The cyclic flats of a q-matroid

In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_{q^m}$$\end{document}Fqm-independence of an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q$$\end{document}Fq-subspace of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {F}_q^n$$\end{document}Fqn and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.


Introduction
The concept of q-matroid may be traced back to Crapo's PhD thesis [9].More recently, the relation between rank-metric codes and q-matroids has led to these combinatorial objects getting a lot of attention from researchers; see for instance [5-8, 13, 14, 17, 22].Indeed, it is well-known that q-matroids generalize F q m -linear rank-metric codes, just as classical matroids generalize linear codes in the Hamming metric.
As in traditional matroid theory, there are many equivalent ways to describe a q-matroid axiomatically, which are called q-cryptomorphisms.A full exposition of these is given in [7], in terms of rank function, independent spaces, flats, circuits, bases, spanning spaces, the closure function, hyperplanes, open spaces etc.
In matroid theory, one of the most crucial objects is the lattice of flats F(M ) of a matroid M , since it uniquely determines the matroid.Another lattice with the same property is the lattice of cyclic sets.This led many researchers to investigate the intersection between these lattices, namely the collection of cyclic flats of the matroid.Cyclic flats have also played several important roles such as in the work of Brylawski, who showed in [4] that the cyclic flats of a matroid, together with their ranks uniquely determine the matroid.Moreover, Eberhardt showed that they provide the Tutte polynomial in [11], and Bonin and de Mier showed in [2] that every lattice is isomorphic to the lattice of cyclic flats of a matroid.Applications to coding theory have been recently investigated: it has been proved that many central invariants in coding theory can be naturally described in terms of the lattice of cyclic flats of the associated matroid; see [12,24].Furthermore, the lattice of cyclic flats of classical polymatroids has also been studied; see [10,25,26].
In this paper we consider a q-analogue of this theory: we define the lattice of cyclic flats of a q-matroid and show that it, along with the ranks of its elements, determines the q-matroid.We furthermore propose a new cryptomorphism of q-matroids based on cyclic flat axioms.We consider the codes associated with representable q-matroids and show that the cyclic spaces of a q-matroid are supports of elements of the corresponding dual code.

Background
In this section, we recall some preliminary notions on q-matroids and rank-metric codes.The following definition of q-matroid is given in terms of a rank function; see [17].Notice that this definition does not require E to be a vector space over a finite field, however, we will assume that a q-matroid is an object defined with respect to an F q -vector space.
Definition 1.1.A q-matroid M is a pair (E, r) where r is an integer-valued function defined on L(E) with the following properties: (R1) Boundedness: 0 ≤ r(A) ≤ dim A, for all A ∈ L(E).
The function r is called the rank function of the q-matroid.
Given a q-matroid (E, r), we define the nullity function ν to be ν : L(E) → Z, A → dim(A) − r(A).
From Definition 1.1, it follows that the nullity has the following properties: (n1) ν(A) ≥ 0, for all A ∈ L(E).

If
x ≤ E and r(x) = 0, then x is called loop of M .A subspace that is not an independent space of (E, r) is called a dependent space of (E, r).We call C ∈ L(E) a circuit if it is itself a dependent space and every proper subspace of C is independent.We write D r and C r to denote the sets of dependent spaces and the circuits of the q-matroid (E, r), respectively.A subspace is called an open space of (E, r) if it is a (vector space) sum of circuits.We write O r to denote the set of open spaces of (E, r).
The closure function of a q-matroid (E, r) is the function defined by x.
Definition 1.4.A subspace A of a q-matroid (E, r) is called a flat or closed space if for all x ∈ L(E) such that x A, we have We write F r to denote the set of flats of the q-matroid (E, r), that is If it is clear from the context, we will simply write I, D, C, O, cl in place of I r , D r , C r , O r , cl r .In [7], several cryptomorphisms of q-matroids have been established, which give equivalent ways of defining a q-matroid.Here, we recall the closure function axioms.The closure function of a q-matroid (E, r) necessarily satisfies the closure function axioms; see [17,Theorem 68].Definition 1.5.Let cl : L(E) → L(E) be a map.We define the following closure axioms.
Similar to the matroid case for any A ∈ L(E), we have that r(A) = r(cl(A)).Finally, we define the restriction and the contraction operations for q-matroids; see [6,17].Definition 1.6.Let M := (E, r) be a q-matroid and A ≤ E be any subspace of E. For every space T ≤ A, we define r where π : E → E/A is the canonical projection.Then the q-matroid M/A := (E/A, r M/A ) is called the contraction of M from A.
We conclude with the notion of dual matroid [17], which we will use in Sections 2 and 4. Definition 1.7.Let M = (E, r) be a q-matroid and consider the function Then r * is a rank function and M * = (E, r * ) is a q-matroid, called the dual q-matroid of M .

