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

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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.


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