# Independent Spaces of $q$-Polymatroids

Heide Gluesing-Luerssen and Benjamin Jany Department of Mathematics,
University of Kentucky, Lexington KY 40506-0027, USA<EMAIL_ADDRESS>HGL
was partially supported by the grant #422479 from the Simons
Foundation.Department of Mathematics, University of Kentucky, Lexington KY
40506-0027, USA<EMAIL_ADDRESS>
(May 3, 2021)

###### Abstract

This paper is devoted to the study of independent spaces of $q$-polymatroids.
With the aid of an auxiliary $q$-matroid it is shown that the collection of
independent spaces satisfies the same properties as for $q$-matroids. However,
in contrast to $q$-matroids, the rank value of an independent space does not
agree with its dimension. Nonetheless, the rank values of the independent
spaces fully determine the $q$-polymatroid, and this fact can be exploited to
derive a cryptomorphism of $q$-polymatroids. Finally, the notions of minimal
spanning spaces, maximally strongly independent spaces, and bases will be
elaborated on.

##  1 Introduction

Thanks to their relation to rank-metric codes, $q$-matroids and
$q$-polymatroids have recently garnered a lot of attention, [1, 2, 3, 4, 5, 6,
7, 10]. Indeed, ${\mathbb{F}}_{q^{m}}$-linear rank-metric codes in
${\mathbb{F}}_{q^{m}}^{n}$ give rise to $q$-matroids, whereas
${\mathbb{F}}_{q}$-linear rank-metric codes induce $q$-polymatroids. This
leads to an abundance of examples of $q$-(poly)matroids. In either case, the
$q$-(poly)matroid induced by a rank-metric code arises via a rank function
which captures the dimension of certain characteristic subspaces of the code
in question. As a consequence, the $q$-(poly)matroid reflects many of the
algebraic and combinatorial properties of the code, such as the generalized
weights [5, 6] and the rank-weight enumerator [1, 10].

For $q$-matroids a variety of cryptomorphic definitions are known [7, 3]. They
are based on independent spaces, bases, circuits, spanning spaces, flats and
many more; see the comprehensive account in [3]. For $q$-polymatroids, most of
these notions have yet to be defined. As to our knowledge the only existing
notion are flats, which have been introduced in [5].

In this paper we introduce the notion of independent spaces and bases for
$q$-polymatroids. As it turns out, the ‘standard notion’ of independence,
namely the equality of rank value and dimension, is too restrictive for
$q$-polymatroids. For this reason we introduce a more general notion of
independence, which is inspired by the analogue for classical polymatroids in
[8, Sec. 11]. In order to derive properties of the independent spaces, we
introduce an auxiliary $q$-matroid on the same ground space (akin to a
construction in [8]). Since the independent spaces of the auxiliary
$q$-matroid coincide with those of the $q$-polymatroid, the latter inherits
all properties known for independent spaces of $q$-matroids — as long as these
properties do not involve the rank function.

The independent spaces naturally give rise to a notion of basis, namely the
maximal-dimensional independent subspaces. Despite the lack of rigidity of the
rank function in a $q$-polymatroid, it turns out that all bases of a subspace
have the same rank value and this value agrees with the rank value of the
subspace. In other words, the rank function restricted to the independent
spaces fully determines the $q$-polymatroid. This result allows us to provide
a cryptomorphism for $q$-polymatroids based on independent spaces: we
characterize the collections of spaces endowed with a rank function on these
spaces that give rise to a $q$-polymatroid with exactly these spaces as
independent spaces and whose rank function restricts to the given one.
Examples show that no such cryptomorphism is possible using only bases,
dependent spaces, or circuits.

We finally turn to spanning spaces. These are the spaces that share the same
rank value as the ground space. It turns out that in a $q$-polymatroid every
minimal spanning space is contained in a basis, but is, in general, not a
basis itself. Thus the notions ‘minimal spanning’ and ‘maximally independent’
do not agree. On the plus side, ‘minimal spanning’ is the dual notion to
‘maximally strongly independent’. This simple fact may be regarded as the
generalization of the duality result for bases in $q$-matroids. The latter
states that in a $q$-matroid a space is a basis if and only if its orthogonal
is a basis of the dual $q$-matroid. It turns out that this equivalence is
never true in a proper $q$-polymatroid and therefore characterizes
$q$-matroids.

Notation: We fix a finite field ${\mathbb{F}}={\mathbb{F}}_{q}$ with $q$
elements and a finite-dimensional ${\mathbb{F}}$-vector space $E$. We write
$V\leq E$ if $V$ is a subspace of $E$ and denote by ${\mathcal{V}}(E)$ the
collection of all subspaces of $E$. The standard basis vectors in
${\mathbb{F}}^{n}$ are denoted by $e_{1},\ldots,e_{n}$. We write $[n]$ for the
set $\\{1,\ldots,n\\}$.

