Geodesic
Semimodular lattices and semibuildings
Journal of Algebraic Combinatorics, Tome 7 (1998) no. 1, pp. 39-51.

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Summary: In a ranked lattice, we consider two maximal chains, or $ldquo$flags $rdquo$ 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 $ldquo$Jordan-Hölder permutation $rdquo$ 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).
Keywords: semimodular lattice, chamber system, Jordan-Hölder permutation 
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     title = {Semimodular lattices and semibuildings},
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Herscovici, David Samuel. Semimodular lattices and semibuildings. Journal of Algebraic Combinatorics, Tome 7 (1998) no. 1, pp. 39-51. http://geodesic.mathdoc.fr/item/JAC_1998__7_1_a3/