A Single Axiom for Matroid Closure Operators

At some point I remembered that groups can be characterised by a single axiom, ^{Footnote¹I first learnt about this from alg-d's YouTube video “Defining groups by one equation” (in Japanese), where the cited reference was Neumann [Neu81]. For more details, see also alg-d's PDF. Single axioms for groups have been studied rather broadly; see, for example, Kunen [Kun92] and McCune [McC93].1} and I wondered whether matroids could also be written using a single axiom. The basis axioms are already essentially a single axiom, so I turned to another axiom system.

Matroids were introduced by Whitney [Whi35] as objects capturing the abstract properties of linear dependence. Standard references include Welsh [Wel76], White [Whi86], and Oxley [Oxl11].

Matroids are known to admit many equivalent definitions, one of which uses a closure operator. For readers from other areas, the phrase “closure operator” may first bring to mind the closure operation in topology. Topological closure and matroid closure are both examples of operations that “add all points forced by a given set.” However, in a topological space the added points are determined by nearness and limits, whereas in a matroid, and especially in a linear matroid, they are determined by linear span. Thus the two notions can be described using the same abstract vocabulary, but the additional axioms they satisfy are different. Topological spaces can also be characterised by closure operators; these are Kuratowski's axioms. Looking into this, I found that attempts to express the four Kuratowski axioms by one condition go back at least to Monteiro [Mon45] ^{Footnote²Strictly speaking, Monteiro [Mon45] gave a single axiom that implies (KCL2)–(KCL4), excluding (KCL1).2} and Pervin [Per64]. This note records the analogous question: can the closure-operator axioms for matroids also be written as a single axiom?

Before getting to matroids, I first recall Pervin's single axiom for topological closure operators.

I am not a specialist in topology, but I understand the closure $c(A)$ of a set $A$ as “the set obtained by adding all points that can be approached arbitrarily closely from $A$.” A point $x \in X$ belongs to $c(A)$ precisely when every neighbourhood of $x$ meets $A$. In other words, even if $x$ itself is not in $A$, if every sufficiently small neighbourhood around $x$ still sees points of $A$, then $x$ lies in the closure of $A$.

In this sense, $c(A)$ can be viewed as the operation that collects all points touching $A$, that is, all points whose every neighbourhood intersects $A$. For example, in the real line $\mathbb{R}$, the closure of the open interval $(0,1)$ is $[0,1]$. The endpoints $0$ and $1$ are not contained in the original set, but every neighbourhood of either endpoint intersects $(0,1)$, so they are included in the closure.

Kuratowski's axioms [Kur22] for such a closure operator are as follows:

These four axioms abstract the basic properties that topological closure should satisfy. Here is a quick intuitive reading.

(KCL1)$c(\emptyset) = \emptyset$(KCL1) Empty-set preservation

There is no way to approach the empty set, so the closure of the empty set is empty. Equivalently, no point has every neighbourhood meeting the empty set.

(KCL2)for every $A \subseteq X$, one has $A \subseteq c(A)$ (extensibility)(KCL2) Extensibility

The points of $A$ are, of course, contained in the closure of $A$. Closure does not delete points from $A$; it adds the points forced by $A$.

(KCL3)for every $A \subseteq X$, one has $c(c(A)) = c(A)$ (idempotency)(KCL3) Idempotency

Once all limit points have been added and the set has been closed, taking closure again adds no new points. Thus closure is an operation of completing a set until it is closed, and closing it again does not change it.

(KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4) Union preservation

A point approachable from a finite union of sets is approachable from one of those sets. In particular, for two sets, the closure of $A \cup B$ is the union of the closures of $A$ and $B$. This reflects the fact that the condition that a neighbourhood meets $A \cup B$ decomposes into meeting $A$ or meeting $B$. More precisely, if $x \notin c(A)$ and $x \notin c(B)$, then there is a neighbourhood $U$ of $x$ disjoint from $A$ and a neighbourhood $V$ of $x$ disjoint from $B$. Then $U \cap V$ is again a neighbourhood of $x$ and is disjoint from $A \cup B$. Hence $x \notin c(A \cup B)$. This gives $c(A \cup B) \subseteq c(A) \cup c(B)$; the reverse inclusion follows from monotonicity.

