Preface

Summary

This book aims at giving a survey of semimodularity and related concepts in lattice theory as well as presenting a number of applications. The book may be regarded as a supplement to certain aspects of vol. 26 (Theory of Matroids), vol. 29 (Combinatorial Geometries), and vol. 40 (Matroid Applications) of this encyclopedia.

Classically semimodular lattices arose out of certain closure operators satisfying what is now usually called the Steinitz–Mac Lane exchange property. Inspired by the matroid concept introduced in 1935 by Hassler Whitney in a paper entitled “On the abstract properties of linear dependence,” Garrett Birkhoff isolated the concept of semimodularity in lattice theory. Matroids are related to geometric lattices, that is, to semimodular atomistic lattices of finite length. The theory of geometric lattices was not foreshadowed in Dedekind's work on modular lattices. Geometric lattices were the first class of semimodular lattices to be systematically investigated.

The theory was developed in the thirties by Birkhoff, Wilcox, Mac Lane, and others. In the early forties Dilworth discovered locally distributive lattices, which turned out to be important new examples of semimodular lattices. These examples are the first cryptomorphic versions of what became later known in combinatorics as antimatroids. The name antimatroid hints at the fact that this combinatorial structure has properties that are very different from certain matroid properties. In particular, antimatroids have the so-called antiexchange property. While matroids abstract the notion of linear independence, antimatroids abstract the notion of Euclidean convexity. The theory was further developed by Dilworth, Crawley, Avann, and others in the fifties and sixties.

During the years 1935–55 and later many ramifications and applications of semimodularity were discovered.