Combinatorial Derived Matroids

Let M be an arbitrary matroid with circuits C ( M ). We propose a deﬁnition of a derived matroid δM that has as its ground set C ( M ). Unlike previous attempts of such a deﬁnition, our deﬁnition applies to arbitrary matroids


Introduction
Let M be an arbitrary matroid. Gian-Carlo Rota [16] and Henry Crapo [6,11] asked, partly independently and with various precise formulations, for a natural definition of a matroid δM that has as its ground set the collection of (co)circuits of M . Programmatically, this would mean -among other things -a combinatorialization of notions such as syzygies and resolutions in algebraic geometry. For represented matroids, such a construction is known [15,10], but the combinatorics of δM is in general not uniquely defined by that of M , unless M is projectively unique [14], for example binary. The special case of binary matroids M was carefully studied by Longyear [12]. Another related notion is the lattice-theoretic notion of an adjoint geometric lattice to the lattice of flats of M [3], but this does also not exist for all matroids. Moreover, when an adjoint geometric lattice exists, it is not necessarily unique [1]. We propose a purely combinatorial construction of δM , defined via the rank function on M , and via an operation that resembles a closure operation on the collection of dependent sets.
To our knowledge, the first actual definition in the literature of a derived matroid is by Longyear [12], in the case of binary matroids. However, these definitions are predated by predictions of their basic properties, made by Crapo [6], and more specific questions asked by Rota [16]. These predictions and questions, as well as some later work in the field, were formulated in two dual versions, with the ground set of the derived matroid being either the circuits or the cocircuits of the underlying matroid. For consistency, in this paper we reformulate all such results, whenever feasible, to the circuit-based definitions.
We start by introducing the background on matroids and earlier work on derived matroids in Sections 2 and 3, respectively. In Section 4, we introduce our construction and give several cryptomorphic descriptions of it. We then give several examples of combinatorial derived matroids in Section 5. In Section 6 we show that the combinatorial derived matroid is connected if and only if the matroid is connected. In Section 7 we investigate the independent sets and rank of our construction, and in particular show that the rank is bounded from above by |M | − r(M ). Section 8 is devoted to comparing our construction to the earlier definitions discussed in Section 3, and in particular deriving some conditions under which they are equal. Finally, in Section 9 we point out some directions for further research.

Preliminaries on matroids
In this paper, we use several cryptomorphic definitions of matroids. Our construction of the derived matroid will start with a matroid defined in terms of circuits, and construct the derived matroid in terms of the dependent sets of the latter. We refer the reader to [13] for a reference on matroid theory. (D3) if D 1 , D 2 ∈ D and D 1 ∩ D 2 / ∈ D, then (D 1 ∪ D 2 )\{e} ∈ D for all e ∈ D 1 ∩ D 2 .
We call the members of D the dependent sets of M . The subsets of E that are not in D are thus independent. An inclusion-minimal dependent set is called a circuit, and the family C of circuits of a matroid M is characterized by the axioms (C1) ∅ / ∈ C; (C2) if C 1 , C 2 ∈ C and C 1 ⊆ C 2 then C 1 = C 2 ; (C3) if C 1 , C 2 ∈ C are distinct and e ∈ C 1 ∩ C 2 , then there exists C 3 ∈ C such that C 3 ⊆ (C 1 ∪ C 2 )\{e}.
Notation 2. Unless explicitly mentioned otherwise, E denotes a finite set, and its power set is denoted by 2 E . The set of natural numbers N includes zero. If A and B are finite sets and F is a field, then F A×B is the set of matrices with rows indexed by A and columns indexed by B. We write F m×B as a shorthand for F {1,2,...,m}×B , and similarly F A×n as a shorthand for F A×{1,2,...,n} . It follows from property (D2) that rank and nullity are both increasing functions on 2 E , and from (D3) that the rank function is submodular and the nullity function is supermodular, meaning that r(S) + r(T ) r(S ∩ T ) + r(S ∪ T ) and n(S) + n(T ) n(S ∩ T ) + n(S ∪ T ) for all S, T ⊆ E [13]. Let R be a matrix with entries in the field F. The matroid associated to R is a matroid M (R) whose ground set is the set of columns of R and a subset of columns is said to independent if the corresponding submatrix is full-rank. A matroid M is called representable over the field F if there exists some R ∈ F m×E such that M (R) ∼ = M . We sometimes refer to the row space of R as a code representing M .
We will also need the notion of fundamental circuits with respect to a fixed basis B of M . Uniqueness follows as n(B ∪ {e}) = 1, and since B is independent it follows that e ∈ C eB .

Earlier definitions of derived matroid
Several authors have investigated structures similar to the combinatorial derived matroid we are about to define. As explained before, the idea goes back to Crapo and Rota. We will describe four approaches to the question and explain how these are related.

Longyear
Longyear [12] introduced the notion of the derived matroid when M is a binary matroid. The primary motivation in [12] was to display the essential structure of a binary matroid in a computationally more tractable way. To do so, Longyear defined the notion of circuit bases, and correspondingly dependent sets of circuits, in terms of Kirkhoff sums, also known as iterated symmetric difference, and equivalent to binary sums. The Kirkhoff sum of the sets A 1 , . . . A k is the set It follows from the definition that a set A ⊆ C(M ) is dependent in δM precisely if it has a non-empty subset A ⊆ A such that the Kirkhoff sum of the circuits in A is empty. In other words, A ⊆ A should be such that every element in E is be contained in an even number of the sets in A . Longyear showed that under this definition, the derived matroid of a binary matroid is also a binary matroid. However, for a non-binary matroid, the set of circuit bases have no reason to constitute the set of basis for a matroid. Definition 6. Let M be a binary matroid. The matroid whose ground set is C(M ) and whose bases are the circuit bases in M is called the Longyear derived matroid, and is denoted δ L (M ).