##  2 Basic Notions of $q$-Polymatroids

In this section we define $q$-polymatroids and present some basic properties.
We also introduce the main class of examples, namely $q$-polymatroids induced
by rank-metric codes. The section is based on the material in [5].

######  Definition 2.1.

Set ${\mathcal{V}}={\mathcal{V}}(E)$. A _$q$ -rank function_ on $E$ is a map
$\rho:{\mathcal{V}}\longrightarrow{\mathbb{Q}}_{\geq 0}$ satisfying:

  * (R1)

Dimension-Boundedness: $0\leq\rho(V)\leq\dim V$ for all $V\in{\mathcal{V}}$;

  * (R2)

Monotonicity: $V\leq W\Longrightarrow\rho(V)\leq\rho(W)$ for all
$V,W\in{\mathcal{V}}$;

  * (R3)

Submodularity: $\rho(V+W)+\rho(V\cap W)\leq\rho(V)+\rho(W)$ for all
$V,W\in{\mathcal{V}}$.

A _$q$ -polymatroid ($q$-PM) on $E$_ is a pair ${\mathcal{M}}=(E,\rho)$, where
$\rho:{\mathcal{V}}\longrightarrow{\mathbb{Q}}_{\geq 0}$ is a $q$-rank
function. The value $\rho(E)$ is called the _rank_ of the $q$-PM. A number
$\mu\in{\mathbb{Q}}_{>0}$ is called a _denominator_ of $\rho$ (and
${\mathcal{M}}$) if $\mu\rho(V)\in{\mathbb{N}}_{0}$ for all
$V\in{\mathcal{V}}$. In that case we call the map $\tau_{\mu}:=\mu\rho$ the
_induced integer $\rho$-function_ w.r.t. $\mu$. The smallest denominator is
called the _principal denominator_. A $q$-PM with principal denominator $1$
(i.e., $\rho(V)\in{\mathbb{N}}_{0}$ for all $V$) is called a _$q$ -matroid_.
If $(E,\rho)$ is the _trivial_ $q$-PM, i.e., $\rho$ is the zero map, we
declare $1$ to be its principal denominator.

We will often make use of the induced integer $\rho$-function $\tau_{\mu}$ for
a given denominator $\mu$. Clearly $\tau_{\mu}$ is also monotonic and
submodular, and instead of (R1) it satisfies $0\leq\tau_{\mu}(V)\leq\mu\dim V$
for all $V\in{\mathcal{V}}$.

The above definition appears in various forms in the literature. In [6, Def.
4.1] of Gorla et al. the same definition occurs with the only difference that
the rank function may assume arbitrary real numbers. Next, a $q$-matroid in
the sense of Jurrius/ Pellikaan [7] is exactly a $q$-matroid as defined above.
Finally, for any $r\in{\mathbb{N}}$ a $(q,r)$-polymatroid as in [10, Def. 2]
by Shiromoto and [4, Def. 1] by Ghorpade/Johnson and [1, Def. 1] by Byrne et
al. can be turned into a $q$-PM with denominator $r$ by dividing the rank
function by $r$. Conversely, given a $q$-PM $(E,\rho)$ with denominator $\mu$,
then $(E,\mu\rho)$ is a $(q,\lceil\mu\rceil)$-polymatroid in the sense of
these papers.

It is possible for a $q$-PM that no nonzero subspace attains the upper bound
in (R1). In [5] we call such $q$-PMs non-exact and discuss some basic facts
about non-exact $q$-PMs.

######  Remark 2.2 ([5, Rem. 2.3(3)]).

Let $(E,\rho)$ be a non-trivial $q$-PM with principal denominator $\mu$. Then
the set of all denominators of $(E,\rho)$ is $\mu{\mathbb{N}}$.

The following basic properties are presented for $q$-matroids in [7, Prop. 6
and 7]. They hold true for $q$-PMs as well, and the proofs are identical to
those in [7].

######  Proposition 2.3.

Let $(E,\rho)$ be a $q$-PM.

  * (a)

Let $V,W\in{\mathcal{V}}(E)$. Suppose
$\rho(V+\mbox{$\langle{x}\rangle$})=\rho(V)$ for all $x\in W$. Then
$\rho(V+W)=\rho(V)$.

  * (b)

Let $V\in{\mathcal{V}}(E)$ and $X,Y\in{\mathcal{V}}(E)$ be $1$-dimensional
spaces such that $\rho(V)=\rho(V+X)=\rho(V+Y)$. Then $\rho(V+X+Y)=\rho(V)$.

In order to discuss duality as well as some details on independent spaces we
need the following notions of equivalence. They extend [6, Def. 4.4]. $q$-PMs
are scaling-equivalent if they differ only by a scalar factor of the rank
functions and an isomorphism between the ground spaces, and they are
equivalent if the scalar factor is $1$.

