Lattice and order properties of the poset of regions in a hyperplane arrangement

Algebra Universalis

Volumn 50, Issue 2, 2003, Pages 179-205

  Reading, Nathan ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Bounded lattice; Cayley lattice; Congruence normal; Congruence uniform; Coxeter group; Critical pair; Doubling; Hyperplane arrangement; Order dimension; Order quotient; Permutation lattice; Poset of regions; Semi distributive lattice; Simplicial; Subcritical pair

Indexed keywords

EID: 1542508681     PISSN: 00025240     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00012-003-1834-0     Document Type: Article

References (24)

1. H. Barcelo and E. Ihrig, Modular elements in the lattice L(A) when A is a real reflection arrangement, Selected papers in honor of Adriano Garsia, Discrete Math. 193 (1998), 61-68.
2. A. Björner, P. Edelman and G. Ziegler, Hyperplane Arrangements with a Lattice of Regions, Discrete Comput. Geom. 5 (1990), 263-288.
3. N. Caspard, The lattice of permutations is bounded, Internat. J. Algebra Comput 10 (2000), 481-489.
4. N. Caspard, G. Le Conte de Poly-Barbut and M. Morvan, Cayley lattices of finite Coxeter groups are bounded, Adv. in Appl. Math., to appear.
5. I. Chajda and V. Snášel, Congruences in Ordered Sets, Math. Bohem. 123 (1998), 95-100.
6. A. Day, Splitting lattices generate all lattices, Algebra Universalis 7 (1977), 163-169.
7. A. Day, Congruence normality: the characterization of the doubling class of convex sets, Algebra Universalis 31 (1994), 397-406.
8. V. Duquenne and A. Cherfouh, On Permutation Lattices, Math. Social Sci. 27 (1994), 73-89.
9. n Dissected by Hyperplanes, Trans. Amer. Math. Soc. 283, (1984), 617-631.
10. P. Edelman and R. Jamison, The theory of convex geometries, Geom. Dedicata 19 (1985), 247-270.
11. S. Flath, The order dimension of multinomial lattices, Order 10 (1993), 201-219.
12. R. Freese, J. Ježek and J. Nation, Free lattices, Mathematical Surveys and Monographs, 42, American Mathematical Society, 1995.
13. G. Grätzer, General Lattice Theory, Second edition, Birkhäuser Verlag, Basel, 1998.
14. B. Grünbaum, Arrangements of hyperplanes, Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing, 41-106. Louisiana State Univ., Baton Rouge, La., 1971.
15. J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge Univ. Press, 1990.
16. M. Jambu and L. Paris, Combinatorics of Inductively Factored Arrangements, European J. Combin. 16 (1995), 267-292.
17. G. Le Conte de Poly-Barbut, Sur les Treillis de Coxeter Finis (French), Math. Inf. Sci. hum. 32, 125 (1994), 41-57.
18. R. McKenzie, Equational bases and nonmodular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1-43.
19. P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300. Springer-Verlag, Berlin, 1992.
20. I. Rabinovitch and I. Rival, The Rank of a Distributive Lattice, Discrete Math. 25 (1979), 275-279.
21. N. Reading, Order dimension, strong Bruhat order and lattice properties for posets, Order 19 (2002), 73-100.
22. N. Reading, Lattice congruences of the weak order, in preparation.
23. W. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in the Mathematical Sciences, The Johns Hopkins Univ. Press, 1992.
24. G. Ziegler, Combinatorial construction of logarithmic differential forms, Adv. Math. 76 (1989), 116-154.