Left Operation is Distributive over Idempotent Operation

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Theorem

Let $\struct {S, \circ, \leftarrow}$ be an algebraic structure where:

$\leftarrow$ is the left operation
$\circ$ is any arbitrary binary operation.


Then:

$\leftarrow$ is distributive over $\circ$

if and only if:

$\circ$ is idempotent.


Proof

From Left Operation is Right Distributive over All Operations:

$\forall a, b, c \in S: \paren {a \circ b} \leftarrow c = \paren {a \leftarrow c} \circ \paren {b \leftarrow c}$

for all binary operations $\circ$.


It remains to show that $\leftarrow$ is left distributive over $\circ$ if and only if $\circ$ is idempotent.


Necessary Condition

Let $\circ$ be idempotent.

Then:

\(\ds a \leftarrow \paren {b \circ c}\) \(=\) \(\ds a\) Definition of Left Operation
\(\ds \) \(=\) \(\ds a \circ a\) Definition of Idempotent Operation
\(\ds \) \(=\) \(\ds \paren {a \leftarrow b} \circ \paren {a \leftarrow c}\) Definition of Left Operation

Thus $\leftarrow$ is left distributive over $\circ$.

$\Box$


Sufficient Condition

Let $\leftarrow$ be left distributive over $\circ$.

Let $a \in S$ be arbitrary.

Then:

\(\ds a\) \(=\) \(\ds a \leftarrow \paren {b \circ c}\) Definition of Left Operation
\(\ds \) \(=\) \(\ds \paren {a \leftarrow b} \circ \paren {a \leftarrow c}\) Definition of Left Distributive Operation
\(\ds \) \(=\) \(\ds a \circ a\) Definition of Left Operation

Hence $\circ$ is idempotent.

$\blacksquare$


Sources

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