  Let $S$ be a nonnegative semiring of the real line, called here a positive
semiring. We study factorizations in both the additive monoid $(S,+)$ and the
multiplicative monoid $(S\setminus\{0\}, \cdot)$. In particular, we investigate
when, for a positive semiring $S$, both $(S,+)$ and $(S\setminus\{0\}, \cdot)$
have the following properties: atomicity, the ACCP, the bounded factorization
property (BFP), the finite factorization property (FFP), and the half-factorial
property (HFP). It is well known that in the context of cancellative and
commutative monoids, the chain of implications HFP $\Rightarrow$ BFP and FFP
$\Rightarrow$ BFP $\Rightarrow$ ACCP $\Rightarrow$ atomicity holds. Here we
construct classes of positive semirings wherein both the additive and
multiplicative structures satisfy each of these properties, and we also give
examples to show that, in general, none of the implications in the previous
chain is reversible.
