Origins and basic concepts

TL;DRAbstract

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 dif

Chat with Paper

AI Agents for this Paper