半环和超环上模的原正合范畴
Proto-exact categories of modules over semirings and hyperrings
摘要：
\emph{Proto-exact categories}由Dyckerhoff和Kapranov提出，是Quillen精确范畴的推广，它为在\emph{non-additive}环境下定义代数K理论和Hall代数提供了一个框架。这种形式主义非常适合于研究对象具有强烈组合味道的范畴。在本文中，我们证明了半环和超环上的模范畴--在热带几何中获得突出地位的代数结构--具有原正合结构。在第一部分中，我们证明了半环上的模范畴具有原正合结构;幂等元半环上的模与拟阵有强联系。证明了代数格范畴$\mathcal{L}$具有原正合结构，并进一步证明了由有限格构成的$\mathcal{L}$的子范畴等价于作为原正合范畴的有限$\mathbb{B}$ -模范畴，其中$\mathbb{B}$是\emph{Boolean semifield} .我们还从这个角度讨论了$\mathcal{L}$与几何格（简单拟阵）之间的一些关系。第二部分证明了超环上的模范畴具有原正合结构。在\emph{Krasner hyperfield} $\mathbb{K}$上的有限模的情况下，有限$\mathbb{K}$ -模和有限关联几何之间的一个众所周知的关系产生了精确序列的组合解释。
英文摘要：
\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathbb{B}$-modules as proto-exact categories, where $\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations between $\mathcal{L}$ and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the \emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite $\mathbb{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
