Cancellable Semiring with Unity is Additive Semiring
Theorem
Let $\struct {S, *, \circ}$ be a cancellable semiring with unity $1_S$.
Then the distributand $*$ is commutative.
That is to say, $\struct {S, *, \circ}$ is also an additive semiring.
Proof
Let $\struct {S, *, \circ}$ be a semiring, all of whose elements of $S$ are cancellable for $*$.
We expand the expression $\paren {a * b} \circ \paren {c * d}$ using the distributive law in two ways:
\(\ds \paren {a * b} \circ \paren {c * d}\) | \(=\) | \(\ds \paren {\paren {a * b} \circ c} * \paren {\paren {a * b} \circ d}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ c} * \paren {b \circ c} * \paren {a \circ d} * \paren {b \circ d}\) |
\(\ds \paren {a * b} \circ \paren {c * d}\) | \(=\) | \(\ds \paren {a \circ \paren {c * d} } * \paren {b \circ \paren {c * d} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {a \circ c} * \paren {a \circ d} * \paren {b \circ c} * \paren {b \circ d}\) |
So, by the fact that all elements of $\struct {S, *}$ are cancellable (and thus, a fortiori, are $a \circ c$ and $b \circ d$), we have:
\(\ds \paren {a \circ c} * \paren {b \circ c} * \paren {a \circ d} * \paren {b \circ d}\) | \(=\) | \(\ds \paren {a \circ c} * \paren {a \circ d} * \paren {b \circ c} * \paren {b \circ d}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {b \circ c} * \paren {a \circ d}\) | \(=\) | \(\ds \paren {a \circ d} * \paren {b \circ c}\) |
As this is true for all $a, b, c, d \in S$, it is true in particular if $c = d = 1_S$.
Thus it is clear that $b * a = a * b$, which is exactly to say that $*$ is commutative.
Hence the result, by definition of additive semiring.
$\blacksquare$