Cyclic Spaces and Cyclic Flats
This section is devoted to the introduction of the q-analogue of cyclic flats and to present some of the properties of these objects.This will be the starting point for establishing a description of q-matroids in terms of cyclic flats.For the remainder, M = (E, r) will denote an arbitrary but fixed q-matroid with ground space E and rank function r.

Cyclic Spaces
We first define what it means for a space to be a cyclic subspace of a q-matroid.If it is clear from the context, we will write Cyc := Cyc r for the q-matroid M = (E, r).
Further, this will occur if both x and y are subspaces of D. Suppose that z D. Then we may assume that x D and so r(z and so Cyc(A) = P x∈Cyc(A) x .To see that the second equality holds, note that for any x ≤ L(E), we have x ∈ Cyc(A) if and only if x is contained A and in the closure of every member of Hyp(A), which holds if and only if x ≤ Proposition 2.5.Let A ∈ L(E).The following are equivalent.
(2) For all x ≤ A, we have that r(x + B) = r(B) for all B ∈ Hyp(A).
and let X ∈ Hyp(C).We claim that r(C) = r(X).Clearly, either X ∩C 1 = C 1 , or X ∩C 1 has codimension 1 in C 1 .Suppose the latter case, so there exists y ≤ C 1 such that y + X = C. Therefore, since C 1 is cyclic, we have that r(y + (X ∩ C 1 )) = r(X ∩ C 1 ) and hence by Lemma 2.6 we have r(X) = r(y + X) = r(C).On the other hand, if , since otherwise we arrive at the contradiction: As before, X ∩ C 2 ∈ Hyp(C 2 ) implies that r(C) = r(X).It follows that C 1 + C 2 is a cyclic space.Now apply the same argument, iteratively, to arrive at the required result.Now consider the following operator.If it is clear from the context, we will write cyc := cyc r for the q-matroid M .We say that From Lemma 2.7, we see that cyc(A) is cyclic and is the unique maximal cyclic subspace of A.
One well-known construction of a q-matroid arises from the generator matrix of an F q mlinear code; see [14,17].Let G be a k × n matrix over F q m and for every U ∈ L(F n q ), let A U be a matrix whose columns form a basis of U .Then the map is the rank function of a q-matroid, which we denote by M [G].
Let G be the following matrix with entries in F 8 , Let r be the rank function of M [G] and let A 1 := e 2 , e 3 ∈ L(F 4 2 ).Then r(A 1 ) = 1 and for all B ∈ Hyp(A 1 ), we have that r(B) = 1 and hence A 1 is cyclically closed.On the other hand, A 2 := e 1 + e 2 , e 3 , e 4 is not cyclically closed, indeed cyc(A 2 ) = e 1 + e 2 + e 4 , e 3 .
We now list some basic properties of the cyclic operator.
Theorem 2.10.For every A, B ∈ L(E), the cyclic operator satisfies the following properties.
Proof.It is immediate by Definition 2.8 that (cyc1) holds.Let A, B ∈ L(E).From (cyc1), cyc(A) ≤ A ≤ B and hence cyc(A) is a cyclic subspace of B, which must therefore be contained in cyc(B), by definition of the cyclic closure.From (cyc1) and (cyc2), we have that cyc(cyc(A)) ≤ cyc(A).Since cyc(A) is itself cyclic and contained in cyc(A), we have Therefore, (cyc3) holds.
