Condition for Invertibility in Power Structure on Associative or Cancellable Operation

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Theorem

Let $\struct {S, \circ}$ be a unital magma.

Let $\struct {\powerset S, \circ_\PP}$ denote the power structure of $\struct {S, \circ}$.

Let identity element $e \in S$ be an identity element of $\struct {S, \circ}$.

Let $\circ$ be either:

an associative operation
a cancellable operation.


Let $X \subseteq S$ be a subset of $S$.

Then:

$X$ is invertible for $\circ_PP$

if and only if:

there exists an element $x \in S$ which is invertible for $\circ$ such that $X = \set x$.


Proof

First we note that from Identity Element for Power Structure, the algebraic structure $\struct {\powerset S, \circ_\PP}$ has an identity element $J = \set e$.


Sufficient Condition

Let $X$ be invertible for $\circ_PP$.

Then:

\(\ds \exists Y \in \powerset S: \, \) \(\ds X \circ_\PP Y\) \(=\) \(\ds J\) Definition of Invertible Element
\(\ds \leadsto \ \ \) \(\ds \set {x \circ y: x \in X, y \in Y}\) \(=\) \(\ds \set e\) Definition of Operation Induced on Power Set
\(\ds \leadsto \ \ \) \(\ds \exists x \in X: \exists y \in S: \, \) \(\ds x \circ y\) \(=\) \(\ds e\)


Similarly:

\(\ds \exists Y \in \powerset S: \, \) \(\ds Y \circ_\PP X\) \(=\) \(\ds J\) Definition of Invertible Element
\(\ds \leadsto \ \ \) \(\ds \set {y \circ x: x \in X, y \in Y}\) \(=\) \(\ds \set e\) Definition of Operation Induced on Power Set
\(\ds \leadsto \ \ \) \(\ds \exists x \in X: \exists y \in S: \, \) \(\ds y \circ x\) \(=\) \(\ds e\)

That is, there exists $x \in X$ such that:

$x \circ y = e = y \circ x$

That is:

$x$ is invertible for $\circ$
$y$ is the inverse of $x$.


Hence we have $Y \in \powerset S$ such that:

$\forall y \in Y: x \circ y = e = y \circ x$


$\circ$ is Associative

Let $\circ$ be an associative operation.

Aiming for a contradiction, suppose $\exists z \in X$ such that $z \ne x$.

Then we have:

$\forall y \in Y: z \circ y = e = y \circ z$


\(\ds z \circ e\) \(=\) \(\ds z\) Definition of Identity Element
\(\ds \leadsto \ \ \) \(\ds z \circ \paren {y \circ x}\) \(=\) \(\ds z\)
\(\ds \leadsto \ \ \) \(\ds \paren {z \circ y} \circ x\) \(=\) \(\ds z\) as $\circ$ is associative
\(\ds \leadsto \ \ \) \(\ds e \circ x\) \(=\) \(\ds z\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds z\)

This contradicts our supposition that $z \ne x$.

Hence there can be no elements in $X$ apart from $x$.

That is:

$X = \set x$

where $x$ is invertible for $\circ$.

$\Box$


$\circ$ is Cancellable

Let $\circ$ be a cancellable operation.

Aiming for a contradiction, suppose $\exists z \in X$ such that $z \ne x$.

Then we have:

$\forall y \in Y: z \circ y = e = y \circ z$

Then:

\(\ds x \circ y\) \(=\) \(\ds z \circ y\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds z\) as $\circ$ is cancellable

This contradicts our supposition that $z \ne x$.

Hence there can be no elements in $X$ apart from $x$.

That is:

$X = \set x$

where $x$ is invertible for $\circ$.

$\Box$


Necessary Condition

Let there exist an element $x \in S$ which is invertible for $\circ$.

Hence there exist $y \in S$ such that:

$x \circ y = e = y \circ x$

Let $X = \set x$.

We have:

\(\ds X \circ \set y\) \(=\) \(\ds \set {x \circ y}\)
\(\ds \) \(=\) \(\ds \set e\)
\(\ds \) \(=\) \(\ds J\)

and:

\(\ds \set y \circ X\) \(=\) \(\ds \set {y \circ x}\)
\(\ds \) \(=\) \(\ds \set e\)
\(\ds \) \(=\) \(\ds J\)

That is, $X$ is invertible for $\circ_\PP$.

$\blacksquare$


Sources

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