Product of Semigroup Element with Right Inverse is Idempotent/Examples/Double and Half Mappings on Integers
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Examples of Use of Product of Semigroup Element with Right Inverse is Idempotent
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:
- $\paren {\sigma \circ \rho}^2 = \sigma \circ \rho$
Proof
\(\ds \map {\paren {\rho \circ \sigma} } x\) | \(=\) | \(\ds \map \rho {\map \sigma x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \rho {2 x}\) | Definition of $\sigma$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x\) | Definition of $\rho$, as $\map \sigma x$ is even |
\(\ds \map {\paren {\sigma \circ \rho} } x\) | \(=\) | \(\ds \map \sigma {\map \rho x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \sigma {\begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases} }\) | Definition of $\rho$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \begin{cases} x & : x \text { even} \\ 0 & : x \text { odd} \end{cases}\) | Definition of $\sigma$ |
Thus $\map {\paren {\sigma \circ \rho} } x \ne \map {\paren {\rho \circ \sigma} } x$ if and only if $x$ is odd.
So $\rho$ is a right inverse but not a left inverse for $\sigma$.
Then we have that:
- $\map {\paren {\sigma \circ \rho} } x = \begin{cases} x & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$
and it follows that:
- $\map {\paren {\sigma \circ \rho}^2} x = \begin{cases} x & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $10$