Remark 2.12.Every nonzero cyclic space is dependent.Indeed, if A = 0 is cyclic and independent, then all its subspaces have to be independent, so, clearly, A cannot have the same rank as its hyperplanes.Lemma 2.7 shows that every open space (every sum of circuits) is cyclic.We will shortly see a converse.First, recall the following result.Lemma 2.13.[7, Proposition 86] The flats of the q-matroid M = (E, r) are the orthogonal spaces of the open spaces of the dual q-matroid M * .That is, F is a flat of M if and only if In Proposition 2.14 we characterize the cyclic spaces of M .In order to do this, for every integer i consider the following set i.e., the set of subspaces of E with nullity equal to i.
The equivalence of ( 1) and (3) of the following proposition was shown in [15, Lemma 13].Note that in [15,Definition 11], a cyclic space is defined as being minimal with respect to inclusion in N a for some a.
(1) A is cyclic in M .
(2) A is minimal with respect to inclusion in N a .
(3) A ⊥ is a flat of the dual q-matroid M * .
(4) A is the vector space sum of the circuits contained in A.
∈ N a and so r(D) = r(A).Since this holds for arbitrary D we obtain the equivalence of ( 1) and ( 2).
(1) ⇔ (3): A ⊥ is a flat in M * if and only if for all x A ⊥ we have r * (A ⊥ + x) = r * (A ⊥ ) + 1.For any x A ⊥ , we have In particular, since every hyperplane in A has the form A ∩ x ⊥ for x A ⊥ , it follows that A ⊥ is a flat of M * if and only if A is a cyclic space of M .
Clearly, the properties of being a cyclic, a cyclically closed, or an open space of M all coincide.
The following corollary immediately follows from Proposition 2.14.
Corollary 2.15.E is cyclic in M if and only if the dual q-matroid M * does not contain a loop.In particular, The following result which will be crucial for establishing a q-cryptomorphism based on cyclic flats.Lemma 2.17.
Proof.Let ν(A) = a, thus A ∈ N a .We claim that cyc(A) ∈ N a .If A is not cyclic, then there exists a subspace Y A, Y ∈ N a such that Y is cyclic, by Proposition 2.14.From Theorem 2.10, it follows that Y = cyc(Y ) ≤ cyc(A) and so a ≤ ν(cyc(A)) ≤ a, which implies that cyc(A) ∈ N a .
Proof.The statement clearly holds if A is cyclic, so assume that cyc(A) A. Let x ∈ Cyc(A) such that x cyc(A).Let H ∈ Hyp(A) such that cyc(A) ≤ H and x H.By the definition of Cyc(A), we have that r(x + H) = r(H).By Lemma 2.17, which yields a contradiction.It follows that Cyc(A) ⊆ P(cyc(A)).Conversely, since cyc(A) is cyclic, by Proposition 2.5 for any x ≤ cyc(A) and any H ∈ Hyp(cyc(A)) we have that r(x + H) = r(H) and so r(x + H ′ ) = r(H ′ ) for any H ′ ∈ Hyp(A).It follows that P(cyc(A)) ⊆ Cyc(A) and Cyc(cyc(A)) = Cyc(A).The result now follows from Proposition 2.5.
The collection of cyclic spaces of M forms a lattice, such that for every pair of cyclic spaces C 1 , C 2 , the join is defined by . Indeed, by Lemma 2.7, the sum of two cyclic spaces C 1 , C 2 is cyclic.However, the intersection of a pair of cyclic spaces is not cyclic in general: for example the intersection of two circuits is independent and hence not cyclic.In Example 2.19, we provide a specific counterexample.
Example 2.19.Consider F 8 = F 2 3 and let α ∈ F 8 be a primitive element satisfying α 3 = α+1.Let G be the following matrix with entries in F 8 , Let M [G] = (F 5 2 , r) be the q-matroid associated to G.With the aid of the computer algebra system Magma [3], we have checked that M [G] contains 102 cyclic spaces.Among these, consider for instance U = e 2 , e 3 , e 4 , e 5 and V = e 1 + e 4 , e 2 + e 5 , e 3 + e 5 .We have that V ∩ U = e 2 + e 5 , e 3 + e 5 is independent and hence cannot be cyclic by Remark 2.12.

