How the infinite matroid got his axioms

In the first post in this series we raised the question of how to find a good notion of infinite matroids with duality. One way to look at the problem is like this: we can’t just define infinite matroids by using the usual axioms for finite matroids and allowing the ground set to be infinite. This fails horribly, because the various standard axiomatisations give rise to quite different notions. We can make them match up again by adding axioms saying that the matroid is finitary (all circuits are finite), but the class of matroids we get in this way isn’t closed under duality.

In this post we’re going to look at a just-so story about how one might solve this problem. We’ll have to wrestle with the axiomatisations until we get natural modifications of all of them which are once more equivalent, but we mustn’t be too violent: we don’t want to lose closure under minors or duality along the way.

A hopeful place to start is with the following Closure Axioms for the closure operator $\text{cl}$ of a finite matroid with ground set $E$:

(CL1) For all subsets $X$ of $E$, $X \subseteq \text{cl}(X)$.
(CL2) For all subsets $X$ of $E$, $\text{cl}(X) = \text{cl}(\text{cl}(X))$.
(CL3) For all subsets $X$ and $Y$ of $E$ with $X \subseteq Y$ we have $\text{cl}(X) \subseteq \text{cl}(Y)$
(CL4) For any $a$ and $b$ in $E$ and any subset $Y$ of $E$ with $a \in \text{cl}(Y \cup {b}) \setminus \text{cl}(Y)$ we have $b \in \text{cl}(Y \cup {a}) \setminus \text{cl}(Y)$.

We can still consider such operators $\text{cl}: {\cal P}E \to {\cal P}E$ even on an infinite set $E$. They are called Idempotent-Exchange operators, or IE-operators for short (the second axiom is idempotence, and the fourth is called the exchange property) [H69]. This is a good place to start because there are very natural definitions of minors and duality for IE-operators.

The dual of $\text{cl}$ is defined as $\text{cl}^*\colon X \mapsto X \cup \{x \in E | x \not \in \text{cl}(E \setminus (X \cup \{x\})\}$. Thus $\text{cl}^*$ has the following property: for any partition of $E$ as $X \dot \cup Y \dot \cup \{x\}$ we have $x \in \text{cl}^*(X) \leftrightarrow x \not \in \text{cl}(Y)$. This property, together with (CL1), characterises $\text{cl}^*$. Thus $\text{cl}^{**} = \text{cl}$. It is isn’t hard to check that if $\text{cl}$ is the closure operator of a matroid then $\text{cl}^*$ is the closure operator of the dual matroid. $\text{cl}^*$ automatically satisfies (CL1) and it satisfies (CL3) if $\text{cl}$ does. Now (CL4) is just a reformulation of the statement we get by applying (CL2) to $\text{cl}^*$. Thus $\text{cl}^*$ satisfies (CL4) because $\text{cl}$ satisfies (CL2) and vice versa. Minors are also easy to define. Thus for example the restriction of $\text{cl}$ to $E’$ is given by $\text{cl}|_{E’} (X) = \text{cl}(X) \cap E’$.

If we try to relate IE-operators to the objects given by the other axiomatisations then we quickly run into problems. We can begin by saying that a set $I$ is $\text{cl}$-independent as long as there is no $x$ in $I$ with $x \in \text{cl}(I \setminus \{x\})$. Then we would like to define bases as maximal independent sets. The trouble is that there might not be any! For example, if $E$ is infinite and we define $\text{cl}(X)$ to be $X$ when $X$ is finite and $E$ when $X$ is infinite then the independent sets are exactly the finite sets, so there are no maximal independent sets.

How can we fix this? One rather silly idea would be to take the collection of IE-operators which have at least one base. Leaving all other problems aside, this collection isn’t minor-closed. Yet more brutally, we could require that $\text{cl}$ and all of its minors each have at least one base. By definition this class is minor-closed. It isn’t hard to see that it is also closed under duality.

This is in fact the right class of infinite matroids. They were discovered by Higgs, who called them B-Matroids [H69]. Oxley showed that B-matroids are a very natural class: he showed that the class of B-Matroids is the largest class of preindependence spaces on which duality and minors are well defined [O92]. (Preindependence spaces are the things satisfying the usual independence axioms for finite matroids.)

