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

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