Cyclic Flats
In this subsection, we focus on cyclic flats, which are simultaneously cyclic spaces and flats, i.e. spaces that are both open and closed in the q-matroid M .We show that also the collection of cyclic flats of a q-matroid forms a lattice and we prove that this lattice, together with the rank values of the cyclic flats, uniquely determines the q-matroid.Definition 2.20.F ∈ L(E) is a cyclic flat if cyc(F ) = F and cl(F ) = F .In terms of the rank function, a cyclic flat F satisfies the following two properties: We write Z r to denote the collection of cyclic flats of M .If it is clear from the context, we will simply write Z.
The cyclic operator and the closure operator are closely related.Their interaction is also expressed by the following preliminary results.Lemma 2.21.Let X ∈ L(E) be cyclically closed.Then cl(X) ∈ Z.
Proof.Since cl(X) is a flat, we need only to show that it is cyclic.Assume that V ∈ Hyp(cl(X)).If X < V < cl(X), then cl(X) = cl(V ) and in particular r(V ) = r(cl(X)).On the other hand, if Proof.We need to show that for every flat F , cyc(F ) is also a flat.Let H = Hyp(F ).For any A ∈ H and any x ≤ cyc(F ) we have r(x , and in particular is an intersection of flats of M .Then cyc(F ) is itself a flat.
( (3) By taking the orthogonal complements on both sides of Part (2) and applying Part (1) we get the desired result.
The next proposition shows that the collection of cyclic flats of a q-matroid forms a lattice under inclusion, which is not induced by the lattice of subspaces of the q-matroid nor the one of flats.
Proposition 2.24.The set Z of cyclic flats of a q-matroid forms a lattice under inclusion, where for any two cyclic flats F 1 , F 2 the meet is defined by F 1 ∧ F 2 := cyc(F 1 ∩ F 2 ) and the join is defined as Proof.F 1 ∩ F 1 is a flat and cyc(F 1 ∩ F 2 ) is a cyclic flat by Lemma 2.22.If F is a cyclic flat contained in F 1 ∩ F 2 then by Theorem 2.10, (cyc2), we have that F ≤ cyc(F 1 ∩ F 2 ).By Lemma 2.7 and Lemma 2.21, it immediately follows that cl( Cl2) and (Cl3).Finally note that the flat cl( 0 ) is cyclic and it is the unique minimal element of Z while cyc(E) is the unique maximal element of Z.
Combining Proposition 2.24 with Lemma 2.17, we get that for every pair of cyclic flats Brylowski outlined in [4, Proposition 2.1] an algorithm for constructing the lattice of flats of a matroid from its lattice of cyclic flats, along with the ranks of the cyclic flats.In [12,Section 5], the authors also showed how to reconstruct the lattice of flats from the lattice of cyclic flats, along with their ranks.The same construction given in [12] applies in the q-analogue and we give a brief sketch.For each X ∈ L(E), define two cyclic flats where ∨ and ∧ denote the join and meet, respectively of the lattice Z and where the intersection of an empty set of spaces is equal to the whole space E.
The following Lemma for matroids can be read in [12].We include a proof of the q-analogue for the convenience of the reader.Lemma 2.25.Let A ∈ L(E) and Z ∈ L(E) be a cyclic flat of the q-matroid M satisfying r(Z) + dim(A/(A ∩ Z)) = r(A).Then cl(cyc(A)) ≤ Z ≤ cyc(cl(A)).
Proof.For every A, Z ∈ L(E) we have that , where the first and last equalities follow from Lemma 2.17.Since cyc(A) and cyc(A ∩ Z) are minimal with respect to inclusion in N ν(A) and cyc(A ∩ Z) ≤ cyc(A), we must have that cyc Conversely, if cyc(A) ≤ A ∩ Z, then we have that cyc(A) = cyc(A ∩ Z), hence, by Lemma 2.17, we have that ν(A) = ν(A ∩ Z), from which the claim follows.
and by submodularity we have that The two claims show that for any A, Z satisfying r(Z) , and it follows that cl(cyc(A)) ≤ Z ≤ cyc(cl(A)).
Clearly, if X is a cyclic flat then X ∨ = X.It can be shown, using an argument identical to that of [12, Proposition 6, (i) ⇔ (iii)], which depends on Lemma 2.25, that if X ∨ ≤ X, then X is a flat if and only if for every cyclic flat A satisfying This property can be checked if Z and the ranks of its elements are known.Note that if this is the case, that is if X is a flat, then X ∨ = cyc(X).Hence, we draw the following conclusion.
Proposition 2.26.Every q-matroid M is uniquely determined by its lattice of cyclic flats along with their rank values.
Proof.By the algorithm outlined above, the lattice of cyclic flats along with their ranks uniquely determines the lattice of flats of M .By [6], M is uniquely determined by its lattice of flats.
The next example illustrates how the reconstruction algorithm works.
Example 2.27.Consider the finite field Let M [G] = (F 4 2 , r) be the q-matroid associated to G. The only cyclic flat except for 0 is e 2 , e 3 , e 4 ; see Figure 1.We have cyc(E) = e 2 , e 3 , e 4 , which means that (M [G]) * has a loop, by Corollary 2.15.We may apply the reconstruction result from the above discussion to obtain all the flats of M [G].Take for instance the space F = e 1 .Then F ∨ = 0 ≤ F and F ∧ = cyc(E) = e 2 , e 3 , e 4 .The only cyclic flat A, satisfying F ∨ A ≤ F ∧ is cyc(E).We can then verify that (3) is satisfied, i.e.
Hence, we conclude that F = e 1 is a flat and 0 = F ∨ = cyc(F ).Moreover, by applying Lemma 2.17, we have that r(F ) = 1.
We conclude this section by providing a characterization in terms of cyclic flats of a wellknown family of q-matroids, namely the family of uniform q-matroids (c.f [17]).To this end, we denote by 0 Z := cl( 0 ) the minimal element and by 1 Z := cyc(E) the maximal element of the lattice of cyclic flats of M .
The following result characterizes the independent spaces and the circuits of a q-matroid in terms of its cyclic flats.
Lemma 2.28.Let M = (E, r) be a q-matroid, then the following hold.
1.I ∈ L(E) is independent if and only if for every cyclic flat X, dim(I ∩ X) ≤ r(X).

