Set of Closed Subsets of Power Structure of Entropic Structure is Closed

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Theorem

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

Let $\struct {S, \odot}$ be an entropic structure.

Let $\struct {\powerset S, \odot_\PP}$ be the power structure of $\struct {S, \odot}$.


Let $\TT$ be the set of all submagmas of $\struct {S, \odot}$.

Then the algebraic structure $\struct {\TT, \odot_\PP}$ is a submagma of $\struct {\powerset S, \odot_\PP}$ which is itself an entropic structure.


Proof

Recall the definition of subset product:

$A \odot_\PP B = \set {a \odot b: \tuple {a, b} \in A \times B}$


First we show that:

$\forall A, B \in \TT: A \odot_\PP B \in \TT$

Let $A$ and $B$ be arbitrary elements of $\TT$.

Let $a$ and $c$ be arbitrary elements of $A$.

Let $b$ and $d$ be arbitrary elements of $B$.

Then we have:

\(\ds a \odot b\) \(\in\) \(\ds A \odot_\PP B\) Definition of Subset Product
\(\ds c \odot d\) \(\in\) \(\ds A \odot B\) Definition of Subset Product
\(\ds a \odot c\) \(\in\) \(\ds A\) as $\struct {A, \odot}$ is closed
\(\ds b \odot d\) \(\in\) \(\ds B\) as $\struct {B, \odot}$ is closed

Then:

\(\ds \paren {a \odot b} \odot \paren {c \odot d}\) \(=\) \(\ds \paren {a \odot c} \odot \paren {b \odot d}\) Definition of Entropic Structure
\(\ds \) \(\in\) \(\ds A \odot_\PP B\) a priori: $a \odot c \in A$, $b \odot d \in B$

That is:

$A \odot_\PP B$ is closed in $S$.

As $A$ and $B$ are arbitrary, it follows that $\struct {\TT, \odot_\PP}$ is closed in $\powerset S$.

$\Box$


Then we need to show that:

$\forall A, B, C, D \in \TT: \paren {A \odot_\PP B} \odot_\PP \paren {C \odot_\PP D} = \paren {A \odot_\PP C} \odot_\PP \paren {B \odot_\PP D}$

demonstrating the entropic nature of $\struct {\TT, \odot_\PP}$.

We have:

\(\ds \forall A, B, C, D \in \TT: \, \) \(\ds \) \(\) \(\ds \paren {A \odot_\PP B} \odot_\PP \paren {C \odot_\PP D}\)
\(\ds \) \(=\) \(\ds \set {\paren {a \odot b} \odot \paren {c \odot d}: a \in A, b \in B, c \in C, d \in D}\) Definition of Subset Product
\(\ds \) \(=\) \(\ds \set {\paren {a \odot c} \odot \paren {b \odot d}: a \in A, b \in B, c \in C, d \in D}\) Definition of Entropic Structure $\struct {S, \odot}$ is entropic
\(\ds \) \(=\) \(\ds \paren {A \odot_\PP C} \odot_\PP \paren {B \odot_\PP D}\) Definition of Subset Product

and the result follows.

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.12 \ \text{(f)}$
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