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