Dual digraphs of finite meet-distributive and modular lattices
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| Title: | Dual digraphs of finite meet-distributive and modular lattices |
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| Authors: | Andrew Craig, Miroslav Haviar, Klarise Marais |
| Source: | Cubo, Vol 26, Iss 2, Pp 279-302 (2024) |
| Publication Status: | Preprint |
| Publisher Information: | Universidad de La Frontera, 2024. |
| Publication Year: | 2024 |
| Subject Terms: | 06B15, 06C10, 06C05, 05C20, 06A75, 0209 industrial biotechnology, Modular lattices, Desarguesian lattices, modular lattice, Directed graphs (digraphs), tournaments, semimodular lattice, TiRS digraph, Mathematics - Rings and Algebras, 02 engineering and technology, Representation theory of lattices, Semimodular lattices, geometric lattices, 01 natural sciences, finite convex geometry, Rings and Algebras (math.RA), lower semimodular lattice, QA1-939, FOS: Mathematics, meet- distributive lattice, tirs digraph, 0101 mathematics, Mathematics, Generalizations of ordered sets |
| Description: | We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Ploščica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work in this journal on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing three open problems. |
| Document Type: | Article |
| File Description: | application/xml |
| ISSN: | 0719-0646 |
| DOI: | 10.56754/0719-0646.2602.279 |
| DOI: | 10.48550/arxiv.2309.14127 |
| Access URL: | http://arxiv.org/abs/2309.14127 https://doaj.org/article/b8502aa5605d40c4a89374b934624b06 |
| Rights: | CC BY NC SA |
| Accession Number: | edsair.doi.dedup.....e0828d1e35b55adb0670b7ccc2239c39 |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | We describe the digraphs that are dual representations of finite lattices satisfying conditions related to meet-distributivity and modularity. This is done using the dual digraph representation of finite lattices by Craig, Gouveia and Haviar (2015). These digraphs, known as TiRS digraphs, have their origins in the dual representations of lattices by Urquhart (1978) and Ploščica (1995). We describe two properties of finite lattices which are weakenings of (upper) semimodularity and lower semimodularity respectively, and then show how these properties have a simple description in the dual digraphs. Combined with previous work in this journal on dual digraphs of semidistributive lattices (2022), it leads to a dual representation of finite meet-distributive lattices. This provides a natural link to finite convex geometries. In addition, we present two sufficient conditions on a finite TiRS digraph for its dual lattice to be modular. We close by posing three open problems. |
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| ISSN: | 07190646 |
| DOI: | 10.56754/0719-0646.2602.279 |