Semimodular Lattices: Theory and Applications

by Manfred Stern

• ISBN13: 9780521118842
• ISBN10: 0521118840
• Edition: 1st
• Format: Paperback
• Copyright: 2009-09-03
• Publisher: Cambridge University Press

Summary

Lattice theory evolved as part of algebra in the nineteenth century through the work of Boole, Peirce and Schr_der, and in the first half of the twentieth century through the work of Dedekind, Birkhoff, Ore, von Neumann, Mac Lane, Wilcox, Dilworth, and others. In Semimodular Lattices, Manfred Stern uses successive generalizations of distributive and modular lattices to outline the development of semimodular lattices from Boolean algebras. He focuses on the important theory of semimodularity, its many ramifications, and its applications in discrete mathematics, combinatorics, and algebra. The author surveys and analyzes Birkhoff's concept of semimodularity and the various related concepts in lattice theory, and he presents theoretical results as well as applications in discrete mathematics group theory and universal algebra. Special emphasis is given to the combinatorial aspects of finite semimodular lattices and to the connections between matroids and geometric lattices, antimatroids and locally distributive lattices. The book also deals with lattices that are "close" to semimodularity or can be combined with semimodularity, for example supersolvable, admissible, consistent, strong, and balanced lattices. Researchers in lattice theory, discrete mathematics, combinatorics, and algebra will find this book valuable.

Table of Contents

Preface

From Boolean algebras to semimodular lattices

M-symmetric lattices

Conditions related to semimodularity, 0-conditions and disjointness properties

Supersolvable and admissible lattices, consistent and strong lattices

The covering graph

Semimodular lattices of finite length

Local distributivity

Local modularity

Congruence semimodularity

Master reference list

Table of notation

Index

Table of Contents provided by Publisher. All Rights Reserved.