Semimodular Lattices and Semibuildings

  • Authors:

  • David Samuel Herscovici

  • Affiliations:

  • Saint Mary‘s College of California, Department of Mathematics and Computer Science, Moraga, CA 94757

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1998

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Abstract

In a ranked lattice, we consider two maximal chains, or“flags” to be i-adjacent if they are equalexcept possibly on rank i. Thus, a finite rank lattice is achamber system. If the lattice is semimodular, as noted in [9], there is a“Jordan-Hölder permutation” between any two flags. Thispermutation has the properties of an S_n-distance function onthe chamber system of flags. Using these notions, we define aW-semibuilding as a chamber system with certain additionalproperties similar to properties Tits used to characterize buildings. Weshow that finite rank semimodular lattices form anS_n-semibuilding, and develop a flag-based axiomatization ofsemimodular lattices. We refine these properties to axiomatize geometric,modular and distributive lattices as well, and to reprove Tits‘ result thatS_n-buildings correspond to relatively complemented modularlattices (see [16], Section 6.1.5).