C ∈ L(E) is a circuit if and only if
C is a minimal space such that there exists a cyclic flat X satisfying C ≤ X and dim(C) = r(X) + 1.

Proof.
1.I is independent if and only if r(I) = dim(I), in which case every subspace of I is independent.In particular, if I is independent then r(I ∩ X) = dim(I ∩ X) for every cyclic flat X.Then dim(I ∩ X) ≤ r(X) by (R2).Assume now that I is not independent.We will construct a cyclic flat X such that r(X) < dim(I ∩ X).There exists a circuit C ≤ I. Let X := cl(C), which is a cyclic flat by Lemma 2.21.Clearly, C ≤ I ∩ X and so which implies that cl(I ∩ X) = X.In particular, r(X) = r(I ∩ X) ≤ dim(I ∩ X).Furthermore, as I ∩ X contains the circuit C, we have that r(I ∩ X) < dim(I ∩ X).Definition 2.29.Let 1 ≤ k ≤ n.For each U ∈ L(E), define r(U ) := min{k, dim(U )}.Then (E, r) is a q-matroid.It is called the uniform q-matroid on E of rank k and is denoted by U k,n .
Proposition 2.30.Let M = (E, r) be a q-matroid of rank k, with 0 < k < n.Then the following are equivalent.
Proof.That (1) implies ( 2) is immediate from the definition of the uniform q-matroid: the only proper flats of U k,n are the subspaces of E of dimension less than k, which are all independent and hence cannot be cyclic by Remark 2.12.Then 0 Zr = 0 and 1 Zr = cyc(E) = E, the vector space sum of its subspaces of dimension k + 1. Suppose now that (2) holds and assume that M is not the uniform q-matroid.Then there is a subspace I ≤ E of dimension k that is not independent.Since I is not independent, it contains a circuit C, with dim(C) ≤ k.By Lemma 2.28, there exists a cyclic flat X, such that 0 C ≤ X and dim(C) = r(X) + 1, which implies that r(X) ≤ k − 1.In particular, there exists a cyclic flat X / ∈ {0 Z , 1 Z }, which contradicts (2).Therefore, M = U k,n .

