The concept of matroid, due to Hassler Whitney, is fundamental to combinatorics, giving several different ways of encoding/defining and presenting a general notion of “independence”, e.g., linear independence in a vector space, algebraic independence in a field extension, etc.
There is also a similar concept of an oriented matroid; every oriented matroid has an underlying matroid.
A matroid on a set is a closure operator satisfying the exchange axiom: if , then .
Usually when combinatorialists speak of matroids as such, is taken to be a finite set. A typical example is some finite subset of a vector space , taking for any .
Under this definition, a subset is independent if there is a strict inclusion for every strict inclusion (this is the same as requiring for every ). Again under this definition, is a basis if and is independent. A hyperplane is a closed subset (meaning ) that is maximal among proper closed subsets of . It is possible to axiomatize the notion of matroid by taking bases as the primitive notion, or independent sets as the primitive notion, or hyperplanes as the primitive notion, etc. – Rota (after Birkhoff) speaks of cryptomorphism between these differing definitions. Much of the power and utility of matroid theory comes from this multiplicity of definitions and the possibility of moving seamlessly between them; for example, a matroid structure might be easy to detect from the viewpoint of one definition, but not from another.
Any two bases of a matroid have the same cardinality, provided that one of them is finite.
The cardinality of such a basis is called of course the dimension of the matroid. Clearly then a finite matroid has a well-defined dimension.
First, suppose is an independent set and is a finite basis, and suppose there are subsets such that is a basis. We claim that for each , there exists such that is a basis. For, let be of minimum cardinality such that ; we know must be inhabited since ; clearly . So let be an element of . Since by minimality of we have
it follows from the exchange axiom that . Thus , whence
so that “spans” . Also is independent: if and , then
with neither side containing since is independent; whereas if and supposing to the contrary that , we conclude has the same span as . Since already spans, , again impossible since is independent. This proves the claim.
Again assuming independent and is a finite basis, now we show that , which will finish the proof. Let , and suppose on the contrary that there are distinct elements . Set and . Applying the claim above inductively, we have that is a basis for , so in particular spans . Hence , contradicting the independence of .
Vector spaces, algebraic closures, graphs, restrictions, localizations, …
Essentially the very same notion arises in model theory, except instead of being called a matroid it is called a “pregeometry” or “geometry”, and in contrast to combinatorialists, model theorists usually mean infinite matroids. The notion arises in the study of geometry of strongly minimal sets, with applications to stability theory (part of Shelah’s classification theory).
A pregeometry is a (possibly infinite) matroid (given by a set equipped with a closure operator ) that is finitary: for all , if then for some finite subset . A geometry is a pregeometry such that and for every .
(See also geometric stability theory.)
The language of independence, spanning, and basis carry over as before. A maximal independent set spans (i.e., is a basis), and maximal independent sets exist according to Zorn's lemma. Again we have a notion of dimension by the following proposition.
In a pregeometry , any two bases have the same cardinality.
We already proved this in the case where the pregeometry has a finite basis. Otherwise, if is independent and is an infinite basis, then
where the second equality follows from the finitary condition. Since each summand has cardinality less than that of by independence of (noting that is a basis of ), the union on the right has cardinality bounded above by . From it follows that any two bases have the same cardinality.
Mnëv’s universality theorem says that any semialgebraic set in defined over integers is stably equivalent to the realization space of some oriented matroid.
To be written, possibly with some original research.