AI 中文总结
本文建立离散多拟阵带笼收缩的统一理论，证明其与秩函数公式一致、构成伽罗瓦连接，得到多拟阵相关构造的带笼版本，还研究其与洛伦兹多项式等的相互作用。
AI 中文摘要
我们建立离散多拟阵的带笼收缩的统一理论。给定一个多拟阵和一个笼κ，κ-收缩是一种规范的κ-带笼多拟阵，通过将基投影到笼中并保留最大投影基得到。我们证明该构造与显式秩函数公式一致。我们表明，κ-带笼多拟阵到所有多拟阵的包含映射与κ-收缩构成关于弱映射序的伽罗瓦连接。作为应用，我们得到多拟阵并、不交基定理和二分图上归纳的带笼版本；当κ=1时，这些可恢复对应的拟阵构造。我们还研究带笼收缩与洛伦兹多项式、近幂等 tract 上表示的相互作用，每种情况下该构造都保留相关结构。
英文摘要
We develop a unified theory of caged retractions of discrete polymatroids. Given a polymatroid and a cage $κ$, the $κ$-retraction is a canonical $κ$-caged polymatroid obtained by projecting bases into the cage and retaining the maximal projected bases. We prove that this construction agrees with an explicit rank-function formula. We show that the inclusion of the $κ$-caged polymatroids into all polymatroids and the $κ$-retraction form a Galois connection with respect to the weak-map order. As applications, we obtain caged versions of polymatroid union, the disjoint basis theorem, and induction along a bipartite graph. When $κ=\textbf{1}$, these recover the corresponding matroid constructions. We also study how caged retractions interact with Lorentzian polynomials and representations over near-idempotent tracts. In each case, the construction preserves the relevant structure.
