nLab matroid

Idea

The concept of matroid (Whitney 1935) 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.

Definitions

Plain definition

Definition

A matroid on a set $X$ is a closure operator $cl \colon P(X) \to P(X)$ satisfying the exchange axiom: for all $a, b \in X$ and $S \subseteq X$, if $a \in cl(S \cup\{b\}) \setminus cl(S)$, then $b \in cl(S \cup\{a\}) \setminus cl(S)$.

Remark

Without loss of generality, the exchange axiom can be stated as: if $a \in cl(S + \{b\}) \setminus cl(S)$, then $b \in cl(S + \{a\}) \setminus cl(S)$, since the implication is vacuous when $b \in S$.

Remark

When combinatorialists speak of matroids as such, $X$ is usually taken to be a finite set. A typical example is $X$ some finite subset of a vector space $V$, taking $cl(S) \,\coloneqq\, X \cap Span(S)$ for any $S \subseteq X$.

Further terminology:

Definition

Under Def. :

1. A subset $S \subseteq X$ is called independent if there is a strict inclusion $cl(T) \subset cl(S)$ for every strict inclusion $T \subset S$ (this is the same as requiring $x \notin cl(S\setminus \{x\})$ for every $x \in S$).

2. A subset $S \subset X$ is called a basis if $cl(S) = X$ and $S$ is independent.

3. A hyperplane is a closed subset $S$ (meaning $cl(S) = S$) that is maximal among proper closed subsets of $X$.

It is possible to axiomatize the notion of matroid by taking bases (Def. ) 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.

Proposition

Any two bases (Def. ) of a matroid $X$ 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.

Proof

First, suppose $A$ is an independent set and $B$ is a finite basis, and suppose there are subsets $A_0 \subseteq A, B_0 \subseteq B$ such that $A_0 \cup B_0$ is a basis. We claim that for each $a \in A \setminus A_0$, there exists $b \in B_0$ such that $A_0 \cup \{a\} \cup (B_0 \setminus \{b\})$ is a basis. For, let $C \subseteq B_0$ be of minimum cardinality such that $a \in cl(A_0 \cup C)$; we know $C$ must be inhabited since $a \notin cl(A \setminus \{a\}) \supseteq cl(A_0)$; clearly $C \cap A_0 = \emptyset$. So let $b$ be an element of $C$. Since by minimality of $C$ we have

$$a \in cl(A_0 \cup (C \setminus \{b\}) \cup \{b\}) \cap \neg cl(A_0 \cup (C \setminus \{b\})),$$

it follows from the exchange axiom that $b \in cl(A_0 \cup (C \setminus \{b\}) \cup \{a\})$. Thus $b \in cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\})$, whence

$$cl(A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}) = cl(A_0 \cup B_0 \cup \{a\}) = X$$

so that $D \coloneqq A_0 \cup (B_0 \setminus \{b\}) \cup \{a\}$ “spans” $X$. Also $D$ is independent: if $x \in D$ and $x \neq a$, then

$$cl(D \setminus \{x\}) \subseteq cl((A_0 \cup B_0) \setminus \{x\})$$

with neither side containing $x$ since $A_0 \cup B_0$ is independent; whereas if $x = a$ and supposing to the contrary that $a \in cl(D \setminus \{a\}) = cl((A_0 \cup (B_0 \setminus \{b\}))$, we conclude $A_0 \cup (B \setminus \{b\})$ has the same span as $D$. Since $D$ already spans, $b \in cl(A_0 \cup (B_0 \setminus \{b\}))$, again impossible since $A_0 \cup B_0$ is independent. This proves the claim.

Again assuming $A$ independent and $B$ is a finite basis, now we show that $card(A) \leq card(B)$, which will finish the proof. Let $n = card(B)$, and suppose on the contrary that there are distinct elements $a_1, \ldots, a_{n+1} \in A$. Set $A_0 = \emptyset$ and $B_0 = B$. Applying the claim above inductively, we have that $\{a_1, \ldots, a_i\} \cup (B \setminus \{b_1, \ldots, b_i\})$ is a basis for $1 \leq i \leq n$, so in particular $\{a_1, \ldots, a_n\}$ spans $X$. Hence $a_{n+1} \in cl(\{a_1, \ldots, a_{n}\})$, contradicting the independence of $A$.

Model-theoretic geometry

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).

Definition

A pregeometry (in model theory) is a (possibly infinite) matroid (Def. , given by a set $X$ equipped with a closure operator $cl$) that is finitary:

• for all $S \subseteq X$, if $x \in cl(S)$ then $x \in cl(S_0)$ for some finite subset $S_0 \subseteq S$.

A geometry is a pregeometry such that, in addition:

1. $cl(\varnothing) = \varnothing$ (the empty subset is closed),

2. $cl(\{x\}) = \{x\}$ for every $x \in X$ (all singleton subsets are closed).

(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.

Proposition

In a pregeometry $(X, cl)$, any two bases have the same cardinality.

Proof

We already proved this in the case where the pregeometry has a finite basis. Otherwise, if $A$ is independent and $B$ is an infinite basis, then

$$A = A \cap X = A \cap \bigcup_{B_0 \subseteq B\; finite} cl(B_0) = \bigcup_{B_0 \subseteq B\; finite} A \cap cl(B_0)$$

where the second equality follows from the finitary condition. Since each summand $A \cap cl(B_0)$ has cardinality less than that of $B_0$ by independence of $A$ (noting that $B_0$ is a basis of $cl(B_0)$), the union on the right has cardinality bounded above by $card(B)$. From $card(A) \leq card(B)$ it follows that any two bases have the same cardinality.

