Transversal theory : an account of some aspects of combinatorial mathematics / L. Mirsky.

Format:

Book

Author/Creator:

Mirsky, L. (Leonid)

Series:

Mathematics in science and engineering ; 75.

Mathematics in science and engineering ; 75

Language:

English

Subjects (All):

Combinatorial analysis.

Mathematical analysis.

Physical Description:

1 online resource (271 p.)

Place of Publication:

New York : Academic Press, 1971.

Language Note:

English

Summary:

Transversal theory; an account of some aspects of combinatorial mathematics

Contents:

Front Cover; Transversal Theory: An account of some aspects of combinatorial mathematics; Copyright Page; Contents; Preface; Chapter 1. Sets, Topological Spaces, Graphs; 1.1 Sets and mappings; 1.2 Families; 1.3 Mapping theorems and cardinal numbers; 1.4 Boolean atoms; 1.5 The lemmas of Zorn and Tukey; 1.6 Tychonoff's theorem; 1.7 Graphs; Notes on Chapter 1; Chapter 2. Hall's Theorem and the Notion of Duality; 2.1 Transversals, representatives, and representing sets; 2.2 Proofs of the fundamental theorem for finite families; 2.3 Duality; Notes on Chapter 2

Chapter 3. The Method of 'Elementary Constructions'3.1 'Elementary constructions'; 3.2 Transversal index; 3.3 Further extensions of Hall's theorem; 3.4 A self-dual variant of Hall's theorem; Notes on Chapter 3; Chapter 4. Rado's Selection Principle; 4.1 Proofs of the selection principle; 4.2 Transfinite form of Hall's theorem; 4.3 A theorem of Rado and Jung; 4.4 Dilworth's decomposition theorem; 4.5 Miscellaneous applications of the selection principle; Notes on Chapter 4; Chapter 5. Variants, Refinements, and Applications of Hall's Theorem; 5.1 Disjoint partial transversals

5.2 Strict systems of distinct representatives5.3 Latin rectangles; 5.4 Subsets with a prescribed pattern of overlaps; Notes on Chapter 5; Chapter 6. Independent Transversals; 6.1 Pre-independence and independence; 6.2 Rado's theorem on independent transversals; 6.3 A characteristic property of independence structures; 6.4 Finite independent partial transversals; 6.5 Transversal structures and independence structures; 6.6 Marginal elements; 6.7 Axiomatic treatment of the rank function; Notes on Chapter 6; Chapter 7. Independence Structures and Linear Structures; 7.1 A hierarchy of structures

7.2 Bases of independence spaces7.3 Totally admissible sets; 7.4 Set-theoretic models of independence structures; Notes on Chapter 7; Chapter 8. The Rank Formula of Nash-Williams; 8.1 Sums of independence structures; 8.2 Disjoint independent sets; 8.3 A characterization of transversal structures; 8.4 Symmetrized form of Rado's theorem on independent transversals; Notes on Chapter 8; Chapter 9. Links of Two Finite Families; 9.1 The notion of a link; 9.2 Common representatives; 9.3 The criterion of Ford and Fulkerson; 9.4 Common representatives with restricted frequencies

9.5 An insertion theorem for common transversals9.6 Harder results for a single family; Notes on Chapter 9; Chapter 10. Links of Two Arbitrary Families; 10.1 The theorem of Mendelsohn and Dulmage and its interpretations; 10.2 Systems of representatives with repetition; 10.3 Common systems of representatives with defect; 10.4 Common transversals of two families; 10.5 Common transversals of maximal subfamilies; Notes on Chapter 10; Chapter 11. Combinatorial Properties of Matrices; 11.1 The language of matrix theory; 11.2 Theorems of König, Frobenius, and Rado

11.3 Diagonals of doubly-stochastic matrices

Notes:

Description based upon print version of record.

Includes bibliography: p. 236-246.

ISBN:

1-282-29043-6

9786612290435

0-08-095584-3

OCLC:

316549564