Identity Element for Power Structure
Theorem
Let $\struct {S, \circ}$ be a magma whose underlying set $S$ is non-empty.
Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.
Then:
- a subset $J$ of $S$ is an identity element of $\struct {\powerset S, \circ_\PP}$
- there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.
Proof
Sufficient Condition
Let $J \subseteq S$ such that $J$ is an identity element of $\struct {\powerset S, \circ_\PP}$.
We have:
\(\ds \forall A \in \powerset S: \, \) | \(\ds J \circ_\PP A\) | \(=\) | \(\ds A\) | Definition of Identity Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set {j \circ a: j \in J, a \in A}\) | \(=\) | \(\ds A\) | Definition of Operation Induced on Power Set | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a \in A: \forall j \in J: \, \) | \(\ds j \circ a\) | \(=\) | \(\ds a\) |
That is, the elements of $J$ are all left identities for $\circ$.
Similarly:
\(\ds \forall A \in \powerset S: \, \) | \(\ds A \circ_\PP J\) | \(=\) | \(\ds A\) | Definition of Identity Element | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set {a \circ j: j \in J, a \in A}\) | \(=\) | \(\ds A\) | Definition of Operation Induced on Power Set | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a \in A: \forall j \in J: \, \) | \(\ds a \circ j\) | \(=\) | \(\ds a\) |
That is, the elements of $J$ are all right identities for $\circ$.
So we have established that $J$ contains at least one element which is both a left identity and a right identity.
From Left and Right Identity are the Same, there is only one element of $J$, and it is the necessarily unique identity element for $\circ$.
That is:
- there exists an identity element $e$ of $\struct {S, \circ}$, such that $J = \set e$.
$\Box$
Necessary Condition
Let $\struct {S, \circ}$ have an identity element $e$.
Let $J = \set e$.
Then we have:
\(\ds \forall A \in \powerset S: \, \) | \(\ds A \circ_\PP J\) | \(=\) | \(\ds \set {a \circ j: j \in J, a \in A}\) | Definition of Operation Induced on Power Set | ||||||||||
\(\ds \) | \(=\) | \(\ds \set {a \circ e: a \in A}\) | Definition of $J$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {a: a \in A}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds A\) |
and similarly:
\(\ds \forall A \in \powerset S: \, \) | \(\ds J \circ_\PP A\) | \(=\) | \(\ds \set {j \circ a: j \in J, a \in A}\) | Definition of Operation Induced on Power Set | ||||||||||
\(\ds \) | \(=\) | \(\ds \set {e \circ a: a \in A}\) | Definition of $J$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {a: a \in A}\) | Definition of Identity Element | |||||||||||
\(\ds \) | \(=\) | \(\ds A\) |
So:
- $\forall A \in \powerset S: A \circ_\PP J = A = J \circ_\PP A$
and it is seen that $J$ is an identity element for $\circ_\PP$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 9$: Compositions Induced on the Set of All Subsets: Exercise $9.6$