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.
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