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Origins and basic concepts

  • Hassler Whitney
  • Garrett Birkhoff
  • Saunders Mac Lane
  • D. A. Higgs

Abstract

There are many notions of dependence in algebra. Besides linear dependence of vectors, there are algebraic and p-dependence of elements in a field extension. (For a definition of p-dependence, see the commentary on Mac Lane [I. 4] in §1.4.) One of the historical forces behind the discovery of the concept of a matroid in the thirties was the recognition that these notions of dependence share many common properties, the most striking being the fact that the maximal independent sets all have the same cardinality. It was natural, in a decade when the axiomatic method was still a fresh idea, to attempt to find the fundamental properties of dependence common to these notions, postulate them as axioms, and derive their common properties from the axioms in a purely abstract manner. This was done by many. (See §1.5 for a complete survey; it was an early testimony to the naturalness and inevitability of the concept of a matroid that all these axiomatizations, discovered independently by very different mathematicians, are all equivalent.) However, except for Whitney’s work, there was no attempt to go beyond the elementary facts and equivalences. This was perhaps due to the fact that a key example, independence of a set of edges in a graph, and hence a key concept, that of a dual graph, were not available to those approaching matroids from an algebraic point of view. Thus, while future historians of mathematics may debate when the definition of a matroid first appeared, there is no doubt that the theory of matroids began in Whitney’s 1935 paper, “On the abstract properties of linear dependence”. This paper is the first paper reprinted in our anthology.

Keywords

Planar Graph Rank Function Dual Graph Modular Lattice Submodular Function 
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Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Hassler Whitney
    • 1
  • Garrett Birkhoff
    • 1
  • Saunders Mac Lane
    • 2
  • D. A. Higgs
    • 3
  1. 1.Harvard UniversityUSA
  2. 2.Harvard ChicagoUSA
  3. 3.Faculty of MathematicsUniversity of WaterlooWaterlooCanada

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