Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917481320284160 |
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| author | Adaricheva, K. Mata, A. Silberger, S. Zamojska-Dzienio, A. |
| author_facet | Adaricheva, K. Mata, A. Silberger, S. Zamojska-Dzienio, A. |
| contents | We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_22682 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond Adaricheva, K. Mata, A. Silberger, S. Zamojska-Dzienio, A. Rings and Algebras 06A07, 06A15, 52A37, 52C05, 52C07 We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices. |
| title | Conjecture on Maximal Sublattices of Finite Semidistributive Lattices and Beyond |
| topic | Rings and Algebras 06A07, 06A15, 52A37, 52C05, 52C07 |
| url | https://arxiv.org/abs/2507.22682 |