Synopses & Reviews
A survey of semimodularity which presents theory and applications in discrete mathematics, group theory and universal algebra.
Review
"Professor Stern's book is the first to include the latest developments in the theory of semimodular lattices. His clear presentation, and his readable mathematical style, will make his treatise an indispensable vade mecum for lattice theorists, and for all discrete mathematicians everywhere." Mathematical Reviews
Synopsis
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.
Synopsis
Semimodular Lattices: Theory and Applications uses successive generalizations of distributive and modular lattices to outline the development of semimodular lattices from Boolean algebras. It surveys and analyzes Garrett Birkhoff's concept of semimodularity, and he presents theoretical results as well as applications in discrete mathematics, group theory and universal algebra.
Synopsis
In Semimodular Lattices: Theory and Applications 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 book surveys and analyzes Garrett Birkhoff's concept of semimodularity and the various related concepts in lattice theory, and it presents theoretical results as well as applications in discrete mathematics group theory and universal algebra.
Table of Contents
Preface; 1. From Boolean algebras to semimodular lattices; 2. M-symmetric lattices; 3. Conditions related to semimodularity, 0-conditions and disjointedness properties; 4. Supersolvable and admissible lattices, consistent and strong lattices; 5. The covering graph; 6. Semimodular lattices of finite length; 7. Local distributivity; 8. Local modularity; 9. Congruence semimodularity.