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

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 $S$.

Proof

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 $*$

$\blacksquare$