The clincher, which not only settles the fact that B-matroids are the right class to work with but also makes it much easier to work with them, is the fact that they have multiple natural axiomatisations [BDKPW13]. At the start of this post, we set out to find those axiomatisations, and we have already found some suitable Closure Axioms (except that we have not yet precisely formulated `all minors have bases’ as an axiom). How could we go about finding the others? It turns out that, as well as slightly tweaking the axiomatisations of finite matroids, we’ll always need a new `all minors have bases’ axiom.

Let’s start by looking at the Independence Axioms for finite matroids:
(I1) The empty set is independent.
(I2) Every subset of an independent set is independent
(I3) If $I$ and $J$ are independent and $J$ has more elements than $I$ then there is $x \in J \setminus I$ such that $I \cup \{x\}$ is still independent.

(I1) and (I2) don’t need to be changed, but the `more elements than’ clause in (I3) becomes less useful when applied to infinite sets. The key idea here is to replace `$J$ has more elements than $I$’ with the far weaker `$J$ is a maximal independent set but $I$ isn’t’ in (I3). Although it looks weaker, we don’t actually lose anything by making this change, because even after the modification (I1), (I2) and (I3) still axiomatise finite matroids.

The requirement that all minors should have at least one base doesn’t follow from (I1), (I2) and our modified (I3). So we need to add it as a new axiom. Formulating this axiom correctly is a subtle matter, because it seems that in order to have a sensible definition of contraction we must first be able to find bases of the sets we wish to contract. The key point is that any minor can be obtained in such a way that the set $I$ of elements that we contract is independent. The independent sets of such contractions are simply sets whose union with $I$ is independent. So we can formulate `all minors have bases’ as follows:

(IM) For any $I \subseteq X \subseteq E$ with $I$ independent, the collection of independent sets $J$ such that $I \subseteq J \subseteq X$ has a maximal element.

To find a base when we contract an arbitrary set $X$ using this axiom, we have to first find a base of $X$ itself. But (IM) allows us to do that too, so this doesn’t cause any problems. (I1)-(I3) together with (IM) really axiomatise B-matroids. The same idea works for bases: the usual base axioms together with (IM) applied to the collection of subsets of bases gives us another axiomatisation. We can also formulate a fifth Closure Axiom (CLM) in the same spirit, saying that the collection of $\text{cl}$-independent sets satisfies (IM). Once more, the collection of operators satisfying (CL1)-(CL4) and (CLM) is exactly the collection of B-matroids.

We won’t discuss the proof that all of these axiomatisations are equivalent (or more precisely cryptomorphic) here, since it is similar to the proof for finite matroids. Instead, we’ll finish by sketching how to get axiomatisations in terms of circuits or the rank function.

In the derivation of other axiomatisations from the circuit axioms, the circuit elimination axiom gets iterated. If one tries to make these proofs work in an infinite context, it turns out that circuit elimination has to be iterated infinitely often, and the proofs fall apart. What is needed is an axiom which allows for controlled iteration of circuit elimination. More precisely, it is enough to require that infinitely many different edges of the same circuit can all be eliminated simultaneously. This principle, the infinite circuit elimination axiom, can be formulated as follows:

(C3) Let $C$ be a circuit, $z$ an edge of $C$ and $X$ a subset of $C \setminus \{z\}$. For each $x \in X$ let $C_x$ be a circuit with $C_x \cap (X \cup \{z\}) = \{x\}$. Then there is a circuit $C’$ with $$z \in C’ \subseteq \left(C \cup \bigcup_{x \in X}C_x\right)\setminus X\,.$$

If we replace the usual circuit elimination axiom with this one and we add an axiom saying that the collection of sets not including any circuit satisfies (IM) then we get an axiomatisation of infinite matroids in terms of their circuits.

The rank function requires the biggest jump from the finite axiomatisation. In order to get a handle on the relationships between sets of infinite rank we must axiomatise not in terms of the rank function but rather in terms of the relative rank function $r$, where $r(A | B)$ is intended to represent the number of elements we need to add to $B$ in order to span $A$. Tweaking the usual rank axioms to corresponding statements about the relative rank function and, as usual, adding an axiom like (IM) gives yet another axiomatisation of infinite matroids.