Examples

Vector spaces, algebraic closures, graphs, restrictions, localizations, …

• A graphic matroid is a matroid $M$ derived from a simple graph by taking the underlying set of $M$ to be the set of edges $E$, and taking as independent sets of $M$ those $S \subseteq E$ such that the edges of $S$ (and their incident vertices) form a forest, i.e., a graph without cycles.

• A vector matroid is a matroid $M$ derived from a a collection of vectors $E$ in a vector space. Independent sets of $M$ are those $S \subseteq E$ such that $S$ is linearly independent. Providing an equivalence to such a vector matroid is the problem of representability over a specified field.

• An algebraic matroid for a field extension $K/F$ is derived from a collection of elements $E \subset K$ and independent sets of $M$ are those such that $F(S)$ has transcendence degree equal to $\mid S \mid$ so all of $S$ are algebraically independent.

• If $M$ is a (finite) matroid, then the matroid dual $M^\ast$ of $M$ has the same underlying set as $M$, and where a basis in $M^\ast$ is precisely the complement of a basis of $M$. It follows that $M^{\ast\ast} \cong M$ (this is even an equality according to the definition).

• For a topological space $X$ the exchange condition is equivalent to

  $$\forall x,y \in X: x\in cl({y}) \Leftrightarrow y\in cl({x})$$

  In particular, every $T1$-space is a matroid.

Properties

Mnev’s theorem

Mnëv’s universality theorem says that any semialgebraic set in $\mathbb{R}^n$ defined over integers is stably equivalent to the realization space of some oriented matroid.

Combinatorial optimization

(…)

Related concepts

• oriented matroid

References

The original article:

• Hassler Whitney, On the abstract properties of linear dependence, American Journal of Mathematics (The Johns Hopkins University Press) 57 3 (1935) 509-533 [jstor:2371182, MR1507091]

• H. H. Crapo, Gian Carlo Rota, On the foundations of combinatorial theory: combinatorial geometries, MIT Press, Cambridge 1970

On the category of matroids with strong maps between them:

• Chris Heunen, Vaia Patta, The category of matroids, Applied Categorical Structures 26 (2018) 205–237 [arXiv:1512.01390, doi:10.1007/s10485-017-9490-2]

On Hodge theory and intersection cohomology of matroids:

• Tom Braden, June Huh, Jacob P. Matherne, Nicholas Proudfoot, Botong Wang, Singular Hodge theory for combinatorial geometries [arXiv:2010.06088]

• June Huh, Combinatorics and Hodge theory, Proc. Int. Cong. Math. 1 (2022)

  [doi:10.4171/ICM2022/205, pdf, pdf]

On the axiomatization of infinite matroids:

• Henning Bruhn, Reinhard Diestel, Matthias Kriesell, Rudi Pendavingh, Paul Wollan, Axioms for infinite matroids, Advances in Mathematics, 239 (2013) 18–46 [arXiv:1003.3919, doi:10.1016/j.aim.2013.01.011]

Further references:

• Wikipedia matroid, Mnëv’s universality theorem, oriented matroid, matroid minor

• J. Oxley, What is a matroid, pdf

• James G. Oxley, Matroid theory, Oxford Grad. Texts in Math. 1992, 2010

• some papers on Coxeter matroids html

• MathOverflow question Mnëv’s universality corollaries, quantitative versions

• Eric Katz, Sam Payne, Realization space for tropical fans, pdf

• Nikolai E. Mnev, The universality theorems on the classification problem of configuration varieties and convex polytopes varieties, pp. 527-543, in “Topology and geometry: Rohlin Seminar.” Edited by O. Ya. Viro. Lecture Notes in Mathematics, 1346, Springer 1988; A lecture on universality theorem (in Russian) pdf

• Talal Ali Al-Hawary, Free objects in the category of geometries, pdf

• Talal Ali Al-Hawary, D. George McRae, Toward an elementary axiomatic theory of the category of LP-matroids, Applied Categorical Structures 11: 157–169, 2003, doi

• Hirokazu Nishimura, Susumu Kuroda, A lost mathematician, Takeo Nakasawa: the forgotten father of matroid theory 1996, 2009

• William H. Cunningham, Matching, matroids, and extensions, Math. Program., Ser. B 91: 515–542 (2002) doi

• L. Lovász, Matroid matching and some applications, J. Combinatorial Theory B 28, 208–236 (1980)

• A. Björner, M. Las Vergnas, B. Sturmfels, N. White, G.M. Ziegler, Oriented matroids, Cambridge Univ. Press 1993, 2000, view at reslib.com

• Matthew Baker, Matroids over hyperfields, arxiv/1601.01204

• David Marker, Model theory: an introduction, Graduate Texts in Math. 217, Springer-Verlag New York, 2002.

• Tom Braden, Carl Mautner, Matroidal Schur algebras, arxiv/1609.04507

• P. S. Kung, A source book in matroid theory, Birkhäuser, Boston, 1986.

• Jim Geelen, Bert Gerards, Geoff Whittle, Towards a structure theory for matrices and matroids , ICM 2006 Proceedings 827–842 pdf

• I. M. Gelfand, R. M. Goresky, R. D. MacPherson, Combinatorial geometries, convex polyhedra, and Schubert cells, Adv. Math. 63 (1987) 301–316

• Federico Ardila, The geometry of geometries: matroid theory, old and new, arXiv:2111.08726

category: combinatorics, model theory