Dilworth Truncations of Geometric Lattices

Abstract

In the paper “Dependence relations in a semi-modular lattice” [5], Dilworth described a construction which represents the elements of a quasimodular point lattice (i. e., a point lattice satisfying the semimodular axiom above points or atoms) as closed sets of a matroid (i. e., a dependence structure satisfying the exchange property). This representation yields an injection of the quasimodular lattice into the geometric lattice of closed sets of the matroid which preserves the rank and meets, but not necessarily joins. Natural examples of quasimodular lattices can be obtained by taking a geometric lattice L of rank n and identifying all the elements of rank less than a fixed positive integer k. Using Dilworth’s construction, we obtain a geometric lattice D k (L) of rank n − k + 1 which contains a copy of the upper n−k levels of L. The lattice D k (L) is now called the k th Dilworth truncation of L. For example, the 2nd Dilworth truncation of the Boolean algebra of all subsets of an n-element set S is isomorphic to the lattice of partitions on S.