发表机构
Clayton State University; Long Island University（克莱顿州立大学; 长岛大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究统一了群有限子集与域扩张有限维子空间的两种匹配理论，引入独立群概念，推导模阿贝尔独立群的匹配准则，并得到推广Cauchy–Davenport定理的乘积增长界。
AI 中文摘要
我们统一了两种匹配理论，一种针对群的有限子集，另一种针对域扩张中的有限维子空间。为此，我们研究了配备相容有限拟阵结构的群，本文将其称为独立群。应用Rado独立横截定理，我们推导了有限秩集合之间可匹配性的充要秩准则。在模阿贝尔独立群$G$的设定下，我们提出了加性数论中$e$-变换的类似物，推导了结构匹配准则，并通过不存在满足$1<ρ(H)<ρ(G)$且$ρ(H)<\u221e$的子幺半群$H$来刻画全局匹配性质，其中$ρ$表示秩。文中给出了模阿贝尔独立群的例子，并在匹配语境下对其进行了研究。由该匹配理论衍生、但无需借助匹配理论表述的是一个推广了Cauchy–Davenport定理的乘积增长界：我们定义了参数$μ(G)$，并证明对于$G$的所有非空有限秩子集$X,Y$，有$ρ(XY)\geq \min\{μ(G),ρ(X)+ρ(Y)-1\}$。此外，我们证明$ρ(XY)$的下界可由一个稳定平坦子集的$G$的子幺半群控制，这一现象让人联想到Kneser定理。
英文摘要
We unify two matching theories, one for finite subsets of groups and the other for finite-dimensional subspaces in a field extension. To achieve this, we study groups equipped with a compatible finitary matroid structure, termed here independence groups. Applying Rado's independent transversal theorem, we derive necessary and sufficient rank criteria for matchability between finite-rank sets. In the setting of a modular abelian independence group $G$, we develop an analogue of the $e$-transform from additive number theory, derive structural matching criteria, and characterize a global matching property by the absence of a finite-rank submonoid $H$ satisfying $1<ρ(H)<ρ(G)$, where $ρ$ denotes rank. Examples of modular abelian independence groups are given and examined in the matching context. Arising from this matching theory, but formulated without any reference to it, is a product-growth bound that generalizes the Cauchy--Davenport theorem: we define a parameter $μ(G)$ and prove that $ρ(XY)\geq \min\{μ(G),ρ(X)+ρ(Y)-1\}$ for all nonempty finite-rank subsets $X,Y$ of $G$. Furthermore, $ρ(XY)$ is shown to be controlled from below by a submonoid of $G$ that stabilizes a flat, a phenomenon reminiscent of Kneser's theorem.
Comments36 pages. The introduction has been expanded. Comments welcome!
