On the atomic structure of exponential Puiseux monoids and semirings

A Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. We say that a Puiseux monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers, which contains $0$, such that $M$ is generated by the set $\{r^s \mid s \in S\}$. If $M$ is multiplicatively closed then we say that $M$ is an exponential Puiseux semiring. Here we study the atomic properties of exponential Puiseux monoids and semirings. First, we characterize atomic exponential Puiseux monoids, and we prove that the finite factorization property, the bounded factorization property, and the ACCP coincide in this context. Then we proceed to offer a necessary condition and a sufficient condition for an exponential Puiseux monoid to satisfy the ACCP. We conclude by describing the exponential Puiseux monoids that are semirings.