Product of Semigroup Element with Right Inverse is Idempotent/Examples

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Examples of Use of Product of Semigroup Element with Right Inverse is Idempotent

$2 x$ and $\dfrac x 2$ Mappings on Integers

Let $\struct {\Z^\Z, \circ}$ be the semigroup defined such that:

$\Z$ is the set of all mappings on the integers.
$\circ$ denotes composition of mappings.

Let $\rho, \sigma \in \Z^\Z$ such that:

$\forall x \in \Z: \map \rho x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$
$\forall x \in \Z: \map \sigma x = 2 x$

Then:

$\rho$ is a right inverse for $\sigma$

but:

$\rho$ is not a left inverse for $\sigma$

As a result:

$\paren {\sigma \circ \rho}^2 = \sigma \circ \rho$