######  Definition 2.4.

Let $E_{i},\,i=1,2,$ be ${\mathbb{F}}$-vector spaces of the same finite
dimension and let ${\mathcal{M}}_{i}=(E_{i},\rho_{i})$ be $q$-PMs.

  * (a)

${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ are called _scaling-equivalent_ if
there exists an ${\mathbb{F}}$-isomorphism $\alpha\in\mbox{\rm
Hom}_{{\mathbb{F}}}(E_{1},E_{2})$ and $a\in{\mathbb{Q}}_{>0}$ such that
$\rho_{2}(\alpha(V))=a\rho_{1}(V)$ for all $V\in{\mathcal{V}}(E_{1})$.

  * (b)

We call ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ _equivalent_ , denoted by
${\mathcal{M}}_{1}\approx{\mathcal{M}}_{2}$, if there exists an
${\mathbb{F}}$-isomorphism $\alpha\in\mbox{\rm
Hom}_{{\mathbb{F}}}(E_{1},E_{2})$ such that $\rho_{2}(\alpha(V))=\rho_{1}(V)$
for all $V\in{\mathcal{V}}(E_{1})$.

Clearly, both types of equivalences are indeed equivalence relations.

######  Theorem 2.5 ([5, Thm. 2.8] and [6, 4.5–4.7], and [7, Thm. 42] for
$q$-matroids).

Let $\langle{\cdot}\\!\mid\\!{\cdot}\rangle$ be a non-degenerate symmetric
bilinear form on $E$. For $V\in{\mathcal{V}}(E)$ define
$\mbox{$V^{\perp}=\\{w\in
E\mid\mbox{$\langle{v}\\!\mid\\!{w}\rangle$}=0$}\text{ for all }v\in V\\}$.
Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM and set

$\rho^{*}(V)=\dim V+\rho(V^{\perp})-\rho(E).$

Then $\rho^{*}$ is a $q$-rank function on $E$ and
${\mathcal{M}}^{*}=(E,\rho^{*})$ is a $q$-PM. It is called the _dual_ of
${\mathcal{M}}$ with respect to the form
$\langle{\cdot}\\!\mid\\!{\cdot}\rangle$. Furthermore, the bidual
${\mathcal{M}}^{**}=({\mathcal{M}}^{*})^{*}$ satisfies
${\mathcal{M}}^{**}={\mathcal{M}}$, and ${\mathcal{M}}$ and
${\mathcal{M}}^{*}$ have the same set of denominators. Finally, the
equivalence class of ${\mathcal{M}}^{*}$ does not depend on the choice of the
bilinear form. More precisely, if
$\langle\\!\mbox{$\langle{\cdot}\\!\mid\\!{\cdot}\rangle$}\\!\rangle$ is
another non-degenerate symmetric bilinear form on $E$ and
${\mathcal{M}}^{\hat{*}}=(E,\rho^{\hat{*}})$ is the resulting dual $q$-PM,
then ${\mathcal{M}}^{\hat{*}}\approx{\mathcal{M}}^{*}$.

The next result has been proven in [6] for $q$-PMs on ${\mathbb{F}}^{n}$,
endowed with the standard dot product. Thanks to the invariance of the dual,
it generalizes without the need to specify bilinear forms.

######  Proposition 2.6 ([6, Prop. 4.7]).

Let ${\mathcal{M}}=(E,\rho)$ and $\hat{{\mathcal{M}}}=(\hat{E},\hat{\rho})$ be
$q$-PMs. Then ${\mathcal{M}}\approx\hat{{\mathcal{M}}}$ implies
${\mathcal{M}}^{*}\approx\hat{{\mathcal{M}}}^{*}$.

######  Example 2.7 ([7, Ex. 4 and Ex. 47]).

Let ${\mathcal{U}}_{k}(E)=(E,\rho)$ be the _uniform $q$-matroid of rank $k$_,
that is, $\rho(V)=\min\\{k,\dim V\\}$ for all $V\in{\mathcal{V}}(E)$. Then
${\mathcal{U}}_{k}(E)^{*}={\mathcal{U}}_{\dim E-k}(E)$.

The rest of this section is devoted to $q$-PMs induced by rank-metric codes.
This will provide us with a large class of $q$-PMs. We start with collecting
some basic properties of codes in ${\mathbb{F}}^{n\times m}$. As usual, we
endow ${\mathbb{F}}^{n\times m}$ with the rank-metric, defined as $\mbox{${\rm
d}$}(A,B)={\rm rk}(A-B)$.

We only consider _linear rank-metric codes_ , that is, subspaces of the metric
space $(\mbox{${\mathbb{F}}^{n\times m}$},\mbox{${\rm d}$})$. The following is
standard knowledge in the theory of rank-metric codes. For
$V\leq{\mathbb{F}}^{n}$ denote by $V^{\perp}\leq{\mathbb{F}}^{n}$ the
orthogonal space with respect to the standard dot product.

######  Remark 2.8.

Let ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$ be a rank-metric
code.