The Rank Function and Cyclic Flats
In this section, we propose a q-cryptomorphism based on cyclic flats.To this end, we consider a q-matroid (E, r), a lattice Z of subspaces of E and a function r Z on L(E), which satisfies the axioms (R1)-(R3) of a rank function.Next, we show that r(F ) = r Z (F ) for every F ∈ Z and we observe that the spaces in Z are exactly the cyclic flats of the q-matroid (E, r Z ).This section is inspired by Sims' work [23], who proved that any finite lattice is isomorphic to the lattice of cyclic flats of a finite matroid.The same result for matroids was also independently proved by Bonin and de Mier in [2], with a different approach.
Definition 3.1.Let Z be a collection of subspaces of E and suppose that (Z, ≤, ∨, ∧) is a lattice with join and meet operations ∨ and ∧, such that for every Z 1 , Z 2 ∈ Z, we have that , respectively.Let f : Z → Z be a map.We define the following cyclic flat axioms.
(Z1) We have that f (0 Z ) = 0, where 0 Z is the minimal element of Z.
(Z2) For every F, G ∈ Z such that G F , we have: (Z3) For every F, G ∈ Z we have: If (Z, f ) satisfies the cyclic flat axioms, we say that Z is a lattice of cyclic flats with respect to f .
The following preliminary result will be used to show that the lattice of cyclic flats of a q-matroid satisfies (Z1)-(Z3).Lemma 3.2.Let F be a cyclic flat of M and let G < F .Then F/G is a cyclic flat of the q-matroid M/G = (E/G, r M/G ).
and hence F/G is a flat of M/G.
It remains to show that F/G is cyclic.Every hyperplane in F/G can be written as D/G, where D ∈ Hyp(F ) such that D contains G. Since F is cyclic, we have: Proof.(Z1) holds because 0 Z = cl( 0 ), i.e. it is the vector space sum of the loops of E.
To show that (Z2) holds, assume that F and G are two cyclic flats with G < F .By the definition of the rank function r M/G , we have: By Lemma 3.2, F/G is a cyclic flat in M/G and by Remark 2.12, it must be dependent, forcing the inequality to be strict.
(Z3) follows from Lemma 2.17 applied to F ∩ G, combined with submodultarity: As an immediate consequence of Theorem 3.3, we have the following.
Corollary 3.4.Let F, G be two distinct cyclic flats of M , then Proof.Assume that F G. Clearly r(F ∩ G) ≤ r(G) and so by (R2), we have r(F , and is therefore dependent, as observed in Remark 2.12.It follows that We introduce the following function. Definition 3.5.Let Z be a collection of subspaces of E. For every map h : L(E) −→ N 0 , we define h Z : L(E) −→ N 0 to be the function: The remainder of this section will be devoted to proving that if Z is the lattice of cyclic flats of the q-matroid (E, r), then (E, r Z ) is a q-matroid, whose lattice of cyclic flats is equal to Z and for which r Z = r as functions on L(E).
We will need to first prove a number of preliminary results.
Lemma 3.6.Let (Z, f ) be a lattice of cyclic flats.Then, for every F, G ∈ Z, we have Proof.We distinguish three cases.If F ≤ G, then F ∨ G = G and Equation ( 6) clearly holds.If G ≤ F , then the result directly follows from (Z2).Finally, assume that F G and G F .Then apply (Z2) to F ∧ G and F and apply (Z3) to F and G to obtain: Combining these inequalities we get the desired result.
It is straightforward to check that if (Z, f ) is a lattice of cyclic flats of E then f and f Z both agree on Z, as we show in the following proposition.Proposition 3.7.Let (Z, f ) be a lattice of cyclic flats.Then, for every A ∈ Z, we have For the opposite inequality, by applying Lemma 3.6, we have: The following inequalities will be very useful for the next propositions.
Proof.The inequality (7) holds if and only if However, this holds since To see that (8) holds, we note the following: where the last inequality holds since ( Proposition 3.9.Let (Z, f ) be a lattice of cyclic flats.Then f Z satisfies the axioms (R1)-(R3).In particular, (E, f Z ) is a q-matroid.
Proof.(R1) For every A ∈ L(E), we have f Z (A) ≥ 0 being the sum of two non-negative integers.Moreover, since f (0 Z ) = 0, by (Z1), we have that follows by the definition of f Z , since, we have that ≤ f This establishes the submodularity of f Z .Since (R1), (R2) and (R3) hold, we conclude that (E, f Z ) is a q-matroid.Remark 3.10.A function that satisfies (R3) with equality is called modular.Let S be a set and let 2 S be the collection of subsets of S. In [18, Theorem 2.5], Lovász showed that if f, g are two functions defined on 2 S such that f is submodular and g is modular, the convolution defined as is submodular.It is straightforward to check that the same can be said when the two functions are defined on L(E), and g(A) = dim(A) for all A ∈ L(E); see [8,Theorem 24].In Proposition 3.9, we showed that submodularity is also satisfied when the convolution is not taken over all the spaces, as in (5).Thus if f is submodular on L(E), then f Z is obtained as a convolution of f with the dimension function and so inherits the submodularity property from f .Theorem 3.11.Let (Z, f ) be a lattice of cyclic flats.Then Z is the lattice of cyclic flats of the q-matroid M Z := (E, f Z ).