The Oxley-Wang definition
A more general definition was introduced in [15] for represented matroids, which are matroids along with a fixed representation. The derived matroid δM (R) of a representation R of the matroid M is determined by the generator matrix whose column vectors are the minimal support vectors in the dual of the given representation of M . By construction, such derived matroids are representable.
Let M = (E, C) be a matroid on the finite ground set E with the set of circuits C. To motivate our further definitions, we first consider the case of a matroid M represented by a code Q ⊆ F n , with dual code Q ⊥ ⊆ F n for some field F. As is commonplace in coding theory, for S ⊆ [n], we denote by Q| S the projection of Q to the coordinates in S, and by Q(S) the shortened code For a circuit C ∈ C(M ), there is a codeword q C in Q ⊥ with supp(q C ) = C, and such a codeword is unique up to scalar multiple. We call q C a circuit vector of Q on C.
Definition 7. Let Q ⊆ F n be a linear code, and let C be the collection of circuits of the associated matroid. Moreover, let A ∈ F n×C be the matrix that has a circuit vector q C in the column indexed by the circuit C ∈ C. We define the derived code δ(Q) to be the row span of A, and the Oxley-Wang derived matroid δ OW (Q) to be the matroid represented by δ(Q).
Clearly, if v ∈ Q ⊥ is not a circuit vector, then v can be written as the sum of two vectors in Q ⊥ whose support is strictly contained in supp v. Thus, Q ⊥ is generated by its minimal support vectors, i.e. the circuit vectors. It immediately follows that the column space of the matrix A in Definition 7 is Q ⊥ , and so the rank of δ OW is dim We can also present a family of explicit bases for δ OW via fundamental circuits. For a fixed basis B and distinct elements e, f ∈ B, we clearly have e ∈ C f B . It follows that the collection δB = {C eB : e ∈ E \ B} is independent in δ OW . Since |δB| = |E \ B| = n − dim Q, we see that δB is a basis for δ OW (Q) for every basis B of M (Q).

Jurrius-Pellikaan
Independently of the work in [12], Jurrius and Pellikaan [10] defined the derived code of a linear code. This work was then not noticed by Oxley and Wang [15]. Both constructions give a derived matroid for a representable matroid, and we will see that the definitions are dual to each other. The main difference is thus that the following definition from [10] uses the cocircuits as the ground set for the derived code. The construction is based on the construction of a derived hyperplane arrangement given in [5, §5.10].
Definition 8 ([10], Definition 4.1). Let G be a k × n matrix of rank k over a field F. Let J ⊆ {1, . . . , n}. Let g j be the j-th column of G. Let G J be the k × |J| submatrix of G consisting of the columns g j with j in J. Let G J (X) be the k×(|J|+1) matrix obtained by adding the column X = (X 1 , . . . , X k ) T to G J , where X 1 , . . . , X k are variables. Let G i,J be the (k −1)×|J| matrix obtained by deleting the i-th row of G J . Let ∆ J (X) = det(G J (X)) and ∆ i,J = det(G i,J ) in case |J| = k − 1. Let D(G) be the derived matrix of G of size k × n k−1 with entries ∆ i,J with i = 1, . . . , k and J ⊆ {1, . . . , n} of size k − 1 ordered lexicographically. The matrix D 2 (G) is obtained by removing zero columns and columns that are a scalar multiple of another column. The derived code D 2 (C) is the code with generator matrix D 2 (G).
It was shown in [10] that the construction of D 2 (C) is independent of the choice of generator matrix G. Geometrically, D 2 (G) is constructed as follows. For every subset of k − 1 columns of G, check if they are independent in F k . If not, add a zero column to D(G). If they are independent, take the coefficients of a linear form determining this (k − 1)-space and make them into a column of D(G). Then remove zeros and multiple subsets that span the same (k − 1)-space.
From this geometric interpretation, one can see that this construction is the same as δ OW of the dual matroid. Every set of k − 1 columns of G (i.e., elements of M ) that has rank k − 1 contributes to a column in D(G). If another set of k − 1 columns spans the same (k − 1)-space, the corresponding column in D(G) will be removed from D 2 (G). Hence all the columns of D 2 (G) correspond to a hyperplane of the matroid represented by G. Since there is a one-to-one correspondence between hyperplanes of M and circuits of M * , this gives the desired duality relation. Note that in the construction of δ OW one gets a representation that is not of full rank, whereas the definition of D 2 (G) does, by definition, give a full rank matrix.

Adjoint matroid
The construction of an adjoint of a matroid is closely related to that of the derived code. Where the derived code always exists, the adjoint does not. It was noted in [10,Remark 4.12] that the construction of the derived code could not be generalised to matroids.
Let L be a geometric lattice. That is, the lattice of flats of a simple matroid. Intuitively, we construct the adjoint of L as follows: take the diagram of L, flip it upside down, and add some elements to make this lattice geometric again. Flipping the lattice upside down gives a lattice with rank function n(A) = r(L) − r(A), the nullity function of the matroid. This function satisfies the first two of the rank axioms for matroids [13], but it is not submodular. So, we need to add new elements for every pair A, B ∈ L such that n(A ∪ B) + n(A ∩ B) > n(A) + n(B). This is the case for every pair that is not modular in L.
We will now give a formal definition of an adjoint [3].
Definition 9. Let L be a geometric lattice. Let L opp be the opposite lattice of L, that is, the lattice on the same elements with the order relation reversed. We say that L ∆ is an adjoint of L if L ∆ is a geometric lattice such that r(L ∆ ) = r(L opp ), there exists an injective embedding e : L opp → L ∆ , and the points of L opp are in bijection with the points of L ∆ . We write ψ : L → L ∆ for the map that fist takes L to L opp and then applies e.

