| Title:
|
Meet-distributive lattices have the intersection property (English) |
| Author:
|
Mühle, Henri |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
148 |
| Issue:
|
1 |
| Year:
|
2023 |
| Pages:
|
95-104 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
This paper is an erratum of H. Mühle: Distributive lattices have the intersection property, Math. Bohem. (2021). Meet-distributive lattices form an intriguing class of lattices, because they are precisely the lattices obtainable from a closure operator with the so-called anti-exchange property. Moreover, meet-distributive lattices are join semidistributive. Therefore, they admit two natural secondary structures: the core label order is an alternative order on the lattice elements and the canonical join complex is the flag simplicial complex on the canonical join representations. In this article we present a characterization of finite meet-distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite meet-distributive lattice is always a meet-semilattice. (English) |
| Keyword:
|
meet-distributive lattice |
| Keyword:
|
congruence-uniform lattice |
| Keyword:
|
canonical join complex |
| Keyword:
|
core label order |
| Keyword:
|
intersection property |
| MSC:
|
06D75 |
| idZBL:
|
Zbl 07655815 |
| idMR:
|
MR4536312 |
| DOI:
|
10.21136/MB.2022.0072-21 |
| . |
| Date available:
|
2023-02-03T11:22:59Z |
| Last updated:
|
2023-09-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151529 |
| . |
| Reference:
|
[1] Adaricheva, K. V., Gorbunov, V. A., Tumanov, V. I.: Join-semidistributive lattices and convex geometries.Adv. Math. 173 (2003), 1-49. Zbl 1059.06003, MR 1954454, 10.1016/S0001-8708(02)00011-7 |
| Reference:
|
[2] Armstrong, D.: The sorting order on a Coxeter group.J. Comb. Theory, Ser. A 116 (2009), 1285-1305. Zbl 1211.20033, MR 2568800, 10.1016/j.jcta.2009.03.009 |
| Reference:
|
[3] Bancroft, E.: The shard intersection order on permutations.Available at https://arxiv.org/abs/1103.1910 (2011), 19 pages. |
| Reference:
|
[4] Barnard, E.: The canonical join complex.Electron. J. Comb. 26 (2019), Article ID P1.24, 25 pages. Zbl 07032096, MR 3919619, 10.37236/7866 |
| Reference:
|
[5] Clifton, A., Dillery, P., Garver, A.: The canonical join complex for biclosed sets.Algebra Univers. 79 (2018), Article ID 84, 29 pages. Zbl 06983724, MR 3877464, 10.1007/s00012-018-0567-z |
| Reference:
|
[6] Dilworth, R. P.: Lattices with unique irreducible decompositions.Ann. Math. (2) 41 (1940), 771-777. Zbl 0025.10202, MR 0002844, 10.2307/1968857 |
| Reference:
|
[7] Edelman, P. H.: Meet-distributive lattices and the anti-exchange closure.Algebra Univers. 10 (1980), 290-299. Zbl 0442.06004, MR 0564118, 10.1007/BF02482912 |
| Reference:
|
[8] Freese, R., Ježek, J., Nation, J. B.: Free Lattices.Mathematical Surveys and Monographs 42. American Mathematical Society, Providence (1995). Zbl 0839.06005, MR 1319815, 10.1090/surv/042 |
| Reference:
|
[9] Garver, A., McConville, T.: Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions.J. Comb. Theory, Ser. A 158 (2018), 126-175. Zbl 1427.05235, MR 3800125, 10.1016/j.jcta.2018.03.014 |
| Reference:
|
[10] Garver, A., McConville, T.: Chapoton triangles for nonkissing complexes.Algebr. Comb. 3 (2020), 1331-1363. Zbl 1455.05080, MR 4184048, 10.5802/alco.142 |
| Reference:
|
[11] Mühle, H.: The core label order of a congruence-uniform lattice.Algebra Univers. 80 (2019), Article ID 10, 22 pages. Zbl 07031055, MR 3908324, 10.1007/s00012-019-0585-5 |
| Reference:
|
[12] Mühle, H.: Distributive lattices have the intersection property.Math. Bohem. 146 (2021), 7-17. Zbl 7332739, MR 4227308, 10.21136/MB.2019.0156-18 |
| Reference:
|
[13] Mühle, H.: Noncrossing arc diagrams, Tamari lattices, and parabolic quotients of the symmetric group.Ann. Comb. 25 (2021), 307-344. Zbl 07360380, MR 4268292, 10.1007/s00026-021-00532-9 |
| Reference:
|
[14] Petersen, T. K.: On the shard intersection order of a Coxeter group.SIAM J. Discrete Math. 27 (2013), 1880-1912. Zbl 1296.05211, MR 3123822, 10.1137/110847202 |
| Reference:
|
[15] Reading, N.: Noncrossing partitions and the shard intersection order.J. Algebr. Comb. 33 (2011), 483-530. Zbl 1290.05163, MR 2781960, 10.1007/s10801-010-0255-3 |
| Reference:
|
[16] Reading, N.: Noncrossing arc diagrams and canonical join representations.SIAM J. Discrete Math. 29 (2015), 736-750. Zbl 1314.05015, MR 3335492, 10.1137/140972391 |
| Reference:
|
[17] Reading, N.: Lattice theory of the poset of regions.Lattice Theory: Special Topics and Applications Birkhäuser, Basel (2016), 399-487. Zbl 1404.06004, MR 3645055, 10.1007/978-3-319-44236-5_9 |
| Reference:
|
[18] Whitman, P. M.: Free lattices.Ann. Math. (2) 42 (1941), 325-330. Zbl 0024.24501, MR 0003614, 10.2307/1969001 |
| Reference:
|
[19] Whitman, P. M.: Free lattices. II.Ann. Math. (2) 43 (1942), 104-115. Zbl 0063.08232, MR 0006143, 10.2307/1968883 |
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