Semimodular Lattices: Theory and Applications

ISBN-13: 9780521461054 / Angielski / Twarda / 1999 / 388 str.

Manfred Stern; G. -C Rota; B. Doran

Opis

Lattice theory evolved as part of algebra in the nineteenth century through the work of Boole, Peirce and Schroder, 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.

Recenzje

'… a very well organized book … a pleasure to read … will certainly become a standard source.' Horst Szambien, Zentralblatt MATH

Spis treści

Preface; 1. From Boolean algebras to semimodular lattices; 2. M-symmetric lattices; 3. Conditions related to semimodularity, 0-conditions and disjointness 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; Master reference list; Table of notation; Index.