×

The fundamental theorem of finite semidistributive lattices. (English) Zbl 07383336

It is known that a finite poset \(L\) is a distributive lattice if and only if it is isomorphic to \(Downset(P)\) for some finite poset \(P\). The authors proved a similar result for semidistributive lattices. Theorem 2.1. A finite poset \(L\) is a semidistributive lattice if and only if there exists a set \(S\) with some additional structure, such that \(L\) is isomorphic to the admissible subsets of\(S\) ordered by inclusion. In this case, \(S\) and its additional structure are uniquely determined by \(L\). All of these concepts are defined in the paper. The authors study also an infinite case and prove a number of theorems on semidistributive lattices and the mentioned construction.

MSC:

08B05 Equational logic, Mal’tsev conditions
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B15 Representation theory of lattices
06D75 Other generalizations of distributive lattices
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Adaricheva, KV; Nation, JB; Grätzer, GA; Wehrung, F., Classes of semidistributive lattices, Lattice Theory: Special Topics and Applications, 59-101 (2016), Cham: Birkhäuser, Cham · Zbl 1477.06043
[2] Adaricheva, KV; Nation, JB, Lattices of algebraic subsets and implicational classes. In: Grätzer, G.A., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, 102-151 (2016), Cham: Birkhäuser, Cham · Zbl 1375.06005
[3] Asai, S.: Semibricks. Int. Math. Res. Not. 2020(16), 4993-5054 (2020) · Zbl 1467.16009
[4] Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras, vol. 1. Cambridge University Press, Cambridge (2006) · Zbl 1092.16001
[5] Barbut, M., Note sur l’algèbre des techniques d’analyse hiérarchique appendix to L’analyse hiérarchique by Benjamin Matalon, 125-146 (1965), Paris: Gauthier-Villars, Paris
[6] Barnard, E., The canonical join complex, Electron. J. Combin., 26, 1, 25 (2019) · Zbl 07032096
[7] Barnard, E.; Carroll, AT; Zhu, S., Minimal inclusions of torsion classes, Algebr. Comb., 2, 5, 879-901 (2019) · Zbl 1428.05314
[8] Birkhoff, G., Rings of sets, Duke Math. J., 3, 3, 443-454 (1937) · Zbl 0017.19403
[9] Birkhoff, G.: Lattice theory. Corrected reprint of the 1967 third edition. American Mathematical Society Colloquium Publications 25. American Mathematical Society, Providence (1979)
[10] Caspard, N., Santocanale, L., Wehrung, F.: Permutahedra and associahedra. In: Grätzer, G.A., Wehrung, F. (eds.) Lattice Theory: Special Topics and Applications, vol. 2, pp. 215-286. Birkhäuser, Cham (2016) · Zbl 1401.06003
[11] Day, A., Characterizations of finite lattices that are bounded-homomorphic images of sublattices of free lattices, Canad. J. Math., 31, 1, 69-78 (1979) · Zbl 0432.06007
[12] Day, A.; Nation, JB; Tschantz, S., Doubling convex sets in lattices and a generalized semidistributivity condition, Order, 6, 2, 175-180 (1989) · Zbl 0695.06005
[13] Demonet, L., Iyama, O., Reading, N., Reiten, I., Thomas, H.: Lattice theory of torsion classes: Beyond tau-tilting theory (2018). arXiv:1711.01785v2
[14] Freese, R.; Ježek, J.; Nation, JB, Free Lattices (1995), Providence: American Mathematical Society, Providence
[15] Ganter, B.; Wille, R., Formal Concept Analysis: Mathematical Foundations (1999), Berlin: Springer, Berlin · Zbl 0909.06001
[16] Garver, A.; McConville, T., Lattice properties of oriented exchange graphs and torsion classes, Algebr. Represent. Theory, 22, 43-78 (2019) · Zbl 1408.16011
[17] Gierz, G.; Hofmann, K.; Keimel, K.; Lawson, J.; Mislove, M.; Scott, D., Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications (2003), Cambridge: Cambridge University Press, Cambridge · Zbl 1088.06001
[18] Grätzer, GA, Lattice Theory: Foundation (2011), Basel: Springer, Basel · Zbl 1233.06001
[19] Iyama, O.; Reading, N.; Reiten, I.; Thomas, H., Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math., 154, 6, 1269-1305 (2018) · Zbl 1443.16016
[20] Jónsson, B., Sublattices of a free lattice, Can. J. Math., 13, 256-264 (1961) · Zbl 0132.26201
[21] Markowsky, G., The factorization and representation of lattices, Trans. Am. Math. Soc., 203, 185-200 (1975) · Zbl 0302.06011
[22] Markowsky, G., Primes, irreducibles and extremal lattices, Order, 9, 3, 265-290 (1992) · Zbl 0778.06007
[23] Mizuno, Y., Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type, Math. Z., 277, 3-4, 665-690 (2014) · Zbl 1355.16008
[24] Ore, O., Galois connexions, Trans. Am. Math. Soc., 55, 493-513 (1944) · Zbl 0060.06204
[25] Reading, N., Lattice and order properties of the poset of regions in a hyperplane arrangement, Algebra Universalis, 50, 179-205 (2003) · Zbl 1092.06006
[26] Reading, N., Lattice congruences of the weak order, Order, 21, 4, 315-344 (2004) · Zbl 1097.20036
[27] Reading, N., Noncrossing partitions and the shard intersection order, J. Algebraic Combin., 33, 4, 483-530 (2011) · Zbl 1290.05163
[28] Reading, N., Noncrossing arc diagrams and canonical join representations, SIAM J. Discrete Math., 29, 2, 736-750 (2015) · Zbl 1314.05015
[29] Reading, N.; Grätzer, GA; Wehrung, F., Lattice theory of the poset of regions, Lattice Theory: Special Topics and Applications, 399-487 (2016), Cham: Birkhäuser, Cham · Zbl 1404.06004
[30] Stanley, RP, Enumerative Combinatorics (2012), Cambridge: Cambridge University Press, Cambridge · Zbl 1247.05003
[31] Thomas, H.; Williams, N., Rowmotion in slow motion, Proc. Lond. Math. Soc. (3), 119, 5, 1149-1178 (2019) · Zbl 1459.06010
[32] Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts, in Ordered sets (Banff, Alta., 1981), pp. 445-470, Reidel, Boston (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.