Semimodular lattices and semibuildings
The author studies relationships between maximal chains (``flags'') in finite rank semimodular lattices. He defines a semibuilding over a Coxeter group \(W\) as a chamber system with a \(W\)-distance function and with some additional properties similar to those used by Tits to define \(W\)-buildings. He shows that finite rank semimodular lattices form an \(S_n\)-semibuilding, and develop a flag-based axiomatization of semimodular lattices. The main results consider the lattice \(L(B)\) of so-called ``\(j\)-equivalence classes'' for a semibuilding \(B\) and characterize upper semimodularity, modularity, relatively complementarity, geometricity and distributivity of \(L(B)\).
scientific article; zbMATH DE number 1139616
scientific article