operations
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}$

demonstrating non-associativity.

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