Condition for Operation to be Left Distributive over Constant Operation

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Theorem

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

Let $\sqbrk c$ be the constant operation for some $c \in S$.


Then:

$\circ$ is left distributive over $\sqbrk c$

if and only if:

$\forall x \in S: x \circ c = c$


Proof

Sufficient Condition

Let $\circ$ be left distributive over $\sqbrk c$.

\(\ds \forall x, y, z \in S: \, \) \(\ds c\) \(=\) \(\ds \paren {x \circ y} \sqbrk c \paren {x \circ z}\) Definition of Constant Operation
\(\ds \) \(=\) \(\ds x \circ \paren {y \sqbrk c z}\) Definition of Left Distributive Operation
\(\ds \) \(=\) \(\ds x \circ c\) Definition of Constant Operation

That is:

$\forall x \in S: x \circ c = c$

$\Box$


Necessary Condition

Let:

$\forall x \in S: x \circ c = c$

Then:

\(\ds \forall x, y, z \in S: \, \) \(\ds \paren {x \circ y} \sqbrk c \paren {x \circ z}\) \(=\) \(\ds c\) Definition of Constant Operation
\(\ds \) \(=\) \(\ds x \circ c\) by hypothesis
\(\ds \) \(=\) \(\ds x \circ {y \sqbrk c z}\) Definition of Constant Operation

That is, $\circ$ is left distributive over $\sqbrk c$.

$\blacksquare$


Sources

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