Right Operation is Left Distributive over All Operations
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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 left distributive over $\circ$.
Proof
By definition of the right operation:
\(\ds a \rightarrow \paren {b \circ c}\) | \(=\) | \(\ds b \circ c\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \rightarrow b} \circ \paren {a \rightarrow c}\) |
The result follows by definition of left distributivity.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Exercise $16.23 \ \text{(a)}$