超自适应图与实现问题
Super adaptable graphs and the realization problem
- Universitat Autònoma de Barcelona（巴塞罗那自治大学）
- Universidad de Zaragoza（萨拉戈萨大学）
- Universidad de Cádiz（加的斯大学）
- Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文引入超自适应幺半群，证明可数者可由冯·诺依曼正则环实现，并建立抽象幺半群、分离图与正则环的系统框架，给出正则幺半群作为图幺半群的完整刻画。
AI中文摘要：
我们引入并研究了超自适应幺半群族，该族包含许多其他幺半群，特别是所有有限生成的锥形细化幺半群，以及可数的、素生成的、正则的锥形细化幺半群。我们的主要结果表明，任何可数的超自适应幺半群都可以由冯·诺依曼正则环实现，从而肯定地回答了该类的实现问题。即使对于可数的、素生成的、正则的锥形细化幺半群这一子类，该结果也是新的，我们证明这类幺半群既可以由冯·诺依曼正则环实现，也可以由纯无限的、实秩零的C*-代数实现。为了证明这一结果，我们开发了一个系统框架，连接了三个基本结构：抽象幺半群、分离图（一类特定的着色图）和冯·诺依曼正则环。通过在这些类之间建立函子性联系，我们将幺半群的若干代数性质转化为组合和环论概念。这不仅使我们能够证明主要实现结果，还获得了超自适应幺半群的若干刻画。作为一个特殊应用，这些刻画可用于提供哪些正则幺半群作为图幺半群出现的完整描述。据我们所知，这项工作提供了文献中第一个自包含的处理，其中完整的实现过程——从抽象幺半群通过I-系统和分离图到正则环和C*-代数——在单一文本中按顺序并完整详细地展开。
英文摘要：
We introduce and study the family of super adaptable monoids, which includes, among many others, all conical refinement monoids that are either finitely generated, or countable, primely generated, and regular. Our main result shows that any countable super adaptable monoid can be realized by a von Neumann regular ring, thus answering affirmatively the Realization problem for this class. The result is new even for the subclass of countable, primely generated, regular conical refinement monoids, which we show can be realized by both a von Neumann regular ring and a purely infinite, real rank zero C*-algebra. To prove this result, we develop a systematic framework that connects three fundamental structures: abstract monoids, separated graphs (a specific class of colored graphs), and von Neumann regular rings. By establishing functorial connections among these classes, we translate several algebraic properties of monoids into combinatorial and ring-theoretic notions. This allows us not only to prove the main realization result, but also to obtain several characterizations of super adaptable monoids. As a particular application, these can be used to provide a complete characterization of which regular monoids arise as graph monoids. To our knowledge, this work offers the first self-contained treatment in the literature where the complete realization process ---from abstract monoids through I-systems and separated graphs to regular rings and C*-algebras--- is developed sequentially and in full detail within a single text.