For
The construction of the derived code by Jurrius and Pellikaan gives in fact an adjoint: the map ψ defined in [10,Definition 4.15] satisfies the properties of the map ψ in Definition 9.

Construction of the combinatorial derived matroid
We are now ready to present our definition of the combinatorial derived matroid δM of a matroid M with circuit set C. In fact, we will make two such definitions, one of which defines the circuits and the other of which defines the dependent sets of δM . The definitions are very similar, but a priori different, and we will introduce them both before we prove that they indeed yield the same matroid.
For motivation, we first assume that M is represented by some code Q over a field F. Like in Section 3.2, for each set S of M we define Q ⊥ (S) = {q ∈ Q ⊥ : supp(q) ⊆ S}, and for each circuit C of M we let q C be a vector in Q ⊥ with supp(q C ) = C, noting that q C is unique up to scalar multiple. For any collection A ⊆ C of circuits, we have where span denotes the linear span over F. Hence, if then the dual codewords q C : C ∈ A are linearly dependent regardless of the chosen representation Q, and so we would like them to be dependent in the combinatorial derived matroid. Now by standard duality arguments, where Q| S is the projection of Q to the coordinates in S. Thus dim(Q ⊥ (S)) = |S| − dim(Q| S ) = n(S), so the condition (1), which is sufficient for dependence of {q C : C ∈ A} can be rewritten as n(∪ C∈A C) < |A|.
For an arbitrary, not necessarily representable matroid, we build the definition of dependence of A ⊆ C in the combinatorial derived matroid on condition (2). First we define two operations on sets of circuits that we will later use to construct the dependent sets of the combinatorial derived matroid.
Definition 11. Let C be the set of circuits of some matroid, and let A ⊆ C. Then we define the collection and denote min A ⊆ C the collection of inclusion minimal sets in A.
the electronic journal of combinatorics 30(2) (2023), #P2.8 For any matroid with ground set C, circuit set B and collection of dependent sets A, we must have min A = B and ↑B = A. Observe that, by definition, A ⊆ ↑A and A ⊆ (A) for every A ⊆ 2 C . The operation : 2 C → 2 C is designed to guarantee properties (C3) and (D3) in the matroid axioms.
Definition 12. Let M be a matroid, and C = C(M ) its collection of circuits. Define the collection Inductively, we let A i+1 = ↑ (A i ) for i 1, and Note that the sequences A i and B i are both increasing and contained in the finite set 2 C . Hence, we have that A 0 ⊆ A and A = A n for some n 0, and analogously B 0 ⊆ B and B = B n for some n 0.
Definition 13. Let M be a matroid with circuits C. Then the combinatorial derived matroid δM is a matroid with ground set C(M ) and dependent sets A. Alternatively, we can define δM by its circuits B.
We will prove, in Propositions 17 and 19, that these definitions gives indeed a matroid and that the two definitions are the same. After this, we will define yet another way to construct the circuit set B, in Proposition 20.
Remark 14. Rewriting axiom (D3) in the logically equivalent form: , as in Definition 12, we guarantee to not construct any small dependent sets when not necessary. This appears as a desirable property for the constructed derived matroid to be "generic". Indeed, consequently choosing to include D 1 ∩ D 2 in A for all prescribed dependent sets D 1 , D 2 ∈ A would result in a rather less interesting matroid -often even with rank 0.
Notation 15. When A ⊆ C, we write n(A) for n(∪ C∈A C) in the matroid M and supp(A) for C∈A C, which stands for the support of A.
Proof. This is clear if A ∈ A i , so assume it is not. Moreover, since the inclusion minimal Proposition 17. For any matroid M = (E, C) the collection A is the collection of dependent sets of some matroid with ground set C.
Proof. We will confirm that δM = (C, A) satisfies the axioms for dependent sets of a matroid as given in Definition 1.