One point to keep in mind is that the last axiom concerns finite unions. For infinite unions, it is not true in general that

For instance, in $\mathbb{R}$, if $A_{n} = \{ 1/n \}$, then each $A_{n}$ is closed, but the closure of $\bigcup_{n \geq 1} A_{n}$ also contains the new point $0$.

Pervin combined these four Kuratowski axioms into the following single condition:

Proof

(Necessity) If $c$ is a Kuratowski closure operator, then (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P) holds.

Take arbitrary $A, B \subseteq X$. By empty-set preservation (KCL1)$c(\emptyset) = \emptyset$(KCL1), $c(\emptyset) = \emptyset$, so

$$c(A \cup B) \setminus c(\emptyset) = c(A \cup B).$$

By extensibility (KCL2)for every $A \subseteq X$, one has $A \subseteq c(A)$ (extensibility)(KCL2), $A \subseteq c(A)$, and by idempotency (KCL3)for every $A \subseteq X$, one has $c(c(A)) = c(A)$ (idempotency)(KCL3), $c(c(B)) = c(B)$. Therefore

$$A \cup c(A) \cup c(c(B)) = c(A) \cup c(B).$$

By union preservation (KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4),

$$c(A) \cup c(B) = c(A \cup B).$$

Combining these equalities gives

$$A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset).$$

Hence (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P) holds.

(Sufficiency) If $c$ satisfies (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P), then $c$ is a Kuratowski closure operator.

It is enough to prove (KCL1)$c(\emptyset) = \emptyset$(KCL1)–(KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4).

Empty-set preservation (KCL1)$c(\emptyset) = \emptyset$(KCL1)

Put $A = B = \emptyset$ in (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P). Then

$$\emptyset \cup c(\emptyset) \cup c(c(\emptyset)) = c(\emptyset) \setminus c(\emptyset) = \emptyset.$$

The left-hand side is $c(\emptyset) \cup c(c(\emptyset))$, so

$$c(\emptyset) \cup c(c(\emptyset)) = \emptyset.$$

In particular, $c(\emptyset) \subseteq \emptyset$. Since $\emptyset \subseteq c(\emptyset)$ always holds, we get

$$c(\emptyset) = \emptyset.$$

Extensibility (KCL2)for every $A \subseteq X$, one has $A \subseteq c(A)$ (extensibility)(KCL2)

Let $A \subseteq X$ be arbitrary. Put $B = \emptyset$ in (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P). Then

$$A \cup c(A) \cup c(c(\emptyset)) = c(A) \setminus c(\emptyset).$$

Since we have already shown $c(\emptyset) = \emptyset$, we also have $c(c(\emptyset)) = c(\emptyset) = \emptyset$. Hence

$$A \cup c(A) = c(A).$$

Therefore $A \subseteq c(A)$.

Idempotency (KCL3)for every $A \subseteq X$, one has $c(c(A)) = c(A)$ (idempotency)(KCL3)

Let $B \subseteq X$ be arbitrary. Put $A = \emptyset$ in (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P). Then

$$\emptyset \cup c(\emptyset) \cup c(c(B)) = c(B) \setminus c(\emptyset).$$

Since $c(\emptyset) = \emptyset$ has already been shown, we obtain

$$c(c(B)) = c(B).$$

Thus $c(c(B)) = c(B)$ for every $B \subseteq X$.

Union preservation (KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4)

Let $A, B \subseteq X$ be arbitrary. By (P)$\forall A, B \subseteq X, \, A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset)$.(P),

$$A \cup c(A) \cup c(c(B)) = c(A \cup B) \setminus c(\emptyset).$$

We have already proved $c(\emptyset) = \emptyset$, $A \subseteq c(A)$, and $c(c(B)) = c(B)$. Hence the left-hand side is $c(A) \cup c(B)$ and the right-hand side is $c(A \cup B)$. Therefore

$$c(A) \cup c(B) = c(A \cup B),$$

so union preservation (KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4) holds.

