Exchange Properties for Reduced Decompositions in Modular Lattices
Joseph P. S. Kung · Birkhäuser Boston eBooks · 1990
In his paper [3], Dilworth studied “the manner in which the irreducibles of two decompositions can replace each other” in a modular lattice. Thus this paper anticipated papers on the closely related area of basis exchange properties in matroids or geometric lattices. This relationship is most transparent for modular lattices of finite rank. For such lattices, replacement properties are “local” in the sense that they can be decided by looking only at intervals of the form [a, u a ], where u a is the join all the elements covering a. This follows from the following variant (cf. [5]) of a result in group theory due to Burnside [2] and Frattini [4]