有限半分配格的基本定理
The fundamental theorem of finite semidistributive lattices
AI总结：
本文证明了有限半分配格的基本定理，通过构造带有额外结构的集合Sha，揭示了半分配格的结构特征及其与表示论中扭对的联系。
摘要：
我们证明了一个有限半分配格的基本定理（FTFSDL），其形式为：一个偏序集L是有限半分配格当且仅当存在一个带有额外结构的集合Sha，使得L与Sha的可容许子集按包含关系有序同构；此时，Sha及其额外结构由L唯一确定。Sha上的额外结构是表示论中扭对概念的组合抽象，并在超平面排列区域的偏序集中具有几何意义。我们展示了FTFSDL如何澄清格论中的许多构造，如标准连接表示和通过商映射，以及半分配性质与其他主要格类的相互作用。许多结果也适用于无限格。
英文摘要：
We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.
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