Right Self-Distributive Operation with Left Identity is Idempotent
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Theorem
Let $\struct {S, \circ}$ be an algebraic structure.
Let $\circ$ be right self-distributive.
Let $\struct {S, \circ}$ have a left identity.
Then $\circ$ is an idempotent operation.
Proof
Let the left identity of $\struct {S, \circ}$ be $e_L$.
We have:
\(\ds \forall a, b, c \in S: \, \) | \(\ds \paren {a \circ b} \circ c\) | \(=\) | \(\ds \paren {a \circ c} \circ \paren {b \circ c}\) | Definition of Right Self-Distributive Operation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall c \in S: \, \) | \(\ds \paren {e_L \circ e_L} \circ c\) | \(=\) | \(\ds \paren {e_L \circ c} \circ \paren {e_L \circ c}\) | In particular, it holds for $e_L$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall c \in S: \, \) | \(\ds c\) | \(=\) | \(\ds c \circ c\) | Definition of Left Identity |
The result follows by definition of idempotent operation.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.23 \ \text{(c)}$