Proof.From Proposition 3.9, we have that M Z is a q-matroid.Let F ∈ Z.We show that F is a flat of M Z .Let x ∈ L(E) be such that x F .Then there exists G ∈ Z such that If F = G, then F < F ∨ G and so by (Z2) and Lemma 3.6, we have that Therefore, In both cases we see that f Z (F + x) > f Z (F ), so F is a flat of M Z .Now, let D ∈ Hyp(F ) and let G ∈ Z be such that If F = G, then again by (Z2) and Lemma 3.6, we have that We conclude that f Z (D) = f Z (F ) and hence F is cyclic in M Z .Finally, we show that every cyclic flat of This implies that F ≤ G. Since F is also a flat, for every x F , we have that which implies that x G.These together show that F = G and, in particular, F ∈ Z.
We summarize the previous results as the following corollary, which gives in turn a new q-cryptomorphism.Corollary 3.12.Let (E, r) be a q-matroid and (Z, ≤, ∨, ∧) be a lattice of subspaces of E.
(1) If Z is the lattice of cyclic flats of (E, r) then (Z, r) satisfies the cyclic flat axioms (Z1)-(Z3).In particular, (E, r Z ) is a q-matroid satisfying r Z (A) = r(A) for all A ∈ Z.
(2) If (Z, f ) satisfies the cyclic flat axioms (Z1)-(Z3) then Z is the collection of cyclic flats of the q-matroid (E, f Z ).In particular, Z = Z f Z .
( There is a correspondence between equivalence classes of non-degenerate [n, k] q m /q rankmetric codes and equivalence classes of [n, k] q m /q systems.We briefly explain this connection; for more details we refer the interested reader to [1,21].
Let C be an [n, k] q m /q non-degenerate rank-metric code with generator matrix G. Then the F q -span U of the columns of G is an [n, k] q m /q system, it is isomorphic to F n q and we call it the q-system associated to C. Conversely, if U ≤ F k q m is an [n, k] q m /q -system and G ∈ F k×n q m is a matrix whose columns form a basis of U , then clearly, the rows of G generate an [n, k] q m /q rank-metric code.
We recall the following result which provides a natural description of the supports of codewords of C in the q-system U associated to C. Theorem 4.4.[19, Theorem 3.1] Let C be an [n, k] q m /q non-degenerate rank-metric code and let U be the F q -span of the columns of a generator matrix G. Consider the isomorphism For every u ∈ F k q m we have that In the following, we use the terminology "linear basis" of a subspace V ≤ F n q to refer to a basis of V as a vector space.This is to distinguish to the notion of basis in the q-matroid sense.Definition 4.5.Let C be an [n, k] q m /q rank-metric code with generator matrix G and U be the F q -span of the columns of G.
i.e. the vectors in one (and hence in any) linear basis of V are linearly independent over F q m .Thanks to Definition 4.5, we immediately obtain the q-analogue of a well-known result in classical matroid theory; see for instance [20,Theorem 1.1.1].
Theorem 4.6.Let G ∈ F k×n q m be a full rank matrix whose columns are linearly independent over F q and let U be the F q -span of the columns of G. Let ψ G : Proof.Since the columns of G are linearly independent over F q , we have that ψ Clearly, (U , r ′ ) is a q-matroid and indeed is equivalent to M [G].The fact that I is the collection of independent spaces of (U , r ′ ) follows from the observation that for every U ≤ U , we have that Let C be the [n, k] q m /q non-degenerate rank-metric code generated by a matrix G ∈ F k×n q , and let M C = (F n q , r) be its associated q-matroid.Recall that M * C = M C ⊥ , where C ⊥ is the dual code of C; see [17].The rest of this section is devoted to establishing the connection between the supports of the codewords of C and M * C .
0 Dimension 5 : F 10 := F 5 2 .The lattice of flats of M C ⊥ can be found in Figure 3.
Finally, using Magma, we found that there are exactly 33 minimal codewords in C. The supports of these codewords are circuits in M C ⊥ .Consider Z := e 1 + e 5 , e 2 + e 5 , e 3 + e 5 , e 4 + e 5 and observe that Z is a cyclic space that is not a circuit (since its rank is 2 and it is not the support of a minimal codeword of C).Z contains exactly 3 circuits, namely e 1 + e 3 , e 2 + e 5 , e 2 + e 3 + e 4 + e 5 , e 1 + e 5 and e 1 + e 4 , e 2 + e 4 .It is easy to see that Z is actually equal to the sum of the circuits it contains, as stated in Corollary 4.13.