  * (a)

The _rank distance_ of ${\mathcal{C}}$ is defined as $\mbox{${\rm d}_{\rm
rk}$}({\mathcal{C}})=\min\\{{\rm rk}(M)\mid M\in{\mathcal{C}}\setminus 0\\}$.
If $d=\mbox{${\rm d}_{\rm rk}$}({\mathcal{C}})$, then
$\dim({\mathcal{C}})\leq\max\\{m,n\\}(\min\\{m,n\\}-d+1)$, which is known as
the _Singleton bound_. If
$\dim({\mathcal{C}})=\max\\{m,n\\}(\min\\{m,n\\}-d+1)$, then ${\mathcal{C}}$
is called an _MRD code_.

  * (b)

The _dual code_ of ${\mathcal{C}}$ is defined as
${\mathcal{C}}^{\perp}=\\{M\in\mbox{${\mathbb{F}}^{n\times m}$}\mid{\rm
tr}(MN\mbox{${}^{\sf T}$})=0\text{ for all }N\in{\mathcal{C}}\\}$, where ${\rm
tr}(\cdot)$ denotes the trace of the given matrix.

  * (c)

For $V\in{\mathcal{V}}({\mathbb{F}}^{n})$ we set ${\mathcal{C}}(V,{\rm
c})=\\{M\in{\mathcal{C}}\mid\mbox{\rm colsp}(M)\leq V\\}$, where $\mbox{\rm
colsp}(M)$ denotes the column space of $M$. Then $\mbox{${\mathbb{F}}^{n\times
m}$}(V,{\rm c})^{\perp}=\mbox{${\mathbb{F}}^{n\times m}$}(V^{\perp},{\rm c})$
and [9, Lem. 28]

$\displaystyle\dim{\mathcal{C}}(V^{\perp},{\rm c})$
$\displaystyle=\dim{\mathcal{C}}-m\dim V+\dim{\mathcal{C}}^{\perp}(V,{\rm
c}).$

The following $q$-PMs induced by rank-metric codes appeared first in [6]. The
statement in (2.1) is immediate with Proposition 2.8(c).

######  Definition/Proposition 2.9 ([6, Thm. 5.3]).

For a nonzero rank-metric code ${\mathcal{C}}\leq{\mathbb{F}}^{n\times m}$
define

$\rho_{\rm c}:{\mathcal{V}}({\mathbb{F}}^{n})\longrightarrow{\mathbb{Q}}_{\geq
0},\quad V\longmapsto\frac{\dim{\mathcal{C}}-\dim{\mathcal{C}}(V^{\perp},{\rm
c})}{m}.$

Then $\rho_{\rm c}$ is a $q$-rank function with denominator $m$. The
$q$-polymatroid ${\mathcal{M}}_{\rm
c}({\mathcal{C}}):=({\mathbb{F}}^{n},\rho_{\rm c})$ is called the _(column)
polymatroid_ of ${\mathcal{C}}$. Its rank is $\dim{\mathcal{C}}/m$. The rank
function satisfies

$\rho_{\rm c}(V)=\dim V-\frac{1}{m}\dim{\mathcal{C}}^{\perp}(V,{\rm c}).$
(2.1)

Analogously we can define the row polymatroid of the rank-metric code, which
then has denominator $n$. It is of course the same as the column polymatroid
of the transposed code, and thus it suffices to consider column polymatroids.

The denominator $m$ is in general not principal; see for instance (a) of 2.11
below.

The rank distances of a code and its dual are closely related to the $q$-PM.
The following is immediate with Proposition 2.9 and also appeared already in
[6, Prop. 6.2], [7, Lem. 30], and [5, Rem. 3.8].

######  Remark 2.10.

Let ${\mathcal{C}}\leq{\mathbb{F}}^{n\times m}$ be a nonzero rank-metric code
with rank-distance $d$ and let $d^{\perp}$ be the rank distance of
${\mathcal{C}}^{\perp}$. Then for any $V\in{\mathcal{V}}({\mathbb{F}}^{n})$ we
have

$\rho_{\rm c}(V)=\begin{cases}\dim V,&\text{ if }\dim
V<d^{\perp},\\\\[4.30554pt] \frac{\dim{\mathcal{C}}}{m},&\text{ if }\dim
V>n-d.\end{cases}$

For MRD codes we can give more detailed information about the associated
$q$-PMs.

######  Proposition 2.11 ([6, Cor. 6.6], [5, Thm. 3.12]).

Let ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$ be an MRD code with
$\mbox{${\rm d}_{\rm rk}$}({\mathcal{C}})=d$.

  * (a)

Let $n\leq m$. Then ${\mathcal{M}}_{\rm
c}({\mathcal{C}})={\mathcal{U}}_{n-d+1}({\mathbb{F}}^{n})$, that is,
${\mathcal{M}}_{\rm c}({\mathcal{C}})$ is the uniform $q$-matroid of rank
$n-d+1$. Thus ${\mathcal{M}}_{\rm c}({\mathcal{C}})$ depends only on
$(n,d,|{\mathbb{F}}|)$.

