Edited by Neil White
Publisher: Cambridge University Press
Print Publication Year: 1986
Online Publication Date:May 2010
Online ISBN:9780511629563
Hardback ISBN:9780521309370
Paperback ISBN:9780521092029
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511629563.013
Subjects: Discrete Mathematics Information Theory and Coding
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It was shown in Chapter 2 that the notion of rank function provides a cryptomorphic theory of matroids. The semimodularity property of the rank is essentially equivalent to the basis-exchange or circuit-elimination axioms and thus is central to matroid theory.
In this chapter we shall consider a more general class of functions on subsets of a finite set E that are semimodular and nondecreasing, but not necessarily nonnegative, normalized, or unit-increased.
The relationships between semimodular functions and matroids have been known and studied since the very beginning of matroid theory. Many results have been found and sometimes independently rediscovered. This chapter presents a unifying theory that attempts to explain most of these known results as derived essentially from Dilworth's fundamental theorem about the embedding of a point lattice into a geometric lattice. Dilworth's original proof was based on a construction of a rank function from a semimodular function, which is a special case of what will be more generally defined in this chapter as expansions. The main thrust of the exposition is to study the properties and applications of expansions.
GENERAL PROPERTIES OF SEMIMODULAR FUNCTIONS
In the most general setting that will be of interest we shall consider a point lattice L and classes of integer-valued functions defined on L.
pp. i-iv
pp. v-viii
List of Contributors : Read PDF
pp. ix-x
Series Editor's Statement : Read PDF
pp. xi-xii
pp. xiii-xiv
pp. xv-xviii
Chapter 1 - Examples and Basic Concepts : Read PDF
pp. 1-28
Chapter 2 - Axiom Systems : Read PDF
pp. 29-44
Chapter 3 - Lattices : Read PDF
pp. 45-61
Chapter 4 - Basis-Exchange Properties : Read PDF
pp. 62-75
Chapter 5 - Orthogonality : Read PDF
pp. 76-96
Chapter 6 - Graphs and Series-Parallel Networks : Read PDF
pp. 97-126
Chapter 7 - Constructions : Read PDF
pp. 127-223
Chapter 8 - Strong Maps : Read PDF
pp. 224-253
Chapter 9 - Weak Maps : Read PDF
pp. 254-271
Chapter 10 - Semimodular Functions : Read PDF
pp. 272-297
Appendix of Matroid Cryptomorphisms : Read PDF
pp. 298-312
pp. 313-316
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS : Read PDF
pp. 317-318