交换群与三元组拓扑
Commuting Groups and the Topos of Triads
摘要：
本文的目的是阐明三元组拓扑与新黎曼PLR-群之间的关系。为此，我们首先发展了一些广义区间系统的理论：1）我们证明了众所周知的事实，即每对对偶群是同构的左和右正则表示的一些群体（凯莱定理），2）给定一个简单的传递群行动，我们展示了如何构造的对偶群，和3）给定两个对偶群，我们展示了如何容易地构造子对偶群。这种次对偶群结构的例子包括科恩的六音系统和八音系统。然后，我们列举了所有的Z_{12} -子集下的三元幺半群是不变的，并承认一个简单的传递PLR-子群的行动，其最大的三元覆盖。作为一个推论，我们实现了所有四个六声系统和所有三个八声系统作为Lawvere-Tierney升级的辅音三和弦。
英文摘要：
The goal of this article is to clarify the relationship between the topos of triads and the neo-Riemannian PLR-group. To do this, we first develop some theory of generalized interval systems: 1) we prove the well known fact that every pair of dual groups is isomorphic to the left and right regular representations of some group (Cayley's Theorem), 2) given a simply transitive group action, we show how to construct the dual group, and 3) given two dual groups, we show how to easily construct sub dual groups. Examples of this construction of sub dual groups include Cohn's hexatonic systems, as well as the octatonic systems. We then enumerate all Z_{12}-subsets which are invariant under the triadic monoid and admit a simply transitive PLR-subgroup action on their maximal triadic covers. As a corollary, we realize all four hexatonic systems and all three octatonic systems as Lawvere--Tierney upgrades of consonant triads.
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。
