笛卡尔闭簇 II：与代数和自相似性的联系
Cartesian closed varieties II: links to algebra and self-similarity
AI总结：
本文证明笛卡尔闭簇中的匹配对 [B|M] 等价于布尔限制幺半群和充足源étale拓扑范畴，从而建立其与 Cuntz 代数、Leavitt 路代数等自相似结构所用广群之间的联系，并开启从逻辑视角研究这些广群性质的方向。
摘要：
本文是研究笛卡尔闭簇的系列文章中的第二篇。在第一篇文章中，我们证明了每一个非退化的有限元笛卡尔簇都是配备有布尔代数 B 和幺半群 M 作用的集合簇，其中 B 和 M 相互作用形成我们称之为匹配对 [B|M] 的结构。在本文中，我们证明了这样的匹配对 [B|M] 等价于布尔限制幺半群，也等价于充足源étale拓扑范畴；这些都是布尔逆幺半群和充足étale拓扑广群的推广，后者被用于编码自相似结构，例如 Cuntz 和 Cuntz--Krieger $C^\ast$-代数、Leavitt 路代数以及与自相似群作用相关联的 $C^\ast$-代数。我们解释并举例说明了这些联系，并启动了从相关簇的逻辑视角来理解此类广群的拓扑性质和代数性质的研究计划。
英文摘要：
This paper is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair [B|M]. In this paper, we show that such pairs [B|M] are equivalent to Boolean restriction monoids and also to ample source-étale topological categories; these are generalisations of the Boolean inverse monoids and ample étale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz--Krieger $C^\ast$-algebras, Leavitt path algebras and the $C^\ast$-algebras associated to self-similar group actions. We explain and illustrate these links, and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.
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。
