Left Self-Distributive Operation with Right Identity is Idempotent

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Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be left self-distributive.

Let $\struct {S, \circ}$ have a right identity.


Then $\circ$ is an idempotent operation.


Proof

Let the right identity of $\struct {S, \circ}$ be $e_R$.


We have:

\(\ds \forall a, b, c \in S: \, \) \(\ds a \circ \paren {b \circ c}\) \(=\) \(\ds \paren {a \circ b} \circ \paren {a \circ c}\) Definition of Left Self-Distributive Operation
\(\ds \leadsto \ \ \) \(\ds \forall a \in S: \, \) \(\ds a \circ \paren {e_R \circ e_R}\) \(=\) \(\ds \paren {a \circ e_R} \circ \paren {a \circ e_R}\) In particular, it holds for $e_R$
\(\ds \leadsto \ \ \) \(\ds \forall a \in S: \, \) \(\ds a\) \(=\) \(\ds a \circ a\) Definition of Right Identity

The result follows by definition of idempotent operation.

$\blacksquare$


Sources

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