提出用'上升'度量格结构的分配性，揭示现实数据格中交分配性较弱。
Rises for Measuring Local Distributivity in Lattices
- 引入'上升'概念，通过覆盖概念属性变化衡量格的分配性
- 证明格可分配当且仅当无非单位上升，实证发现交分配性高而合分配性低
- 适用于形式概念分析中的数据格研究，适合逻辑与数据结构学者
分配性是格论中一个经典且广泛研究的概念。在数据挖掘，特别是形式概念分析（FCA）中，格结构常表现出高度的分配性，但目前尚无标准化度量方法。本文提出在（概念）格中使用'上升'来评估分配性。上升反映覆盖概念中属性或对象数量的变化。我们证明：一个格是分配的当且仅当不存在非单位上升。进一步，将上升与经典的交-和并-分配性联系起来。实证发现，真实数据生成的概念格具有高度的并分配性，但交分配性明显较弱。此外，我们研究了并分配性在有序集层面的表现。
原文摘要 · Abstract (English)
Distributivity is a well-established and extensively studied notion in lattice theory. In the context of data analysis, particularly within Formal Concept Analysis (FCA), lattices are often observed to exhibit a high degree of distributivity. However, no standardized measure exists to quantify this property. In this paper, we introduce the notion of rises in (concept) lattices as a means to assess distributivity. Rises capture how the number of attributes or objects in covering concepts change within the concept lattice. We show that a lattice is distributive if and only if no non-unit rises occur. Furthermore, we relate rises to the classical notion of meet- and join distributivity. We observe that concept lattices from real-world data are to a high degree join-distributive, but much less meet-distributive. We additionally study how join-distributivity manifests on the level of ordered sets.
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