The Arithmetic of Semirings Part I: Ideals

We study ideals in the semiring $\mathbb{N}$ of natural numbers, with a focus on those which are lost when extending from $\mathbb{N}$ to $\mathbb{Z}$. This leads to a new perspective on the classical theory of numerical semigroups, including the introduction of a natural multiplicative structure. We prove that unique factorization of ideals fails in $\mathbb{N}$ on several levels, introduce a handful of new tropical multiplicative invariants of numerical semigroups, characterize integral closures of ideals in terms of Newton polygons, and analyze the behavior of classical numerical semigroup invariants with respect to the product operation.