Thus $c$ is a Kuratowski closure operator.

End of proof□

At first sight, Pervin's single axiom may look rather artificial. Nevertheless, it can be viewed as embedding Kuratowski's four axioms into one equation. In this way, Pervin's axiom simultaneously requires “the empty set generates no extra points,” “each set contains itself,” “taking closure twice changes nothing,” and “closure is compatible with finite unions.”

To understand matroid closure operators, it is helpful to keep linear matroids in mind, so I briefly recall them before formally introducing the term “matroid.” Consider a family of vectors $(v_e)_{e \in E}$ indexed by a finite set $E$. For $A \subseteq E$, define

Thus $\cl(A)$ is the set of all elements of $E$ whose vectors can be generated as linear combinations of the vectors indexed by $A$. The matroid obtained in this way is called a linear matroid. This viewpoint, abstracting linear dependence, goes back to Whitney [Whi35], where matroids originated. For basic examples, including linear matroids, see Oxley [Oxl11, Chap. 1].

In a linear matroid, $a \in \cl(A)$ means that the vector $v_{a}$ corresponding to $a$ lies in the subspace spanned by $\{ v_{x} : x \in A \}$. Thus closure in a matroid is not the topological operation of adding limit points; it is the linear-algebraic operation of adding all elements already generated by $A$.

The closure-operator definition of matroids is as follows:

Unlike topological closure, matroid closure does not in general require $\cl(\emptyset) = \emptyset$. For example, in a linear matroid, an element corresponding to the zero vector is already generated by the empty set and therefore lies in $\cl(\emptyset)$. Such an element is called a loop.

With the example of linear matroids in mind, the closure axioms for matroids can be read as follows.

(MCL1)for every $A \subseteq E$, one has $A \subseteq \cl(A)$ (extensibility)(MCL1) Extensibility

Each element of $A$ can, of course, be generated by $A$ itself. Hence

$$A \subseteq E \cap \span \{ v_{a} : a \in A \} = \cl(A).$$

Since closure adds elements generated by the original set $A$, it does not lose any element of $A$.

(MCL2)for every $A \subseteq B \subseteq E$, one has $\cl(A) \subseteq \cl(B)$ (monotonicity)(MCL2) Monotonicity

If $A \subseteq B$, then every vector generated from $A$ can also be generated from $B$. Therefore

$$\span \{ v_{a} : a \in A \} \subseteq \span \{ v_{b} : b \in B \},$$

and $\cl(A)\subseteq \cl(B)$ follows. Adding more available generators cannot reduce what can be generated.

(MCL3)for every $A \subseteq E$, one has $\cl(\cl(A)) = \cl(A)$ (idempotency)(MCL3) Idempotency

The set $\cl(A)$ already collects all elements generated by $A$. Therefore, adding the elements of $\cl(A)$ as generators does not change the spanned subspace. That is,

$$\span \{ v_{a} : a \in \cl(A) \} = \span \{ v_{a} : a \in A \},$$

and so $\cl(\cl(A))=\cl(A)$. This expresses the fact that adding elements that are already generated does not increase the generating power.

(MCL4)for every $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$, one has $e \in \cl(A \cup \{ a \})$ (Mac Lane–Steinitz exchange property) (MCL4) Mac Lane–Steinitz exchange property

The exchange property abstracts a symmetry of dependence from linear algebra. Suppose $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$. In a linear matroid, this means that $v_{a}$ can be generated using the vectors corresponding to $A$ together with $v_{e}$, but cannot be generated from $A$ alone. Thus there are $w \in \span\{ v_{x} \mathrel{:} x \in A \}$ and a scalar $\lambda$ such that

$$v_{a} = w + \lambda v_{e}.$$

If $\lambda = 0$, then $v_{a} \in \span\{ v_{x} \mathrel{:} x \in A \}$, contradicting $a \in \cl(A)$. Hence $\lambda \neq 0$. Therefore

$$v_{e} = \lambda^{-1}(v_{a} - w),$$

so $e \in \cl(A \cup \{ a \})$.

