User:Alecscooper/Sandbox

Let $\left({S, *, \circ}\right)$ be a semiring, all of whose  elements of $S$ are  cancellable for $*$ with at either a right identity or a left identity (or both) for the distributor.

We will demonstrate the case where $\left({S, *, \circ}\right)$ has a right identity by expanding $\left({x * y}\right) \circ \left({r * r}\right)$ using the distributive law in two ways:

But we also have

Thus we have $x * \left({ x * y }\right) * y = x * \left({ y * x }\right) * y$.

But our semiring is cancellable, so this simplifies to $x * y = y * x$, which is exactly what we wanted to show.

Similarly, if $r$ was a left identity, then the above would follow similarly by expanding $\left({ r * r }\right) \circ \left({ x * y }\right)$.