Product of Semigroup Element with Right Inverse is Idempotent/Examples/Double and Half Mappings on Integers

From ProofWiki
Jump to navigation Jump to search

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

Retrieved from "https://proofwiki.org/w/index.php?title=Product_of_Semigroup_Element_with_Right_Inverse_is_Idempotent/Examples/Double_and_Half_Mappings_on_Integers&oldid=437837"