Right Operation is Distributive over Idempotent Operation

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Theorem

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

$\rightarrow$ is the right operation
$\circ$ is any arbitrary binary operation.


Then:

$\rightarrow$ is distributive over $\circ$

if and only if

$\circ$ is idempotent.


Proof

From Right Operation is Left Distributive over All Operations:

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

for all binary operations $\circ$.


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


Necessary Condition

Let $\circ$ be idempotent.

Then:

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

Thus $\rightarrow$ is right distributive over $\circ$.

$\Box$


Sufficient Condition

Let $\rightarrow$ be right distributive over $\circ$.

Let $c \in S$ be arbitrary.

Then:

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

Hence $\circ$ is idempotent.

$\blacksquare$


Sources

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