Set of Idempotent Elements of Entropic Structure is Closed

Theorem

Let $\struct {S, \odot}$ be an algebraic structure such that $\odot$ is entropic.

Let $T \subseteq S$ be the set of idempotent elements of $S$.

Then $\struct {T, \odot}$ is closed.

Proof

\(\ds \forall a, b \in T: \, \)

\(\ds \paren {a \odot b} \odot \paren {a \odot b}\)

\(=\)

\(\ds \paren {a \odot a} \odot \paren {b \odot b}\)

Definition of Entropic Operation

\(\ds \)

\(=\)

\(\ds a \odot b\)

Definition of Idempotent Operation

\(\ds \leadsto \ \ \)

\(\ds a \odot b\)

\(\in\)

\(\ds T\)

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.27 \ \text {(b)}$