If
Next, we will study the circuits of the derived matroid δM that we just defined. The collection of circuits is by definition min A, and we will show in Proposition 19 that this is equal to B, as constructed in Definition 12.
Lemma 18 (Circuits of δM ). Let A ∈ min A have depth i + 1 1. Then there exist sets Note that there is a choice involved for A 1 and A 2 : we pick them such that there is no We now need to show that A 1 , A 2 ∈ min A. For contradiction, assume A 1 is not a circuit of δM . Then it contains a circuit A 3 A by the assumption on A 1 , which again contradicts the assumption that A ∈ min A. The claim follows.
Proposition 19. Let M be a matroid and let B ⊆ 2 C be constructed as in Definition 12. Then B is the collection of circuits of δM .
Proof. We need to show that B = min A, and we will show by induction that min B i = min A i for every i.
By definition, we have min A 0 = B 0 . Now Lemma 18 shows that the sets in min A i+1 \ A i are precisely those that can be obtained from sets in min A i via the operation. Since by definition we have B i+1 = B i , the claim follows by induction.
From our previous results, it follows that we have the next construction of the circuits of the combinatorial derived matroid.
Proposition 20 (Construction of circuits of δM ). The set of circuits of δM can be constructed iteratively as follows: 3. The sequence E i terminates, and its limit equals the collection B of circuits of δM .
Proof. From Proposition 19, it suffices to show that min B i = E i for every i 0. This follows by induction, if we can show that every element of depth i + 1 in min B can be written as , by the assumption on B 1 , which is again a contradiction. The claim follows.
We conclude this section by showing that the combinatorial derived matroid is a simple matroid, so it has no circuits of size 1 and 2, but if M is connected, it does have a circuit of size 3.
Lemma 21. Let M be a matroid. Then δM is simple, that is, there are no dependent sets of size 1 or 2.
Proof. By Lemma 16 it suffices to show that there are no sets of size 1 or 2 in A 0 . If Since C 1 and C 2 are not contained in one another, C 1 ∩ C 2 is independent in M hence has nullity 0. So, by supermodularity of the nullity function, we get that n(A) = n( Proposition 22 (Triangles in δM ). Suppose that M is a connected matroid with at least two circuits. Then every element of δM is contained a triangle, that is, a circuit of size 3.
Proof. Let C and D be two circuits of M with C ∩ D = ∅. Then we have hence A ∈ A 0 . In fact, A is a circuit of δM , since by Lemma 21 there are no dependent sets of size 1 or 2 in δM .
Indeed, the circuits C, C as constructed are the first two circuits in an ear decomposition of M , as in [4]. We only construct them explicitly for completeness.
Remark 23. By Lemma 18, every circuit in δM can be written as Proposition 22 shows that the converse of this statement does not hold: if N is a matroid such that every circuit can be written as C = (C 1 ∪ C 2 )\{e}, there does not necessarily exists some M such that δM = N . A counterexample is given by letting N be a uniform matroid of rank at least three: we can clearly write every circuit as C = (C 1 ∪ C 2 )\{e}, but since N does not contain a triangle, it cannot be a combinatorial derived matroid.

Examples
In this section we will give several examples of combinatorial derived matroids and compare them to earlier definitions.
Example 24. Consider the matroid Q 6 from [13, Example 7.2.4] whose geometric representation is given in Figure 1.
Let us construct A 0 . Direct computation shows that | supp(A)| 5 for all A ∈ A 0 . Hence, applying Lemma 16 we know that | supp(A)| 5 for all A ∈ A.
We will show how to construct the circuits of δM with support of cardinality 5 by fixing the excluded element of the support. First, fix the element 1.

Uniform matroids
Example 25. Let M = U (3, 6) be the rank 3 uniform matroid on the ground set [6]. This is a representable matroid whose representations are [6,3] MDS codes. We give two representations Q 1 and Q 2 of M over F 7 below using generator matrices for the dual linear codes: Since this matroid is representable, we can apply the definition of Oxley-Wang derived matroids from [15] for each of its representations. The generator matrices of the represented derived matroids corresponding to G Q ⊥ 1 and G Q ⊥ 2 are: .
Direct computation in Macaulay2 shows that both represented derived matroids have 751 circuits, of which 712 belong to the intersection of the two matroids. Let δC i denote the set of circuits in the corresponding represented derived matroid. We want to study δC 1 δC 2 . It follows that |δC i − δC j | = 39. Further computation shows that in both cases |δC i − δC j | contains 3 sets of cardinality 3 and 36 sets of cardinality 4. The sets of cardinality 3 are independent in the other represented derived matroid, whereas the sets of cardinality 4 are dependent sets but not minimal in the other matroid. Specifically, the sets that are circuits in δM (Q 1 ) but are independent in δM (Q 2 ) are given by the columns This shows that there is a bijection of sets between the two matroids, but they are not isomorphic, because in δM (Q 2 ), the sets {1, 2, 4}, {1, 3, 5} have a non-empty intersection but all circuits of cardinality three in δC 1 δC 2 are disjoint. So different representations of the same matroid can give non-isomorphic represented derived matroids in the sense of [15]. Let us now compare these two represented derived matroids with the combinatorial derived matroid from Definition 13. Let A ∈ A 0 . For future record we note that the represented derived matroids δM (Q 1 ) and δM (Q 2 ) have 32256 dependent sets. If |A| 2, then |A| n(A) which contradicts (3), so |A| 3. Since M = U (3, 6), 2 n(A) 3. Therefore, any set A ⊂ C with |A| 4 will be an element of A 0 . There are dependent sets A ∈ A 0 of this type. Therefore, there are 32252 elements in A 0 . The set A 0 is already closed under the operations and ↑ as we will show in Corollary 27, so A = A 0 . We conclude that both δ(Q 1 ) and δ(Q 2 ) are not the combinatorial derived matroid, since they don't have the same number of dependent sets. Next, we will find a representation of M such that the derived and combinatorial derived matroid are the same. By Proposition 26, we get that δU (3, 6) = (C, A 0 ), so Corollary 41 applies to U (3, 6). Thus, it suffices to find a representation Q such that δ OW (Q) has precisely 32252 dependent sets, which is efficiently done by randomly constructed matrices over large enough fields.
The combinatorial derived matroid is representable over a sufficiently large field extension of F 7 . For example, the following is a generator matrix of U (3, 6) over F 49 , where a is a primitive element over F 7 : Direct computation confirms that its represented derived matroid has 32252 dependent sets, and thus indeed equals the combinatorial derived matroid. Furthermore, the combinatorial derived matroid of U (3, 6) is representable over any field of characteristic 0. For example, the following integral matrix has the reprentable derived matroid equal to the combinatorial derived matroid: Both (4) and (5) were generated as random matrices in the corresponding fields, which coincides with the intuitive interpretation of the combinatorial derived matroid being the most common matroid.
Proposition 26. Let M = U (k, n) be a uniform matroid. Then the combinatorial derived matroid δM is generated by the set Moreover, for all A ∈ A 0 it holds that k + 2 | supp(A)| n and |A| 3.
Proof. Since the collection of circuits of U (k, n) is exactly the collection of sets of cardinality k + 1, we have n(A) = | supp(A)| − k for all A ⊆ C and thus A 0 is of the described form. If |A| = 1 then | supp(A)| = k + 1 so A / ∈ A 0 . If |A| = 2 then | supp(A)| k + 2 so also A / ∈ A 0 . Therefore |A| 3 for all A ∈ A 0 and this implies k +2 | supp(A)| n.
In other words, A 0 can be constructed by taking any subset S ⊆ [n] such that |S| k + 2. Any set A ⊆ C such that supp(A) = S and |A| > |S| − k, will be an element of A 0 . In particular, any set A ⊂ C with |A| n − k + 1 will be an element of A 0 . The lower bound |A| 3 from Proposition 26 is sharp because δU (k, n) contains a triangle, as shown in Proposition 22.
We randomly generated matrices R ∈ Z k×n . Such R should be sufficiently generic, so with probability 1, M (R) = U (k, n). Furthermore, we conjecture that δ OW M (R) = δU (k, n). Then in Table 1 we present the empirical distribution of sizes of circuits of δ OW M (R), which we expect to be indicative of the distribution of the sizes of circuits of δU (k, n).  Proof. We will show that A 0 as defined in Proposition 26 is invariant under the operations from Definition 11. The case k n − 2 is immediate, so assume k = n − 3.
and hence supp((A 1 ∪A 2 )\{C}) = n−1. This shows that derived dependent sets of cardinality 3 supported on n elements of the ground set of U (n − 3, n) cannot be constructed via the operation . So (↑A 0 ) = A 0 . We conclude that A = A 0 .

