Condition for Constant Operation to be Distributive over Another Operation

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

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

Then:
 * $\sqbrk c$ is distributive over $\circ$


 * $c \circ c = c$
 * $c \circ c = c$

Sufficient Condition
Let $\sqbrk c$ be distributive over $\circ$


 * $c \circ c = c$

Necessary Condition
Let $\forall c \circ c = c$.

Then:

and:

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