'Researchers working in lattice theory will surely welcome this excellent and up-to-date reference book.' Acta. Sci. Math.
'I recommend the book highly to all interested readers, both experts and non-experts.' Stefan E. Schmidt, Bulletin of the London Mathematical Society
'… a very well organized book … a pleasure to read … will certainly become a standard source.' Horst Szambien, Zentralblatt MATH
Über das Produkt
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.
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.