  * (b)

Let $n\geq m$. Then ${\mathcal{M}}_{\rm c}({\mathcal{C}})$ satisfies

$\rho_{\rm c}(V)=\left\\{\begin{array}[]{cl}\dim V,&\text{if }\dim V\leq
m-d+1,\\\\[2.15277pt] \frac{n(m-d+1)}{m},&\text{if }\dim V\geq
n-d+1,\end{array}\right.$

and $\rho_{\rm c}(V)\geq\max\\{1,\,(\dim V)/m\\}(m-d+1)$ if $\dim
V\in[m-d+2,\,n-d]$. Thus, if $m=n-1$, then ${\mathcal{M}}_{\rm
c}({\mathcal{C}})$ depends only on $(n,d,|{\mathbb{F}}|)$.

Equivalence of codes, in the usual sense, translates into equivalence of the
associated $q$-PMs.

######  Proposition 2.12 ([5, Prop. 3.5] and [6, Prop. 6.7]).

Let ${\mathcal{C}},\,{\mathcal{C}}^{\prime}\leq\mbox{${\mathbb{F}}^{n\times
m}$}$ be rank-metric codes.

  * (a)

Suppose ${\mathcal{C}},\,{\mathcal{C}}^{\prime}$ are _equivalent_ , i.e.,
${\mathcal{C}}^{\prime}=X{\mathcal{C}}Y:=\\{XMY\mid M\in{\mathcal{C}}\\}$ for
some $X\in{\rm GL}_{n}({\mathbb{F}})$ and $Y\in{\rm GL}_{m}({\mathbb{F}})$.
Then ${\mathcal{M}}_{\rm c}({\mathcal{C}})$ and ${\mathcal{M}}_{\rm
c}({\mathcal{C}}^{\prime})$ are equivalent via $\beta\in\mbox{\rm
Hom}_{{\mathbb{F}}}({\mathbb{F}}^{n},{\mathbb{F}}^{n})$ given by
$x\mapsto(X\mbox{${}^{\sf T}$})^{-1}x$.

  * (b)

Let $n=m$ and suppose ${\mathcal{C}},\,{\mathcal{C}}^{\prime}$ are
_transposition-equivalent_ , that is,
${\mathcal{C}}^{\prime}=X{\mathcal{C}}\mbox{${}^{\sf T}$}Y$ for some
$X,Y\in{\rm GL}_{n}({\mathbb{F}})$, and where ${\mathcal{C}}\mbox{${}^{\sf
T}$}=\\{M\mbox{${}^{\sf T}$}\mid M\in{\mathcal{C}}\\}$. Then
${\mathcal{M}}_{\rm c}({\mathcal{C}}\mbox{${}^{\sf T}$})$ and
${\mathcal{M}}_{\rm c}({\mathcal{C}}^{\prime})$ are equivalent via $\beta$,
where $\beta$ is as in (a).

Occasionally, we will consider ${\mathbb{F}}_{q^{m}}$-linear rank-metric
codes, which we introduce as follows. Recall that
${\mathbb{F}}={\mathbb{F}}_{q}$. Let $\omega$ be a primitive element of the
field extension ${\mathbb{F}}_{q^{m}}$ and
$\psi:{\mathbb{F}}_{q^{m}}\longrightarrow{\mathbb{F}}_{q}^{m}$ be the
coordinate map with respect to the basis $(1,\omega,\ldots,\omega^{m-1})$.
Extending $\psi$ entry-wise, we obtain, for any $n$, an isomorphism
$\Psi:{\mathbb{F}}_{q^{m}}^{n}\longrightarrow{\mathbb{F}}_{q}^{n\times m}$
that maps $(c_{1},\,\cdots,\,c_{n})$ to the matrix with rows
$\psi(c_{1}),\,\ldots,\,\psi(c_{n})$. Let
$f=x^{m}-\sum_{i=0}^{m-1}f_{i}x^{i}\in{\mathbb{F}}[x]$ be the minimal
polynomial of $\omega$ over ${\mathbb{F}}$ and

$\Delta_{f}=\begin{pmatrix}&1&&\\\ &&\ddots&\\\ &&&1\\\
f_{0}&f_{1}&\cdots&f_{m-1}\end{pmatrix}\in{\rm GL}_{m}({\mathbb{F}}_{q})$
(2.2)

be the companion matrix of $f$.

Given a code ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$, we can now
describe linearity of the code
$\Psi^{-1}({\mathcal{C}})\subseteq{\mathbb{F}}_{q^{m}}^{n}$ over
${\mathbb{F}}_{q^{m}}$ or a subfield thereof as follows; see also [5, Sec. 3].
The matrix $X$ below simply allows for arbitrary bases of
${\mathbb{F}}_{q^{m}}$ over ${\mathbb{F}}_{q}$.