Graphical matroids
We will next give an example of the derived matroid of the graphical matroid M (K 4 ). We believe that this may be an illuminating example for future studies of graphical matroids in general, and perhaps even more generally of binary matroids. A key feature of the example is that δM (K 4 ) turns out to have strictly fewer dependent sets than the Longyear derived matroid of M (K 4 ). This foreshadows some of our conjectures in Section 8.  Note that these circuits correspond to the cycles abc, abd, acd, bcd, abcd, abdc, acbd in the graph, respectively. Now n(M ) = 3 and n(S) 2 for all S E. Hence, we have that corresponding to the six ways to write a four-cycle in K 4 as the symmetric difference of two triangles. But these six three-element lines form the non-trivial circuits of a matroid on ground set C, namely the non-Fano matroid F − 7 [13, Example 1.5.13], depicted in Figure 3. Thus we have A = A 0 = A 0 , so δ(M (K 4 )) is itself the non-Fano matroid.

Non-representable matroids
In this section we consider the Vámos matroid. It is not representable, so its derived matroid in the sense of Oxley-Wang does not exist. Also, Cheung [3] proved that the Vámos matroid does not have an adjoint. Its combinatorial derived matroid, however, does exist. In our analysis we will particularly focus on studying the sets Example 29 (Vámos matroid). Let M be the Vámos matroid. We summarise our finding here and the detailed computations are located in Appendix A. Vámos matroid is a nonrepresentable and non-algebraic matroid of rank 4 on the ground set [8]. All sets of three or fewer elements are independent and among the 70 sets of four elements only the 5 depicted as gray rectangles in Figure 4  Let us consider ∆A 0 . It does not contain any 3-sets. This is because if |A| = 3 and For any distinct C, C in C, |C ∪ C | 6. On the other hand, that is, A is a 3-set supported on 6 elements of E, so A ∈ A 0 . Let It is shown in Appendix A that ∆A 0 = ∆A 03 , so the only new elements generated with (A 0 ) come from combinations of 3-sets. This is a consequence of (3) and does not propagate to subsequent iterations. For example, there exist new 4-sets in ∆A 1 that are generated by A 1 ∈ A 1 and A 2 ∈ ∆A 0 with 3 |A 1 | 4 and |A 2 | = 4. The large number of 4-sets makes it computationally infeasible to directly check whether all of them are dependent.
We will return to the Vámos matroid in Example 36, where we will find circuits in δ(M ) that fail to satisfy some conditions that hold for circuits in Oxley-Wang derived matroids.