I hope this has helped to explain why the axioms for infinite matroids look the way they do. Next time we’ll get a better feel for the meaning of these axioms by looking at some examples.

[BDKPW13] H. Bruhn, R. Diestel, M. Kriesell, R. A. Pendavingh and P. Wollan, Axioms for Infinite Matroids. Advances in Mathematics, 239, p18-46
[H69] D. A. Higgs, Matroids and Duality. Colloq. Maths 20 (1969), p215-220
[O92] J. G. Oxley, Infinite Matroids. Matroid Applications (ed. N. White). Encyc. Math. Appl. 40 (CUP 1992), p73-90.

Growth Rates V

Hi everyone, happy new year and sorry for my recent absence/lateness!

Today I am going to say some things about growth rates of minor-closed classes, but in more concrete cases that I’ve been writing about previously. To recap on a definition that has come up many times on this blog, the growth rate function of a minor-closed class of matroids is the definition capturing the answer to a question that Joseph Kung called the ‘primary concern’ of extremal matroid theory: given a minor-closed class $\mathcal{M}$ of matroids, what is the maximum number of elements that a simple rank-$r$ matroid in $\mathcal{M}$ can have? For each nonnegative integer $n$, the growth rate function for $\mathcal{M}$ at $n$ is defined as follows:

$h_{\mathcal{M}}(n) = \max\{|M|: M \in \mathcal{M} \text{ simple}, r(M) \le n\}$.

For instance, for the class $\mathcal{G}$ is the class of graphic matroids we have $h_{\mathcal{G}}(n) = \binom{n+1}{2}$. In fact, the growth rate theorem [GKW09] says that every minor-closed class of matroids with finite growth rate function either has growth rate function in $O(n)$, or contains the graphic matroids (and therefore has growth rate function at least $\binom{n+1}{2}$ for all $n$). Thus, the graphic matroids have the smallest possible growth rate function that is not just linear in $n$.

There are other classes of matroids with growth rate function exactly $\binom{n+1}{2}$ for all $n$, or for all but a few $n$. They necessarily contain the graphic matroids, but can be strictly larger. Several were mentioned in a post by Irene last year; they include the regular matroids, and the class $\mathrm{Ex}(\mathrm{AG}(3,2))$ of binary matroids with no $\mathrm{AG}(3,2)$-minor. The former result was shown by Heller [H57] and the latter by Kung et al [KMPR13]. Given this (and Irene asked a question along these lines in her post), it is natural to ask for a characterisation of exactly which classes of matroids grow like the graphic ones:

Problem 1: Characterise the minor-closed classes of matroids $\mathcal{M}$ that satisfy $h_{\mathcal{M}}(n) = \binom{n+1}{2}$ for all but finitely many $n$.

or, more concretely, we could ask to characterise the minors whose exclusion gives such a class:

Problem 2: Let $\mathcal{O}$ be a finite set of simple matroids and let $\mathcal{M} = \mathrm{Ex}(\mathcal{O})$ be the class of matroids with no minor in $\mathcal{O}$. Characterise when $\mathcal{M}$ satisfies $h_{\mathcal{M}}(n) = \binom{n+1}{2}$ for all but finitely many $n$.

Nice Classes

This post will answer both these questions. Before I state the theorem that does this, I’ll give three examples of minor-closed classes that do grow faster than the graphic matroids.

  • The class $\mcevencycle$ of even cycle matroids having a signed graph representation with a blocking pair (that is, a pair of vertices through which all negative cycles pass). This class contains only binary matroids and has growth rate function $\binom{n+2}{3}-3$.
  • The class $\mcsigned$ of signed-graphic matroids having a signed graph representation with a vertex that is contained in every nonloop negative cycle. This class contains matroids representable over every field of odd order, and has growth rate function $\binom{n+2}{2}-2$
  • The class $\mcfree$ that is the union of the class of graphic matroids and the class of truncations of graphic matroids. There is no finite field over which every matroid in this class is representable, and it has growth rate function $\binom{n+2}{2}$.

The superscripts stand for ‘even cycle’, ‘signed graphic’ and ‘truncation’ respectively.  These are three ‘nice’ classes of matroids that are a bit bigger than the graphic matroids, all arising from well-known classes and constructions. The following theorem resolves Problem 1, and will also be enough to resolve 2.