######  Definition 2.13.

Let ${\mathcal{C}}\leq{\mathbb{F}}_{q}^{n\times m}$ be a rank-metric code
(hence an ${\mathbb{F}}_{q}$-linear subspace). Let $s$ be a divisor of $m$ and
set $M=(q^{m}-1)/(q^{s}-1)$. Then ${\mathcal{C}}$ is _right
${\mathbb{F}}_{q^{s}}$-linear_ if there exists an $X\in{\rm
GL}_{m}({\mathbb{F}}_{q})$ such that the code ${\mathcal{C}}X$ is invariant
under right multiplication by $\Delta_{f}^{M}$. _Left linearity_ over
${\mathbb{F}}_{q^{n}}$ and its subfields is defined analogously.

Obviously, the qualifiers left/right are needed only in the case where
${\mathbb{F}}_{q^{s}}$ is a subfield of both ${\mathbb{F}}_{q^{n}}$ and
${\mathbb{F}}_{q^{m}}$. The following is now easy to verify; see also [5, Sec.
3].

######  Remark 2.14.

Let ${\mathbb{F}}_{q^{s}}$ be a subfield of ${\mathbb{F}}_{q^{m}}$ and
${\mathcal{C}}\leq{\mathbb{F}}_{q}^{n\times m}$ be a right
${\mathbb{F}}_{q^{s}}$-linear rank-metric code. Then $\mu=m/s$ is a
denominator of the column polymatroid ${\mathcal{M}}_{\rm c}({\mathcal{C}})$.
In particular, for $s=m$ the polymatroid ${\mathcal{M}}_{\rm
c}({\mathcal{C}})$ is a $q$-matroid. These are exactly the $q$-matroids
studied in [7].

It should be noted that ${\mathcal{M}}_{\rm c}({\mathcal{C}})$ may be a
$q$-matroid even if ${\mathcal{C}}$ is not right
${\mathbb{F}}_{q^{m}}$-linear. Indeed, in 2.11(a) we saw already that the
polymatroid associated to an MRD code in ${\mathbb{F}}^{n\times m}$ is a
$q$-matroid if $n\leq m$. Furthermore, [5, Thm. 5.5] shows that if
${\mathcal{C}}$ induces a $q$-matroid, then so does every shortening and
puncturing of ${\mathcal{C}}$.

Duality of $q$-PMs (see 2.5) corresponds to duality of codes.

######  Theorem 2.15 ([6, Thm. 8.1]).

Let ${\mathcal{C}}\leq{\mathbb{F}}^{n\times m}$ be a rank-metric code and
${\mathcal{C}}^{\perp}\leq{\mathbb{F}}^{n\times m}$ be its dual. Then
${\mathcal{M}}_{\rm c}({\mathcal{C}})^{*}={\mathcal{M}}_{\rm
c}({\mathcal{C}}^{\perp})$, where ${\mathcal{M}}_{\rm c}({\mathcal{C}})^{*}$
is the dual of ${\mathcal{M}}_{\rm c}({\mathcal{C}})$ w.r.t. the standard dot
product on ${\mathbb{F}}^{n}$.

Another instance of the interplay between polymatroid duality and rank-metric
codes has been presented in [5, Thms. 5.3 and 5.5]. Therein, it is shown that
contraction and deletion of $q$-PMs are mutually dual and correspond to
shortening and puncturing of rank-metric codes.

We close this section with the following example of two MRD codes whose
associated column polymatroids are not equivalent. The parameters are as in
2.11(b), where the interval $[m-d+2,\,n-d]$ is not empty. We will return to
this example later when discussing independent spaces.

######  Example 2.16 ([5, Ex. 3.15]).

In ${\mathbb{F}}_{2}^{5\times 2}$ consider the codes
${\mathcal{C}}_{1}=\mbox{$\langle{A_{1},\dots,A_{5}}\rangle$}$ and
${\mathcal{C}}_{2}=\mbox{$\langle{B_{1},\ldots,B_{5}}\rangle$}$, where

$\displaystyle A_{1}=\begin{pmatrix}1&1\\\ 1&0\\\ 0&0\\\ 1&0\\\
0&0\end{pmatrix},\ A_{2}=\begin{pmatrix}1&1\\\ 1&1\\\ 1&0\\\ 0&1\\\
0&0\end{pmatrix},\ A_{3}=\begin{pmatrix}0&0\\\ 0&0\\\ 1&1\\\ 0&0\\\
0&1\end{pmatrix},\ A_{4}=\begin{pmatrix}0&0\\\ 0&1\\\ 0&0\\\ 0&0\\\
1&1\end{pmatrix},\ A_{5}=\begin{pmatrix}1&0\\\ 0&1\\\ 1&1\\\ 0&0\\\
0&1\end{pmatrix},$ $\displaystyle B_{1}=\begin{pmatrix}1&0\\\ 0&1\\\ 0&0\\\
0&0\\\ 0&0\end{pmatrix},\ B_{2}=\begin{pmatrix}0&0\\\ 1&0\\\ 0&1\\\ 0&0\\\
0&0\end{pmatrix},\ B_{3}=\begin{pmatrix}0&0\\\ 0&0\\\ 1&0\\\ 0&1\\\
0&0\end{pmatrix},\ B_{4}=\begin{pmatrix}0&0\\\ 0&0\\\ 0&0\\\ 1&0\\\
0&1\end{pmatrix},\ B_{5}=\begin{pmatrix}0&1\\\ 0&0\\\ 0&1\\\ 0&0\\\
1&0\end{pmatrix}.$