In words, if adding $e$ is what first makes $a$ generatable, then adding $a$ also makes $e$ generatable.

Proof

(Necessity) If $M = (E, \cl)$ is a matroid, then (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL) holds.

We prove both inclusions in (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL).

($\supseteq$)

The inclusion $\cl(A) \subseteq \cl(A \cup \{ e \})$ follows from monotonicity (MCL2)for every $A \subseteq B \subseteq E$, one has $\cl(A) \subseteq \cl(B)$ (monotonicity)(MCL2). Also $e \in \cl(A \cup \{ e \})$ follows from extensibility (MCL1)for every $A \subseteq E$, one has $A \subseteq \cl(A)$ (extensibility)(MCL1). Next, take $b \in \{ a \in E \mathrel{:} e \in \cl(A \cup \{ a \}) \setminus \cl(A) \}$. Then $e \in \cl(A \cup \{ b \}) \setminus \cl(A)$. Applying the Mac Lane–Steinitz exchange property (MCL4)for every $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$, one has $e \in \cl(A \cup \{ a \})$ (Mac Lane–Steinitz exchange property) (MCL4) to $A$, the element $b$, and the element $e$, we obtain $b \in \cl(A \cup \{ e \})$. Thus the third term on the right-hand side is also contained in $\cl(A \cup \{ e \})$, and $\cl(A \cup \{ e \}) \supseteq \cl(A) \cup \{ e \} \cup \{ a \in E \mathrel{:} e \in \cl(A \cup \{ a \}) \setminus \cl(A) \}$ is proved.

($\subseteq$)

Take $a \in \cl(A \cup \{ e \})$. If $a \in \cl(A) \cup \{ e \}$, then $a$ is already in the right-hand side. Otherwise, $a \notin \cl(A)$ and $a \ne e$. We first claim that $e \notin \cl(A)$. Indeed, if $e \in \cl(A)$, then adding $e$ to both sides of the extensibility inclusion $A \subseteq \cl(A)$ gives $A \cup \{ e \} \subseteq \cl(A) \cup \{ e \} = \cl(A)$. By monotonicity and idempotency,

$$\cl(A \cup \{ e \}) \subseteq \cl(\cl(A))=\cl(A).$$

The reverse inclusion follows from monotonicity, so $\cl(A \cup \{ e \}) = \cl(A)$. This contradicts $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$. Hence indeed $e \notin \cl(A)$. Now the Mac Lane–Steinitz exchange property (MCL4)for every $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$, one has $e \in \cl(A \cup \{ a \})$ (Mac Lane–Steinitz exchange property) (MCL4) gives $e \in \cl(A \cup \{ a \})$, and therefore

$$e \in \cl(A\cup\{a\}) \setminus \cl(A).$$

Thus $a$ belongs to the set

$$\{ a \in E \mathrel{:} e \in \cl(A \cup \{ a \}) \setminus \cl(A)\},$$

and $\cl(A \cup \{ e \}) \subseteq \cl(A) \cup \{ e \} \cup \{ a \in E \mathrel{:} e \in \cl(A \cup \{ a \}) \setminus \cl(A) \}$ follows.

Hence (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL) holds.

(Sufficiency) If $M = (E, \cl)$ satisfies (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL), then $M$ is a matroid.

It is enough to prove (MCL1)for every $A \subseteq E$, one has $A \subseteq \cl(A)$ (extensibility)(MCL1)–(MCL4)for every $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$, one has $e \in \cl(A \cup \{ a \})$ (Mac Lane–Steinitz exchange property) (MCL4).

Extensibility (MCL1)for every $A \subseteq E$, one has $A \subseteq \cl(A)$ (extensibility)(MCL1)

Let $A \subseteq E$ and $a \in A$ be arbitrary. Put $A^{\prime} \coloneqq A \setminus \{ a \}$. Applying the single axiom to $A^{\prime}$ and $a$, we get

$$\cl(A) = \cl(A^{\prime} \cup \{ a \}) = \cl(A^{\prime}) \cup \{ a \} \cup \{ x \in E \mathrel{:} a\in \cl(A^{\prime} \cup \{ x \}) \setminus \cl(A^{\prime})\}.$$

Since the right-hand side contains $a$, we have $a\in\cl(A)$. Hence $A \subseteq \cl(A)$.