Theorem 1 [GN14]: Let $\mathcal{M}$ be a minor-closed class of matroids with growth rate function in $\Theta(n^2)$. Either

  • $h_{\mathcal{M}}(n) = \binom{n+1}{2}$ for all sufficiently large $n$, or
  • $\mathcal{M}$ contains one of $\mcevencycle$, $\mcsigned$, or $\mcfree$.

Note that this ‘or’ is really an ‘exclusive or’, since the latter outcome implies that $h_{\mathcal{M}}(n) \ge \binom{n+2}{2}-3$ for all $n \ge 4$ and therefore that the former outcome does not hold. This theorem gives a very nice answer for problem 1; the classes that grow like the graphic matroids are exactly those that do not contain any of three particular classes that are all too big.

To deal with Problem 2, we just need to understand which minors that $\mathcal{O}$ must contain in order that $\mathrm{Ex}(\mathcal{O})$ does not contain any of $\mcevencycle$, $\mcsigned$ or $\mcfree$. This is just saying that $\mathcal{O}$ should contain a matroid from each of these classes. The classes actually intersect, so a single matroid in $\mathcal{O}$ could serve a dual purpose. For $\mathrm{Ex}(\mathcal{O})$ to be quadratically dense, it is also necessary to have some rank-$2$ uniform matroid in $\mathcal{O}$ and no graphic matroid in $\mathcal{O}$. The answer to Problem 2 is therefore in the following corollary:

Corollary 1: If $\mathcal{O}$ is a finite set of simple matroids and $\mathcal{M} = \mathrm{Ex}(\mathcal{O})$, then the following two statements are equivalent:

  1. $h_{\mathcal{M}}(n) = \binom{n+1}{2}$ for all sufficiently large $n$.
  2. $\mathcal{M}$ contains a rank-$2$ uniform matroid, contains no graphic matroids, and contains a matroid from each of $\mcevencycle$, $\mcsigned$ and $\mcfree$.

This solves the problem in general. We can also obtain specialisations to particular fields. The binary case (that is, where $U_{2,4} \in \mathcal{O}$) says the following:

Corollary 2: If $\mathcal{O}$ is a finite set of simple matroids and $\mathcal{M}$ is the set of binary matroids with no minor in $\mathcal{O}$, then the following two statements are equivalent:

  1. $h_{\mathcal{M}}(n) = \binom{n+1}{2}$ for all sufficiently large $n$.
  2. $\mathcal{O}$ contains no graphic matroids, and contains a nongraphic matroid in $\mcevencycle$.

When $\mathcal{O} = \{\mathrm{AG}(3,2)\}$ we get (for large $n$) the aforementioned result of Kung et al. There is also a similar corollary for matroids over any fixed odd-order finite field, where $\mcevencycle$ is replaced by $\mcsigned$.

Future Work

Theorem 1 should just be the beginning of an effort to characterise quadratic growth rate functions completely. It would be really nice to answer the following question:

Problem 3: Which functions in $\Theta(n^2)$ can occur as growth rate functions?

It follows from Theorem 1 that any function strictly between $\binom{n+1}{2}$ and $\binom{n+2}{2}-3$ cannot. In general, a conjecture of Geelen, Gerards and Whittle [GGW14] suggests that all such functions are quadratic polynomials with half-integral leading coefficient. Matroid structure theory in [GGW14] should (with enough work) give the answer for classes over any fixed finite field, but I would love to know what can be done without appeals to structure theory, since we are probably a long way off full structure theorems for general matroids.

[H57] I. Heller, On linear systems with integral valued solutions, Pacific J. Math. 7 (1957), 1351-1364.

[GGW14] J. Geelen, B. Gerards and G. Whittle, The highly connected matroids in minor-closed classes, arXiv:1312.5102.

[GKW09] J. Geelen, J. P. S. Kung, and G. Whittle, Growth rates of minor-closed classes of matroids, J. Combin. Theory Ser. B, 99(2):420–427, 2009

[GN14] J. Geelen, P. Nelson, Matroids denser than a clique, arXiv:1409.0777

[KMPR13] J. Kung, D. Mayhew, I. Pivotto, and G. Royle, Maximum size binary matroids with no AG(3,2)-minor are graphic, arXiv:1304.2448.