AI 中文总结
本文针对布尔格、分拆格和Tamari格，给出了布尔反链的构造方法，并得出了其数量的简洁公式，为格的关联代数表示理论研究提供了组合计数支撑。
AI 中文摘要
我们称格L中的一个反链为布尔反链，若它以一种特别好的方式生成L中的一个布尔子格。这些纯粹的组合对象在格L的关联代数的表示理论中发挥着作用，正如Rognerud、Yıldırım以及最后一位作者与Klász、Kleinau和Marczinzik的最新结果所表明的那样。鉴于这些动机，自然会提出问题：我们能否在某些格中计数并构造布尔反链？在本文中，我们给出了布尔格、分拆格和Tamari格中布尔反链的构造方法，此外还给出了我们发现的这些布尔反链数量的简洁得惊人的公式。
英文摘要
We say that an antichain in a lattice $L$ is Boolean if it generates a Boolean sublattice in $L$ in a particularly nice way. These purely combinatorial objects play a role in the representation theory of the incidence algebra of the lattice $L$, as indicated by recent results of Rognerud, Yıldırım and of the last author with Klász, Kleinau and Marczinzik. Given these motivations, it is natural to ask: Can we count and construct Boolean antichains in certain lattices? In this paper we give a construction of Boolean antichains in Boolean lattices, partition lattices and Tamari lattices. Furthermore, we share the surprisingly elegant formulas we found for the number of these Boolean antichains.
Comments23 pages, 8 figures, comments welcome; fixed names of referenced environments
