标题： 清除格子负债网络的区域 显示英文标题

标题： Clearing Sections of Lattice Liability Networks

摘要： 现代金融网络涉及超越简单货币债务的复杂义务：多种货币、优先索赔、供应链依赖关系等。 我们提出一个数学框架，通过将经典的Eisenberg-Noe金融清算模型重新表述为格负债网络，从而统一并扩展这些场景。 网络中的每个节点都携带一个完整的可能状态格，而边则编码名义负债。 我们的框架将经典模型的标量值清算向量推广为格值清算部分，在保持优雅的固定点结构的同时，显著扩展了其描述能力。 我们的主要定理证明了这样的网络在乘积序下拥有自身形成完整格的清算部分。 这个结构定理使得在多个领域（包括多货币金融系统、带有自动做市商的去中心化金融、具有资源转换的供应链以及具有复杂授权结构的权限网络）中对均衡进行可处理的分析成为可能。 我们进一步将框架扩展到链完备格以用于期限结构模型，以及用于复杂协商系统的多值映射。 我们的结果展示了格理论如何为理解跨多个领域的复杂网络动态提供一种自然语言，为分析系统性风险、资源分配和网络稳定性建立了统一的数学基础。 显示英文摘要

摘要： Modern financial networks involve complex obligations that transcend simple monetary debts: multiple currencies, prioritized claims, supply chain dependencies, and more. We present a mathematical framework that unifies and extends these scenarios by recasting the classical Eisenberg-Noe model of financial clearing in terms of lattice liability networks. Each node in the network carries a complete lattice of possible states, while edges encode nominal liabilities. Our framework generalizes the scalar-valued clearing vectors of the classical model to lattice-valued clearing sections, preserving the elegant fixed-point structure while dramatically expanding its descriptive power. Our main theorem establishes that such networks possess clearing sections that themselves form a complete lattice under the product order. This structure theorem enables tractable analysis of equilibria in diverse domains, including multi-currency financial systems, decentralized finance with automated market makers, supply chains with resource transformation, and permission networks with complex authorization structures. We further extend our framework to chain-complete lattices for term structure models and multivalued mappings for complex negotiation systems. Our results demonstrate how lattice theory provides a natural language for understanding complex network dynamics across multiple domains, creating a unified mathematical foundation for analyzing systemic risk, resource allocation, and network stability.