Connectedness
As with many other notions about matroids, taking the combinatorial derived matroid commutes with taking the direct sum. That is: the combinatorial derived matroid of a direct sum of two matroids is the direct sum of the combinatorial derived matroid of these matroids. The proofs of the two results below are inspired by the proofs of Proposition 17 and Theorem 18 of [15].
Since A ∈ A 0 , we have that |A| > n(A) with n the nullity function of M . This implies Since nullity is bounded by cardinality, |A 1 | + |A 2 | > n(A 1 ) + n(A 2 ) implies that |A i | > n(A i ) for at least one of i = 1, 2. This means that A i ∈ A 0 (M ). By minimality of A, we get that A = A i . Let n i be the nullity function of M i , then n(A i ) = n i (A i ), so |A i | > n i (A i ) and thus A i ∈ A 0 (M i ). In fact A i has to be in E 0 (M i ), because otherwise it would not be minimal in A 0 (M ). This proves that E 0 (M ) = E 0 (M 1 ) ∪ E 0 (M 2 ).
We now construct the circuits of δM as in Proposition 20. The operation only constructs new sets from pairs of elements of E j (M ) that have nonempty intersection, and so for all j 0, This means that inductively we get In the previous proposition we have seen that if M is not connected, then δM is not connected. We will next prove the opposite: if M is connected, then δM is also connected. In order to do so, we first need a partial result about connectivity among the fundamental circuits of M with respect to a fixed basis B.
Proposition 31. Let B be a basis of a connected matroid M , and let e, f ∈ E − B. Then the circuits C eB and C f B are in the same connected component of δM .
Proof. Assume first that C eB ∩ C f B = ∅. Then C eB ∪ C f B ⊆ B ∪ {e, f }, so n(C eB ∪ C f B ) n(B ∪ {e, f }) = 2, and as C eB ∪ C f B contains two circuits, its nullity is indeed exactly two. By the circuit exchange axiom (C3), we have a third circuit C ⊆ C eB ∪ C f B , and so we have the dependent triangle Since δM is simple, this triangle is indeed a circuit in δM . Now consider the bipartite graph G B (M ) with vertex classes (E \ B) B and edge set If there is a path of length 2 from e to f in this graph, then C eB ∩ C f B = ∅, so C eB and If M has only one basis, we are done. So, let B 1 and B 2 be bases of M . We will use induction on the size of B 1 ∆B 2 , which is always an even number as all bases have the same cardinality.
Assume now that whenever |B 1 ∆B 2 | k, then all fundamental circuits with respect to B 1 or B 2 are elements of the same connected component of δM . Let |B 1 ∆B 2 | = k + 2 and take a ∈ B 1 \B 2 . By the basis exchange axiom, there exists b ∈ B 2 \B 1 , such that B = (B 1 \{a}) ∪ {b} is a basis of M . Let C b be the fundamental circuit of b with respect to B 1 and C a be the fundamental circuit of a with respect to B . Then C a ⊆ B ∪ {a} = B 1 ∪ {b}. The uniqueness of fundamental circuits implies that C a = C b . This implies that all fundamental circuits with respect to B are elements of the same connected component of δM as the fundamental circuits with respect to B 1 . But now B ∆B 2 ⊆ (B 1 ∆B 2 )\{a, b}, so |B ∆B 2 | k and we can apply the induction assumption and conclude that the fundamental circuits with respect to B 2 are elements of the same connected component as those with respect to B , and hence also of same connected component as the fundamental circuits of B 1 .
Applying the procedure above to all bases of M gives that all fundamental circuits of M are elements of the same connected component of δM . Since all circuits are fundamental circuits with respect to some basis, it follows that δM has only one connected component and is thus connected.

Independent sets and rank
In this section we derive some further structural results about the combinatorial derived matroid. In cases where we could not find proofs or counterexamples, we make conjectures.
The following result was proven in [7] for the Oxley-Wang derived matroid. It shows how to find independent sets in δ OW , independent of the representation of M .
Lemma 33 (Lemma 5.4 of [7]). Let M be an F-representable matroid on the ground set [n]. Take any set of circuits S = {C 1 , . . . , C m } such that every circuit C i satisfies Then S is independent in the derived matroid δ OW of every F-representation of M .
One can aim to prove a similar result for the combinatorial derived matroid, also for matroids that are not representable. A first step is the following lemma.
Lemma 34. Let M be a matroid with C its set of circuits. Let S ⊆ C with S / ∈ A 0 and let C ∈ C such that C − supp(S) = ∅. Then S ∪ {C} / ∈ A 0 .
Proof. Since C ∩ supp(S) C, the set C ∩ supp(S) is independent in M and hence n(C ∩ supp(S)) = 0. Because S / ∈ A 0 , we have |S| n(S). Combining this with the supermodularity of nullity, we get that A next step could be to prove a similar result for A i instead of A 0 . Then, because A = A i , we would have the following.
Conjecture 35. Let M be a matroid with C its set of circuits. Let S ⊆ C with S / ∈ A and let C ∈ C such that C − supp(S) = ∅. Then S ∪ {C} / ∈ A.
Note that from the above conjecture Lemma 33 for the combinatorial derived matroid follows. Unfortunately, Conjecture 35 is not true, as the next example, due to Santiago Guzmán-Pro, shows [8].
Let C = adgh and consider S\{C}: it needs to be either independent or dependent in δM . Suppose it is independent, so S\{C} / ∈ A. We have that C − supp(S\{C}) = a. Applying Conjecture 35 now gives that S / ∈ A, which is a contradiction. Now suppose S\{C} ∈ A and let C = def gh. We have that C − supp(S\{C, C }) = d. Applying Conjecture 35 gives that S\{C, C } ∈ A, which is a contradiction since by Lemma 21, S\{C, C } is independent because it has size 2. We conclude that Conjecture 35 cannot be true.
Although this example does not disprove Lemma 33 for the combinatorial derived matroid, it shows that new ideas are needed in order to prove or disprove the statement. Unfortunately, we were not able to prove a more precise statement about the independent sets of δM . We do have the following straightforward observation about the rank of δM .
Lemma 37. Let M be a non-empty matroid of rank k. Then the rank of the combinatorial derived matroid δM is at most n − k.
Proof. The nullity of any subset of C is at most the nullity of M , which is n − k. This means that any subset of cardinality n − k + 1 of C is in A 0 and is thus dependent in δM . Now the rank of δM is the cardinality of its largest independent set, which we just saw is at most n − k.
It is unknown if the rank of the combinatorial derived matroid is always equal to n−k. We give a conjecture and a question towards a solution to this problem.
Conjecture 38. Let M be a matroid such that M * has an adjoint. Then δM is isomorphic to an adjoint of M * . In particular, the rank of δM is equal to n − k.
It is shown in [15,Lemma 3] that if B is a basis of a representable matroid M and R any representation of M , then the set of fundamental circuits with respect to B is a basis of δ OW (R). This motivates the following question.
Question 39. Does there always exist a basis B of M such that a subset of the fundamental circuits with respect to B is a basis in δM ?

