  A PROP is a symmetric monoidal category whose objects are the nonnegative
integers and whose tensor product on objects is addition. A morphism from $m$
to $n$ in a PROP can be visualized as a string diagram with $m$ input wires and
$n$ output wires. For a field $k$, the PROP $\mathrm{FinVect}_k$ where
morphisms are $k$-linear maps is used by Baez and Erbele to study signal-flow
diagrams. We aim to generalize their result characterizing this PROP in terms
of generators and relations by looking at the PROP $\mathrm{Mat}(R)$ of
matrices of values in $R$, where $R$ is a commutative rig (that is, a
generalization of a ring where the condition that each element has an additive
inverse is relaxed). To this end, we show that the category of symmetric
monoidal functors out of $\mathrm{Mat}(R)$ is equivalent to the category of
bicommutative bimonoids equipped with a certain map of rigs; such functors are
called algebras. By choosing $R$ correctly, we will see that the algebras of
the PROP $\mathrm{FinSpan}$ of finite sets and spans between them are
bicommutative bimonoids, while the algebras of the PROP $\mathrm{FinRel}$ of
finite sets and relations between them are special bicommuative bimonoids and
the algebras of $\mathrm{Mat}(\mathbb Z)$ are bicommutative Hopf monoids.
