AI 中文总结
研究自然数半环$\mathbb{N}$中理想，关注从$\mathbb{N}$到$\mathbb{Z}$扩展时丢失的理想，通过新视角带来自然乘法结构，证明理想唯一分解不成立并引入新不变量，刻画整闭包并分析不变量乘积行为。
AI 中文摘要
我们研究自然数半环$\mathbb{N}$中的理想，重点关注从$\mathbb{N}$扩展到$\mathbb{Z}$时丢失的理想。这为经典数值半群理论带来新视角，包括引入自然乘法结构。我们证明理想的唯一分解在$\mathbb{N}$的多个层面上不成立，引入一些新的数值半群热带乘法不变量，用牛顿多边形刻画理想的整闭包，并分析经典数值半群不变量关于乘积运算的行为。
英文摘要
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.
