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.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations: Example $66$