Both codes are MRD with rank distance $d=2$, and ${\mathcal{C}}_{2}$ is
actually a (${\mathbb{F}}_{2^{5}}$-linear) Gabidulin code. Consider the
$q$-PMs ${\mathcal{M}}_{\rm c}({\mathcal{C}}_{1})=({\mathbb{F}}^{5},\rho_{\rm
c}^{1})$ and ${\mathcal{M}}_{\rm
c}({\mathcal{C}}_{2})=({\mathbb{F}}^{5},\rho_{\rm c}^{2})$. From 2.11(b) we
know that $\rho_{\rm c}^{1}(V)=\rho_{\rm c}^{2}(V)=\dim V$ for $\dim V\leq 1$
and $\rho_{\rm c}^{1}(V)=\rho_{\rm c}^{2}(V)=5/2$ if $\dim V\geq 4$. As for
the $2$-dimensional subspaces of ${\mathbb{F}}_{2}^{5}$, it turns out that the
map $\rho_{\rm c}^{1}$ assumes the value $1$ exactly once and the values $3/2$
and $2$ exactly 28 and 126 times, respectively, whereas $\rho_{\rm c}^{2}$
assumes the values $3/2$ and $2$ exactly 31 and 124 times, respectively, and
never takes the value $1$. Similar differences occur for the $3$-dimensional
subspaces. Thus ${\mathcal{M}}_{\rm c}({\mathcal{C}}_{1})$ and
${\mathcal{M}}_{\rm c}({\mathcal{C}}_{2})$ are not equivalent.

##  3 Independent Spaces

We now return to general $q$-PMs and introduce independent spaces. We show
that the collection of independent spaces satisfies properties analogous to
those of independent spaces in $q$-matroids. However, different from the
latter, they do not fully determine the $q$-PM. Only if we also take the rank
values of the independent spaces into account, can we fully recover the
$q$-PM. This will be dealt with in the next section.

Considering the theory of classical matroids and $q$-matroids, one may be
inclined to declare a space $V$ in a $q$-PM $(E,\rho)$ independent if
$\rho(V)=\dim V$. While this is indeed the right notion for $q$-matroids, it
turns out to be too restrictive for $q$-PMs: in many $q$-PMs the only subspace
satisfying $\rho(V)=\dim V$ is the zero space. Nonetheless, the property
$\rho(V)=\dim V$ turns out to play a conceptual role (see also [1]), and we
will return to it in Section 5, where we will call such spaces strongly
independent.

The following definition of independence is inspired by [8, Cor. 11.1.2],
which deals with classical polymatroids.

######  Definition 3.1.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$ (which need
not be principal). A space $I\in{\mathcal{V}}(E)$ is called _$\mu$
-independent_ if

$\rho(J)\geq\frac{\dim J}{\mu}\text{ for all subspaces }J\leq I.$

$I$ is called _$\mu$ -dependent_ if it is not $\mu$-independent. A _$\mu$
-circuit_ is a $\mu$-dependent space for which all proper subspaces are
$\mu$-independent. A $1$-dimensional $\mu$-dependent space is called a _$\mu$
-loop_. We define
${\mathcal{I}}_{\mu}={\mathcal{I}}_{\mu}({\mathcal{M}})=\\{I\in{\mathcal{V}}(E)\mid
I\text{ is $\mu$-independent}\\}$. If $\mu$ is the principal denominator of
${\mathcal{M}}$, we may skip the quantifier $\mu$ and simply use independent,
dependent, loop, circuit, and ${\mathcal{I}}$.

Clearly, if $\hat{\mu}$ is the principal denominator of ${\mathcal{M}}$, then
$\hat{\mu}$-independence implies $\mu$-independence for any denominator $\mu$
of ${\mathcal{M}}$; see 2.2.

