Product on Left with Idempotent Element under Left Self-Distributive Operation is Idempotent

Theorem

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

Let $\circ$ be left self-distributive.

Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.

Then for all $b \in S$, $b \circ a$ is an idempotent element of $\struct {S, \circ}$

Proof

Let $a \in S$ be an idempotent element of $\struct {S, \circ}$.

We have:

\(\ds \forall b \in S: \, \)

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

\(=\)

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

Definition of Left Self-Distributive Operation

\(\ds \)

\(=\)

\(\ds b \circ a\)

Definition of Idempotent Element

The result follows by definition of idempotent element.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.23 \ \text{(d)}$