Previous |  Up |  Next

Article

Keywords:
meet-distributive lattice; congruence-uniform lattice; canonical join complex; core label order; intersection property
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.
References:
[1] Adaricheva, K. V., Gorbunov, V. A., Tumanov, V. I.: Join-semidistributive lattices and convex geometries. Adv. Math. 173 (2003), 1-49. DOI 10.1016/S0001-8708(02)00011-7 | MR 1954454 | Zbl 1059.06003
[2] Armstrong, D.: The sorting order on a Coxeter group. J. Comb. Theory, Ser. A 116 (2009), 1285-1305. DOI 10.1016/j.jcta.2009.03.009 | MR 2568800 | Zbl 1211.20033
[3] Bancroft, E.: The shard intersection order on permutations. Available at https://arxiv.org/abs/1103.1910 (2011), 19 pages.
[4] Barnard, E.: The canonical join complex. Electron. J. Comb. 26 (2019), Article ID P1.24, 25 pages. DOI 10.37236/7866 | MR 3919619 | Zbl 07032096
[5] Clifton, A., Dillery, P., Garver, A.: The canonical join complex for biclosed sets. Algebra Univers. 79 (2018), Article ID 84, 29 pages. DOI 10.1007/s00012-018-0567-z | MR 3877464 | Zbl 06983724
[6] Dilworth, R. P.: Lattices with unique irreducible decompositions. Ann. Math. (2) 41 (1940), 771-777. DOI 10.2307/1968857 | MR 0002844 | Zbl 0025.10202
[7] Edelman, P. H.: Meet-distributive lattices and the anti-exchange closure. Algebra Univers. 10 (1980), 290-299. DOI 10.1007/BF02482912 | MR 0564118 | Zbl 0442.06004
[8] Freese, R., Ježek, J., Nation, J. B.: Free Lattices. Mathematical Surveys and Monographs 42. American Mathematical Society, Providence (1995). DOI 10.1090/surv/042 | MR 1319815 | Zbl 0839.06005
[9] Garver, A., McConville, T.: Oriented flip graphs of polygonal subdivisions and noncrossing tree partitions. J. Comb. Theory, Ser. A 158 (2018), 126-175. DOI 10.1016/j.jcta.2018.03.014 | MR 3800125 | Zbl 1427.05235
[10] Garver, A., McConville, T.: Chapoton triangles for nonkissing complexes. Algebr. Comb. 3 (2020), 1331-1363. DOI 10.5802/alco.142 | MR 4184048 | Zbl 1455.05080
[11] Mühle, H.: The core label order of a congruence-uniform lattice. Algebra Univers. 80 (2019), Article ID 10, 22 pages. DOI 10.1007/s00012-019-0585-5 | MR 3908324 | Zbl 07031055
[12] Mühle, H.: Distributive lattices have the intersection property. Math. Bohem. 146 (2021), 7-17. DOI 10.21136/MB.2019.0156-18 | MR 4227308 | Zbl 7332739
[13] Mühle, H.: Noncrossing arc diagrams, Tamari lattices, and parabolic quotients of the symmetric group. Ann. Comb. 25 (2021), 307-344. DOI 10.1007/s00026-021-00532-9 | MR 4268292 | Zbl 07360380
[14] Petersen, T. K.: On the shard intersection order of a Coxeter group. SIAM J. Discrete Math. 27 (2013), 1880-1912. DOI 10.1137/110847202 | MR 3123822 | Zbl 1296.05211
[15] Reading, N.: Noncrossing partitions and the shard intersection order. J. Algebr. Comb. 33 (2011), 483-530. DOI 10.1007/s10801-010-0255-3 | MR 2781960 | Zbl 1290.05163
[16] Reading, N.: Noncrossing arc diagrams and canonical join representations. SIAM J. Discrete Math. 29 (2015), 736-750. DOI 10.1137/140972391 | MR 3335492 | Zbl 1314.05015
[17] Reading, N.: Lattice theory of the poset of regions. Lattice Theory: Special Topics and Applications Birkhäuser, Basel (2016), 399-487. DOI 10.1007/978-3-319-44236-5_9 | MR 3645055 | Zbl 1404.06004
[18] Whitman, P. M.: Free lattices. Ann. Math. (2) 42 (1941), 325-330. DOI 10.2307/1969001 | MR 0003614 | Zbl 0024.24501
[19] Whitman, P. M.: Free lattices. II. Ann. Math. (2) 43 (1942), 104-115. DOI 10.2307/1968883 | MR 0006143 | Zbl 0063.08232
Partner of
EuDML logo