Rises for Measuring Local Distributivity in Lattices
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912456161361920 |
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| author | Abdulla, Mohammad Hille, Tobias Dürrschnabel, Dominik Stumme, Gerd |
| author_facet | Abdulla, Mohammad Hille, Tobias Dürrschnabel, Dominik Stumme, Gerd |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rises for Measuring Local Distributivity in Lattices Abdulla, Mohammad Hille, Tobias Dürrschnabel, Dominik Stumme, Gerd Artificial Intelligence Discrete Mathematics Combinatorics Rings and Algebras 06B99 G.2.1 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. |
| title | Rises for Measuring Local Distributivity in Lattices |
| topic | Artificial Intelligence Discrete Mathematics Combinatorics Rings and Algebras 06B99 G.2.1 |
| url | https://arxiv.org/abs/2506.23168 |