Comparison of definitions
In this section, we present some partial results relating our construction to previous notions of derived matroids from the literature. In Section 8.1, we relate δM to the Oxley-Wang derived matroid δ OW (R), where R is a representation of M , and as a special case of this to the Longyear derived matroid δ L (M ) when M is binary. In Section 8.2, we compare the lattice of flats F(δM ) to adjoint lattices of the lattice of cyclic sets U(M ) ∼ = F(M * ). All results in this section are admittedly rather preliminary, and mainly serve as inspiration for further research.

Oxley and Wang
The relation between the combinatorial derived matroid and the Oxley-Wang derived matroid is not entirely straightforward. In particular, Example 5.5 shows that when M = M (K 4 ) and F has characteristic two, there are representations of M over F, but none of these representations yields δ OW (R) ∼ = δM , even if we replace F by an extension field. However, we hope that in the precise sense of Conjecture 43, the combinatorial derived matroid has less dependencies than any Oxley-Wang derived matroid. Our first results in this direction are the following. Proof. Let A ⊆ C. For a representation Q of M and C ∈ A, denote by q C the circuit vector supported on C as in Section 3.2. If A ∈ A 0 , then we have |A| > n(supp A) = dim(Q ⊥ (supp S)) dim span{q C : C ∈ A}.
Thus the vectors {q C : C ∈ A} are linearly dependent, so A is dependent in δ OW (Q).
This shows that for matroids such that δM = (C, A 0 ), the combinatorial derived matroid is generic, in a precise sense. In particular, this applies to uniform matroids U (k, n) with k n − 3, by Corollary 27. We also expect this to apply to many other naturally occurring matroids of high rank. The simple Lemma 40 also shows that if δM = (C, A 0 ) we can characterize representations for which the Oxley-Wang derived matroid equals the combinatorial derived matroid from mere enumerative properties of δ OW (Q).
Corollary 41. If δM = (C, A 0 ) and Q is a representation of M such that δ OW (Q) has the same number of dependent sets as δM , then δ OW (Q) = δM .
Proof. By Lemma 40 we have D(δM ) = A 0 ⊆ D(δ OW (Q)), so since the two sets are finite, they must be equal if they have the same cardinality. Since δM and δ OW (Q) also have the same ground set C, they must thus be equal.
In general, if M is representable, we conjecture that the combinatorial derived matroid is generic in two different ways; one categorical and one algebraic-geometric. In order to state Conjectures 43 and 45, we need some auxiliary standard definitions.
Definition 42. Let E be a finite set. The weak order on the class M of matroids with ground set E is the partial order with M 1 M 2 if every dependent set in M 1 is also dependent in M 2 .
Conjecture 43. Let M be a representable matroid with set of circuits C. Then δM δ OW (R) for every representation R of M , where the order relation is the weak order on the set of matroids on C.
Note that Lemma 40 proves Conjecture 43 in the case that δM = (C, A 0 ). It is conceivable that Conjecture 43 may be approached with techniques similar to those in [9]. There, the authors studied conditions under which the collection of matroids with some circuits prescribed have a maximal element in the weak order.
The next example shows that it is possible that M is representable over a field F, without any of the representations of M over F (or indeed over any extension field of F) yielding δM as an Oxley-Wang derived matroid.
Example 44. Consider the graphical matroid M = M (K 4 ) as in Example 28. Recall that δM is the non-Fano matroid F − 7 , with dependent sets A = A 0 . We will now look at Oxley-Wang derived matroids of M (K 4 ). By Lemma 40, all sets in A = A 0 are dependent in δ OW (R) for any representation over any field. Moreover, a triple of circuit vectors can only be linearly dependent if every element in E occurs in at least two of the circuits, which follows from Lemma 34. The only possible dependent set in δ OW (R), except for the sets in A, is therefore the triple of 4-circuits {1346, 1256, 2345}. It is straightforward to see that these circuits are linearly dependent in the (projectively unique) representation of M over a field F or characteristic 2, but not in a representation over fields of characteristic = 2. This can intuitively be understood if we interpret a linear combination of the circuit vectors supported on the circuits {1346, 1256, 2345} as a formal sum of the cycles {abcd, abdc, acbd} in the graph. In order for such a formal sum to take the value zero on the edge 6 = cd, the coefficient in front of the (directed) cycles abcd and abdc must be the same, say α. But then the formal sum of cycles traverses the edge ab 2α times in the same direction, and so it can only be zero if 2α = 0, i.e., if char(F) = 2.
Thus, we see that δ OW (R) is the Fano matroid in characteristic 2, and the non-Fano matroid in characteristic = 2. In particular, we get δ(M ) = δ OW (R) for a representation R of M if and only if the representation is in characteristic = 2.
For algebraic-geometric notions of genericity, fix a field F and let F be its algebraic closure. Assume that M is representable over F, with |E(M )| = n and r(M ) = k. We can then define the variety V (M ) = V F (M ) ⊆ F k×n of matrices that represent M . This is clearly an algebraic variety, since it is defined by the vanishing of certain finitely many minors and the non-vanishing of finitely many others. We also have a variety V (N ) ⊆ Note that it is possible that V N (M ) is empty over certain ground fields F. Example 5.5 shows that this is true, in particular, when M = M (K 4 ) and F is the two element field.
Since there are only finitely many possible Oxley-Wang derived matroids, this would imply that V δM (M ) would be a full-dimensional subvariety of V (M ). In particular, assume that F is a finite field and that F /F is a finite field extension whose degree [F : F] tends to infinity. Conjecture 45 would then imply that asymptotically almost all representations R of M over F would satisfy δ OW (R) = δM .

