Semimodular Lattices and Semibuildings
David Samuel Herscovici
DOI: 10.1023/A:1008667127908
Abstract
In a ranked lattice, we consider two maximal chains, or 
flags 
to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a Jordan-Hölder permutation between any two flags. This permutation has the properties of an S n-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an S n-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits' result that S n-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).
flags to be i-adjacent if they are equal except possibly on rank i. Thus, a finite rank lattice is a chamber system. If the lattice is semimodular, as noted in [9], there is a Jordan-Hölder permutation between any two flags. This permutation has the properties of an S n-distance function on the chamber system of flags. Using these notions, we define a W-semibuilding as a chamber system with certain additional properties similar to properties Tits used to characterize buildings. We show that finite rank semimodular lattices form an S n-semibuilding, and develop a flag-based axiomatization of semimodular lattices. We refine these properties to axiomatize geometric, modular and distributive lattices as well, and to reprove Tits' result that S n-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).Pages: 39–51
Keywords: semimodular lattice; chamber system; Jordan-Hölder permutation
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References
1. H. Abels, “The gallery distance of flags,” Order 8 (1991), 77-92.
2. H. Abels, “The geometry of the chamber system of a semimodular lattice,” Order 8 (1991), 143-158.
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17. J. Tits, “Buildings and group amalgamations,” Proceedings of Groups-St. Andrews 1985, London Mathematical Society Lecture Notes Series 121, Cambridge University Press, 1986, pp. 110-127.
2. H. Abels, “The geometry of the chamber system of a semimodular lattice,” Order 8 (1991), 143-158.
3. G. Birkhoff, Lattice Theory, third edition, American Mathematical Society, Providence, RI, 1967.
4. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Transactions of the American Mathematical Society 260 (1980), 159-183.
5. A. Bj\ddot orner, “Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings,” Advances in Mathematics 52 (1984), 173-212.
6. N. Bourbaki, Groupes et alg`ebres de Lie, Hermann, Paris, France, 1968, Chapters 4, 5, and 6.
7. K. Brown, Buildings, Springer-Verlag, 1989.
8. G. Gr\ddot atzer, General Lattice Theory, Birkh\ddot auser Verlag, Basel, Germany, 1978.
9. D.S. Herscovici, “Minimal paths between maximal chains in finite rank semimodular lattices,” J. Alg. Combin. 7 (1998), 17-37.
10. J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, New York, NY, 1990.
11. M. Ronan, Lectures on Buildings, Harcourt Brace Jovanovich, Boston, MA, 1989.
12. R.P. Stanley, Enumerative Combinatorics, Wadsworth & Brooks/Cole, Belmont, CA, 1986, Vol. I.
13. R.P. Stanley, “Supersolvable lattices,” Algebra Universalis 2 (1972), 197-217.
14. R.P. Stanley, “Finite lattices and Jordan-H\ddot older sets,” Algebra Universalis 4 (1974), 361-371.
15. J. Tits, “Buildings of spherical type and finite BN-pairs,” Lecture Notes in Mathematics, Springer-Verlag, Berlin, Germany, 1974, Vol. 386.
16. J. Tits, “A local approach to buildings,” The Geometric Vein. The Coxeter Festschrift, Springer Verlag, 1981, pp. 519-547.
17. J. Tits, “Buildings and group amalgamations,” Proceedings of Groups-St. Andrews 1985, London Mathematical Society Lecture Notes Series 121, Cambridge University Press, 1986, pp. 110-127.