Monotonicity (MCL2)for every $A \subseteq B \subseteq E$, one has $\cl(A) \subseteq \cl(B)$ (monotonicity)(MCL2)

The right-hand side of (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL) always contains $\cl(A)$. Therefore $\cl(A) \subseteq \cl(A \cup \{ e \})$ for every $A$ and $e$. Since $E$ is finite, if $A \subseteq B$, we may add the elements of $B \setminus A$ one at a time and obtain $\cl(A) \subseteq \cl(B)$.

Idempotency (MCL3)for every $A \subseteq E$, one has $\cl(\cl(A)) = \cl(A)$ (idempotency)(MCL3)

By extensibility, $A \subseteq \cl(A)$. Since $E$ is finite, write

$$\cl(A) \setminus A = \{ e_{1}, \dots, e_{n} \}.$$

Put $A_{0} \coloneqq A$ and $A_{i} \coloneqq A_{i-1} \cup \{ e_{i} \}$. By monotonicity, $\cl(A) \subseteq \cl(A_{i-1})$, so in particular $e_{i} \in \cl(A_{i-1})$. From (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL), we have

$$\tag{*} \cl(A_{i-1} \cup \{ e_{i} \}) = \cl(A_{i-1}) \cup \{ e_{i} \} \cup \{ a \in E : e_{i} \in \cl(A_{i-1} \cup \{ a \}) \setminus \cl(A_{i-1})\}.$$

Since $e_{i} \in \cl(A_{i-1})$, the second term $\{ e_{i} \}$ on the right-hand side is already contained in $\cl(A_{i-1})$. Moreover, for every $a \in E$, the fact that $e_{i} \in \cl(A_{i-1})$ implies

$$e_{i} \notin \cl(A_{i-1} \cup \{ a \}) \setminus \cl(A_{i-1}).$$

Hence the third term is empty, and we obtain

$$\cl(A_{i}) = \cl(A_{i-1} \cup \{ e_{i} \}) = \cl(A_{i-1}).$$

Inductively,

$$\cl(\cl(A)) = \cl(A_{n}) = \cl(A_{0}) = \cl(A).$$

Mac Lane–Steinitz exchange property (MCL4)for every $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$, one has $e \in \cl(A \cup \{ a \})$ (Mac Lane–Steinitz exchange property) (MCL4)

Let $A \subseteq E$, $e \in E$, and $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$ be arbitrary.

If $a = e$, then the desired conclusion is $e \in \cl(A \cup \{ e \})$, which follows directly from the assumption $a \in \cl(A \cup \{ e \}) \setminus \cl(A)$. Thus assume $a \neq e$. In the right-hand side of (MCL)$\forall A \subseteq E,\ \forall e \in E, \, \cl(A \cup \{ e \}) = \cl(A) \cup \{ e \} \cup \{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}$.(MCL), the element $a$ belongs neither to $\cl(A)$ nor to $\{ e \}$. Therefore it must belong to

$$\{ x \in E \mathrel{:} e \in \cl(A \cup \{ x \}) \setminus \cl(A) \}.$$

Hence $e \in \cl(A \cup \{ a \}) \setminus \cl(A)$, as required.

End of proof□

The interpretation for linear matroids is also clear: the elements lying in the subspace spanned by $A$ and $e$ are precisely the elements already spanned by $A$, the element $e$ itself, and those elements $a$ for which adding $a$ would conversely make $e$ spanned. The theorem says that this single requirement characterises matroid closure operators.

For general closure spaces with exchange properties, see also Faigle [Fai86].

In this note, as is common, “matroid” means a finite matroid, and my own work also concerns finite matroids. I have therefore not discussed infinite matroids, but I end by noting that finiteness is necessary for the single axiom above.

