ISSN:
1572-9192
Keywords:
semimodular lattice
;
chamber system
;
Jordan-Hölder permutation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 Sn-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 Sn-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 Sn-buildings correspond to relatively complemented modular lattices (see [16], Section 6.1.5).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1008667127908
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