Entropic Idempotent Structure is Self-Distributive
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Theorem
Let $\struct {S, \odot}$ be an algebraic structure such that $\odot$ is both idempotent and entropic.
Then $\struct {S, \odot}$ is a self-distributive structure.
Proof
| \(\ds \forall a, b, c \in S: \, \) | \(\ds \paren {a \odot b} \odot \paren {a \odot c}\) | \(=\) | \(\ds \paren {a \odot a} \odot \paren {b \odot c}\) | Definition of Entropic Operation | ||||||||||
| \(\ds \) | \(=\) | \(\ds a \odot \paren {b \odot c}\) | Definition of Idempotent Operation |
and:
| \(\ds \forall a, b, c \in S: \, \) | \(\ds \paren {a \odot c} \odot \paren {b \odot c}\) | \(=\) | \(\ds \paren {a \odot b} \odot \paren {c \odot c}\) | Definition of Entropic Operation | ||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {a \odot b} \odot c\) | Definition of Idempotent Operation |
Hence the result by definition of self-distributive structure.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.27 \ \text {(a)}$