Footnotes

1. I first learnt about this from alg-d's YouTube video “Defining groups by one equation” (in Japanese), where the cited reference was Neumann [Neu81]. For more details, see also alg-d's PDF. Single axioms for groups have been studied rather broadly; see, for example, Kunen [Kun92] and McCune [McC93]. ↩
2. Strictly speaking, Monteiro [Mon45] gave a single axiom that implies (KCL2)for every $A \subseteq X$, one has $A \subseteq c(A)$ (extensibility)(KCL2)–(KCL4)for every $A, B \subseteq X$, one has $c(A \cup B) = c(A) \cup c(B)$ (union preservation)(KCL4), excluding (KCL1)$c(\emptyset) = \emptyset$(KCL1). ↩
3. For the historical background of the name, see Steinitz [Ste10] and Mac Lane [Mac38]. For the matroid closure axioms used here, Oxley [Oxl11, Section 1.4] is sufficient. ↩

References

1. [Neu81] B. H. Neumann. Another single law for groups. Bull. Austral. Math. Soc., vol. 23, no. 1, pp. 81–102, 1981. doi:10.1017/S0004972700006912 Citation contextI first learnt about this from alg-d's YouTube video “Defining groups by one equation” (in Japanese), where the cited reference was Neumann [Neu81]. For more details, see also alg-d's PDF. Single axioms for groups have been studied rather broadly; see, for example, Kunen [Kun92] and McCune [McC93].↩
2. [Kun92] Kenneth Kunen. Single axioms for groups. J. Automat. Reason., vol. 9, no. 3, pp. 291–308, 1992. doi:10.1007/BF00245293 Citation contextI first learnt about this from alg-d's YouTube video “Defining groups by one equation” (in Japanese), where the cited reference was Neumann [Neu81]. For more details, see also alg-d's PDF. Single axioms for groups have been studied rather broadly; see, for example, Kunen [Kun92] and McCune [McC93].↩
3. [McC93] William W. McCune. Single axioms for groups and abelian groups with various operations. J. Automat. Reason., vol. 10, no. 1, pp. 1–13, 1993. doi:10.1007/BF00881862 Citation contextI first learnt about this from alg-d's YouTube video “Defining groups by one equation” (in Japanese), where the cited reference was Neumann [Neu81]. For more details, see also alg-d's PDF. Single axioms for groups have been studied rather broadly; see, for example, Kunen [Kun92] and McCune [McC93].↩
4. [Whi35] Hassler Whitney. On the Abstract Properties of Linear Dependence. Amer. J. Math., vol. 57, no. 3, pp. 509–533, 1935. doi:10.2307/2371182 Citation contextMatroids were introduced by Whitney [Whi35] as objects capturing the abstract properties of linear dependence. Standard references include Welsh [Wel76], White [Whi86], and Oxley [Oxl11].↩1 Citation contextThus $\cl(A)$ is the set of all elements of $E$ whose vectors can be generated as linear combinations of the vectors indexed by $A$. The matroid obtained in this way is called a linear matroid. This viewpoint, abstracting linear dependence, goes back to Whitney [Whi35], where matroids originated. For basic examples, including linear matroids, see Oxley [Oxl11, Chap. 1].↩2
5. [Wel76] D. J. A. Welsh. Matroid theory. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, vol. No. 8, pp. xi+433, 1976 Citation contextMatroids were introduced by Whitney [Whi35] as objects capturing the abstract properties of linear dependence. Standard references include Welsh [Wel76], White [Whi86], and Oxley [Oxl11].↩
6. [Whi86] Neil White. Theory of Matroids. Cambridge University Press, vol. 26, 1986. doi:10.1017/CBO9780511629563 Citation contextMatroids were introduced by Whitney [Whi35] as objects capturing the abstract properties of linear dependence. Standard references include Welsh [Wel76], White [Whi86], and Oxley [Oxl11].↩
7. [Oxl11] James G. Oxley. Matroid Theory. Oxford University Press, vol. 21, 2011 Citation contextMatroids were introduced by Whitney [Whi35] as objects capturing the abstract properties of linear dependence. Standard references include Welsh [Wel76], White [Whi86], and Oxley [Oxl11].↩1 Citation contextThus $\cl(A)$ is the set of all elements of $E$ whose vectors can be generated as linear combinations of the vectors indexed by $A$. The matroid obtained in this way is called a linear matroid. This viewpoint, abstracting linear dependence, goes back to Whitney [Whi35], where matroids originated. For basic examples, including linear matroids, see Oxley [Oxl11, Chap. 1].↩2 Citation contextFor the historical background of the name, see Steinitz [Ste10] and Mac Lane [Mac38]. For the matroid closure axioms used here, Oxley [Oxl11, Section 1.4] is sufficient.↩3