Final Remarks
q-Polymatroids and their connections to rank-metric codes were introduced in [14] and [22].In [10], it was shown that knowledge of the lattice of cyclic flats, along with the ranks of its elements, is sufficient to determine a polymatroid.It is a natural question to ask whether or not a q-analogue of this result holds.We propose that an answer to this question may require more than a straightforward extension of the results presented here.

Definition 5.1 ([22]
).A (q, r)-polymatroid is a pair M = (E, ρ) for which r ∈ Z and ρ is an integer-valued function on the subspaces of E, satisfying the following axioms.
We can define cyclic spaces along with the cyclic operator as follows.In [10] a rank function on the lattice of cyclic flats is defined via a convolution of the rank function of the polymatroid with a modular function on the underlying Boolean lattice (see Remark 3.10), which is an important device in obtaining the cryptomorphism in the polymatroid case.It is known that a function µ : L(E) → Z is modular if and only if µ( 0 ) = 0 and µ is additive on L(E), that is, if µ(X + Y ) = µ(X) + µ(Y ) for all X, Y ∈ L(E) satisfying X ∩ Y = {0}.However, as the following result shows, the only additive function on L(E) is a constant multiple of the dimension function.
Proposition 5.3.Let µ : L(E) → Z be an additive function.Then there exists an integer ℓ ∈ Z such that µ(A) = ℓ dim(A) for all A ∈ L(E).
Proof.Let x, y ∈ L(E) and choose A ∈ Hyp(E) that contains neither x nor y.Clearly, E = A + x = A + y and by the additivity of µ we have µ(A) + µ(x) = µ(A + x) = µ(E) = µ(A + y) = µ(A) + µ(y), which implies that µ is constant on all the one-dimensional subspaces of E. Hence there exists a constant ℓ ∈ Z, such that µ(A) = ℓ dim(A), for every A ∈ L(E).
Note that modular functions defined on a boolean lattice are not necessarily constant multiples of the cardinality function.Generalizing the results of Section 3 to the q-polymatroid case would be possible if an analogue of [10, Lemma 1] could be obtained.Proposition 5.3 implies that a q-analogue would be given by showing that for every A ∈ L(E) we have that ρ(A) − ρ(cyc ρ (A)) = r • (dim(A) − dim(cyc ρ (A)).However, the following example shows that this is not always the case.Consider the function ρ : L(F 3 3 ) → Z, A → 4 − dim F 3 (C A ), where C A is the subspace of C made of the matrices whose column span is contained in A ⊥ .From [22], M = (F 3 3 , ρ) is a (q, 3)-polymatroid.Consider the 1-dimensional space c = (1, 1, 2) whose rank is 2. It is easy to see that 2 = ρ(c) − ρ(cyc ρ (c)) = 3(dim(c) − dim(cyc ρ (c)) = 3.

Definition 2 . 8 .
The cyclic operator of M is the function defined by cyc r : L(E) → L(E), A → cyc r (A) := C≤A C is cyclic C.

2 .
A circuit C is a minimal dependent space with r(C) = dim(C) − 1.Moreover, by Lemma 2.21, the closure cl(C) is a cyclic flat with the property that r(cl(C)) = r(C).Hence, by setting X = cl(C), we get the statement.

Theorem 3 . 3 .
Let Z be the lattice of cyclic flats of M and let f : Z → Z, be the map defined by f (F ) = r(F ) for all F ∈ Z. Then (Z, f ) is a lattice of cyclic flats with respect to f , i.e. (Z, f ) satisfies (Z1)-(Z3).
) If Z is the lattice of cyclic flats of E, then (E, r) = (E, r Z ).Proof.The first statement of Part (1) follows from Theorem 3.3.The next statement is a consequence of Proposition 3.7 and Theorem 3.9.Part (2) is the statement of Theorem 3.11.Part (3) can be deduced by Parts (1) and (2) combined with Proposition 2.26.

Figure 3 :
Figure 3: Lattice of flats of the matroid M C ⊥ from Example 4.14.It is not difficult to see that all the flats in Figure 3 are the orthogonal complements of the cyclic spaces of Figure 2, as Proposition 2.14 states.Clearly, the cyclic flats of M C ⊥ are the orthogonal complement of the cyclic flats in M C ; see Figure 4.In particular, A ⊥ 10= F 0 , A ⊥ 1 = F 7 , A ⊥ 2 = F 8 and A ⊥ 3 = F 9 .Finally, using Magma, we found that there are exactly 33 minimal codewords in C. The supports of these codewords are circuits in M C ⊥ .Consider Z := e 1 + e 5 , e 2 + e 5 , e 3 + e 5 , e 4 + e 5 and observe that Z is a cyclic space that is not a circuit (since its rank is 2 and it is not the support of a minimal codeword of C).Z contains exactly 3 circuits, namely e 1 + e 3 , e 2 + e 5 , e 2 + e 3 + e 4 + e 5 , e 1 + e 5 and e 1 + e 4 , e 2 + e 4 .It is easy to see that Z is actually equal to the sum of the circuits it contains, as stated in Corollary 4.13.

Figure 4 :
Figure 4: The lattices of the cyclic flats of the q-matroids M C and M C ⊥ from Example 4.14.