# Rises for Measuring Local Distributivity in Lattices

arXiv: https://arxiv.org/abs/2506.23168 · HTML: https://mlsanity.com/paper/2506.23168/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 23 references that are in the map.
- First posted on arXiv 29 Jun 2025.

## Builds on (most cited first, 4 of 4)

- [Discovering Implicational Knowledge in Wikidata](https://mlsanity.com/paper/1902.00916/) (2019)
- [Compressed representation of Learning Spaces](https://mlsanity.com/paper/1407.6327/) (2014)
- [A Repository for Formal Contexts](https://mlsanity.com/paper/2404.04344/) (2024)
- [The Birkhoff completion of finite lattices](https://mlsanity.com/paper/2405.02342/) (2024)

## Abstract (arXiv)

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.