8. [Mon45] António Monteiro. Caractérisation de l'opération de fermeture par un seul axiome. Portugal. Math., vol. 4, pp. 158–160, 1945 Citation contextStrictly speaking, Monteiro [Mon45] gave a single axiom that implies item:K-extensibility–item:union-preserving, excluding item:emptyset-preserving.↩1 Citation contextMatroids are known to admit many equivalent definitions, one of which uses a closure operator. For readers from other areas, the phrase “closure operator” may first bring to mind the closure operation in topology. Topological closure and matroid closure are both examples of operations that “add all points forced by a given set.” However, in a topological space the added points are determined by nearness and limits, whereas in a matroid, and especially in a linear matroid, they are determined by linear span. Thus the two notions can be described using the same abstract vocabulary, but the additional axioms they satisfy are different. Topological spaces can also be characterised by closure operators; these are Kuratowski's axioms. Looking into this, I found that attempts to express the four Kuratowski axioms by one condition go back at least to Monteiro [Mon45] and Pervin [Per64]. This note records the analogous question: can the closure-operator axioms for matroids also be written as a single axiom?↩2
9. [Per64] William J. Pervin. Foundations of General Topology. Academic Press, 1964 Citation contextMatroids are known to admit many equivalent definitions, one of which uses a closure operator. For readers from other areas, the phrase “closure operator” may first bring to mind the closure operation in topology. Topological closure and matroid closure are both examples of operations that “add all points forced by a given set.” However, in a topological space the added points are determined by nearness and limits, whereas in a matroid, and especially in a linear matroid, they are determined by linear span. Thus the two notions can be described using the same abstract vocabulary, but the additional axioms they satisfy are different. Topological spaces can also be characterised by closure operators; these are Kuratowski's axioms. Looking into this, I found that attempts to express the four Kuratowski axioms by one condition go back at least to Monteiro [Mon45] and Pervin [Per64]. This note records the analogous question: can the closure-operator axioms for matroids also be written as a single axiom?↩
10. [Kur22] Casimir Kuratowski. Sur l'opération A de l'Analysis Situs. Fundamenta Mathematicae, vol. 3, no. 1, pp. 182–199, 1922. doi:10.4064/fm-3-1-182-199 https://eudml.org/doc/213290 Citation contextKuratowski's axioms [Kur22] for such a closure operator are as follows:↩
11. [Ste10] Ernst Steinitz. Algebraische Theorie der Körper. J. Reine Angew. Math., vol. 137, pp. 167–309, 1910. doi:10.1515/crll.1910.137.167 Citation contextFor the historical background of the name, see Steinitz [Ste10] and Mac Lane [Mac38]. For the matroid closure axioms used here, Oxley [Oxl11, Section 1.4] is sufficient.↩
12. [Mac38] Saunders Mac Lane. A lattice formulation for transcendence degrees and p-bases. Duke Math. J., vol. 4, no. 3, pp. 455–468, 1938. doi:10.1215/S0012-7094-38-00438-7 Citation contextFor the historical background of the name, see Steinitz [Ste10] and Mac Lane [Mac38]. For the matroid closure axioms used here, Oxley [Oxl11, Section 1.4] is sufficient.↩
13. [Fai86] Ulrich Faigle. Exchange properties of combinatorial closure spaces. Discrete Appl. Math., vol. 15, no. 2-3, pp. 249–260, 1986. doi:10.1016/0166-218X(86)90046-6 Citation contextFor general closure spaces with exchange properties, see also Faigle [Fai86].↩