从指针建筑集获得晶格：广义装饰晶格
Lattices from Pointed Building Sets: Generalized Ornamentation Lattices
AI总结：
本文提出指针建筑集的概念，并通过构造装饰晶格，推广了Tamari晶格等经典晶格的理论，探索了逆极限和群作用等理论方向。
摘要：
我们介绍了一种新的组合结构，称为指针建筑集，可以视为配备兼容关系的晶格家族。对于每个指针建筑集B，我们关联一个完整的晶格O(B)，称为B的装饰晶格。这种构造的特殊情形已经证明在理解三个偏序集家族的结构方面很有用：操作棱柱晶格、仿射Tamari晶格和路径超图的子超图的偏序集。本文的目标是建立这些广义装饰的理论。我们检查了几种自然的指针建筑集类别，这些类别可以恢复经典的晶格，如Tamari晶格、由粗化顺序的拓扑晶格，以及自然标记的偏序晶格。此外，还探讨了几个理论方向，包括逆极限和群作用。值得注意的是，这导致了Tamari晶格逆极限的简单构造，从而得到Tamari晶格的无限类比。
英文摘要：
We introduce a novel combinatorial structure called pointed building sets, which can be viewed as families of lattices equipped with compatibility relations. To each pointed building set $\mathsf{B}$, we associate a complete lattice $\mathcal{O}(\mathsf{B})$, called the ornamentation lattice of $\mathsf{B}$. Special cases of this construction have already proven useful in understanding the structure of three families of posets: operahedron lattices, the affine Tamari lattice, and hypergraphic posets of subhypergraphs of the path hypergraph of an increasing tree. The goal of this paper is to establish the theory of these generalized ornamentations. We examine several natural classes of pointed building sets which recover classical lattices such as the Tamari lattice, the lattice of topologies ordered by coarsening, and the lattice of naturally labeled partial orders. Furthermore, several theoretical directions are explored, including inverse limits and group actions. Notably, this leads to a straightforward construction of inverse limits of Tamari lattices, yielding infinite analogs of the Tamari lattice.
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