Associative Operation/Examples/Associative/x circ a circ y

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Example of Associative Operation

Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is an associative operation.

Let $a \in S$ be an arbitrary element of $S$.

Let $*$ be the operation defined on $S$ by:

$\forall x, y \in S: x * y := x \circ a \circ y$

Then $*$ is associative on $S$.


Let $x, y, z \in S$ be arbitrary.

\(\ds x * \paren {y * z}\) \(=\) \(\ds x \circ a \circ \paren {y \circ a \circ z}\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {x \circ a \circ y} \circ a \circ z\) as $\circ$ is associative
\(\ds \) \(=\) \(\ds \paren {x * y} * z\) Definition of $*$