# 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:

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

 $\ds \paren {a \circ a} \circ b$ $=$ $\ds b \circ b$ $\ds$ $=$ $\ds a$ $\ds a \circ \paren {a \circ b}$ $=$ $\ds a \circ c$ $\ds$ $=$ $\ds b$

demonstrating non-associativity.

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