Adjoint
The construction of the combinatorial derived matroid is iterative. If a set of circuits of M satisfies the nullity condition (3) in Definition 13, the set is in A 0 hence dependent. If not, the only way to check its (in)dependence is by constructing iteratively all dependent sets of δM and see if our set is one of them.
The fact that there are sets for which (in)dependence is hard to check, means it is also difficult to define a closure function of δM . We are not optimistic about finding a definition of closure in δM without some form of iteration.
As a result of the lack of closure in δM , we have not managed to compare the definition of δM , that uses dependent sets, to that of the adjoint (Section 3.4), that uses closed sets. We expect the following to be true.
Conjecture 46. If an adjoint of a matroid M exists, one of them is equal to the combinatorial derived matroid of the dual. Moreover, every adjoint of M is less δM * in the weak order on matroids with ground set C.

Suggestions for further research
We have developed the notion of the combinatorial derived matroid. Contrary to earlier constructions for a derived matroid, our construction exists for all matroids and is unique.
There are many open questions about our construction. Apart from the questions and conjectures already stated in the text, we list here some more ideas for further research.
Question 47. Is there a characterization of the class of matroids N that are isomorphic to the combinatorial derived matroid δM of some matroid M ? By Proposition 4.12, a necessary condition is that every element in N that is contained in a circuit is also contained in a circuit of size 3. Many more necessary conditions like this should probably first be obtained, before a complete characterization is feasible.
Question 48. For which matroids does it hold that δM = M or δM = M * ? The latter property may be more natural, since the rank of δM is bounded from above by the rank of M * . It is also possible to repeat the procedure of taking the combinatorial derived matroid of the dual multiple times, i.e. study the sequence M, δ(M * ), δ(δ(M * ) * ), . . . .
Does this sequence converge? For earlier constructions of the derived matroid, these questions have been studied in [15,Section 3] and [10, Section 4], see also [11].
Question 49. Is there any relation between δM and δM * ?
Question 50. There are matroids that do not have an adjoint, whereas the combinatorial derived matroid always exists. This means one of the properties in Definition 9 and/or Lemma 10 is not in general satisfied by the combinatorial derived matroid. Which one?
Question 51. If X is a representation of M by topological spheres, as in [17,2], does X induce a representation δX of δM by topological spheres?

A Vamos matroid
Let M be the Vámos matroid. It is a non-representable and non-algebraic matroid of rank 4 on the ground set [8]. All sets of three or fewer elements are independent and among the 70 sets of four elements only the 5 depicted as gray rectangles in Figure 4 are circuits.
The ground set of δM . First we calculate the number of elements of the ground set of δM , that is, the number of circuits in M . There are 5 circuits of size 4 in M , and of the 8 5 = 56 sets of size 5 there are 5 · 4 = 20 that contain a circuit of size 4. This gives (56 − 20) + 5 = 41 circuits in M , hence δM is a matroid on 41 elements.
The elements of A 0 . To construct A 0 , first note that all subsets of C of size 0, 1 and 2 are not contained in A 0 by Lemma 21. Moreover, all sets of size at least 5 are in A 0 , since n(M ) = 4. For a 3-set A ⊆ C to be contained in A 0 , we need that n(A) 2, so | supp A| = 6. There are 290 such sets in A 0 .
A 4-set A is in A 0 if and only if its support is of size at most 7. Motivated by Proposition 20, we would like to understand the minimal elements of cardinality 4 in A 0 . If | supp(A)| = 6, any 3-set subset A of A satisfies | supp(A )| = 6 because any 3-set of circuits has support at least 6. But then A is a 3-set on support of size 6 and rank 4, so A is in A 0 . Hence, to include the minimal 4-sets of A 0 , we can require that | supp(A)| = 7, so n(A) = 3 and |A| − n(A) = 1. There are 14656 minimal 4-sets in A 0 .
The elements of ∆A 0 . All elements of ∆A 0 have cardinality 4. This is because all sets of cardinality at least 5 are included in A 0 . On the other hand, the set ∆A 0 does not contain any 3-sets. This is because if |A| = 3 and A ∈ A 1 − A 0 , then A = (A 1 ∪ A 2 )\{C} with C ∈ A 1 ∩ A 2 , |A 1 ∩ A 2 | = 2, A 1 , A 2 in A 0 and |A 1 | = |A 2 | = 3. For any distinct C, C in C, |C ∪ C | 6. On the other hand, | supp A 1 | = | supp A 2 | = 6, so supp A 1 = supp(A 1 ∩ A 2 ) = supp A 2 = supp(A 1 ∪ A 2 ). Hence, 6 | supp A| | supp A 1 ∪ A 2 | = 6, that is, A is a 3-set supported on 6 elements of E, so A ∈ A 0 .
Next we analyse the 4-sets inside ∆A 0 . First, consider