######  Remark 3.2.

Let $(E,\rho)$ be a $q$-PM. Then for all $V\in{\mathcal{V}}(E)$

$\rho(V)=\dim V\Longrightarrow\rho(W)=\dim W\text{ for all }W\leq V.$ (3.1)

Indeed, writing $V=W\oplus Z$ for some complement $Z$ of $W$, we obtain from
submodularity $\dim V=\rho(V)\leq\rho(W)+\rho(Z)\leq\dim W+\dim Z=\dim V$, and
thus we have equality everywhere. As a consequence, the condition
$\rho(V)=\dim V$ implies $\mu$-independence for every denominator $\mu$, and
for $q$-matroids our notion of independence (which is $1$-independence)
coincides with independence as defined in [7, Def. 2], that is: $V\text{ is
$1$-independent }\Longleftrightarrow\rho(V)=\dim V$.

We continue with discussing basis properties of independent spaces. A crucial
difference to $q$-matroids is the following: While for $q$-matroids a space is
independent iff its rank value assumes the maximal possible value, this is not
the case for $q$-PMs. More generally, for $q$-PMs independence is not
characterized by the rank value of the given space; see the examples below. In
this context we also would like to point out that the inequality
$\rho(I)\geq\dim I/\mu$ is not preserved under taking subspaces, which is why
the condition for subspaces is built into our definition.

######  Example 3.3.

  * (a)

Let ${\mathbb{F}}={\mathbb{F}}_{2}$ and consider the code
${\mathcal{C}}\leq{\mathbb{F}}^{3\times 3}$ generated by

$\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&1\end{pmatrix},\ \begin{pmatrix}0&1&1\\\
0&0&0\\\ 0&0&1\end{pmatrix},\ \begin{pmatrix}0&1&1\\\ 1&0&0\\\
0&1&0\end{pmatrix}.$

Let ${\mathcal{M}}={\mathcal{M}}_{\rm
c}({\mathcal{C}})=({\mathbb{F}}^{3},\rho_{\rm c})$ be the associated column
polymatroid. Then for all $V\in{\mathcal{V}}({\mathbb{F}}^{3})\setminus 0$

$\rho_{\rm c}(V)=\left\\{\begin{array}[]{cl}2/3&\text{ if $\dim V=1$ or
$V=\mbox{$\langle{e_{1}+e_{2},\,e_{3}}\rangle$}$,}\\\\[1.72218pt] 1&\text{
otherwise.}\end{array}\right.$

This shows that $3$ is the principal denominator and
${\mathcal{I}}({\mathcal{M}})={\mathcal{V}}({\mathbb{F}}^{3})$, that is, all
spaces are independent. In particular, all $2$-dimensional spaces are
independent, even though they do not assume the same rank value.

  * (b)

Dependent spaces may have a larger rank value than independent spaces of the
same dimension. For instance, let ${\mathbb{F}}={\mathbb{F}}_{2}$ and
${\mathcal{C}}\leq{\mathbb{F}}^{5\times 3}$ be the code generated by the
standard basis matrices $E_{11},\,E_{12},\,E_{23},\,E_{32},\,E_{41},\,E_{42}$.
In the column polymatroid ${\mathcal{M}}_{\rm
c}({\mathcal{C}})=({\mathbb{F}}^{5},\rho_{\rm c})$ the subspace
$I=\mbox{$\langle{e_{2},e_{3}}\rangle$}$ is independent with $\rho_{\rm
c}(I)=2/3$, and the subspace $V=\mbox{$\langle{e_{1}+e_{2},e_{5}}\rangle$}$ is
dependent with $\rho_{\rm c}(V)=1$.

Let us consider the independent spaces of some $q$-PMs from the previous
section.

######  Example 3.4.

  * (a)

Consider the uniform $q$-matroid ${\mathcal{U}}={\mathcal{U}}_{k}(E)$ from
2.7. Then
${\mathcal{I}}_{1}({\mathcal{U}})=\\{V\in{\mathcal{V}}(E)\mid\dim(V)\leq
k\\}$.

  * (b)

Let $m\geq n$ and ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$ be an
MRD code with rank distance $d$. 2.11(a) yields ${\mathcal{M}}_{\rm
c}({\mathcal{C}})={\mathcal{U}}_{n-d+1}({\mathbb{F}}^{n})$ and thus
${\mathcal{I}}_{1}({\mathcal{M}}_{\rm
c}({\mathcal{C}}))=\\{V\in{\mathcal{V}}({\mathbb{F}}^{n})\mid\dim V\leq
n-d+1\\}$ by (a).

  * (c)

Let $m\leq n$ and ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$ be an
MRD code with rank distance $d$. With the aid of 2.11(b) one verifies that
${\mathcal{I}}_{m}({\mathcal{M}}_{\rm
c}({\mathcal{C}}))={\mathcal{V}}({\mathbb{F}}^{n})$, that is, every subspace
is $m$-independent. In particular, the two non-equivalent $q$-PMs
${\mathcal{M}}_{\rm c}({\mathcal{C}}_{1})$ and ${\mathcal{M}}_{\rm
c}({\mathcal{C}}_{2})$ in 2.16 trivially have the same collection of
$2$-independent spaces (and $2$ is the principal denominator). This shows that
the collection of independent spaces does not determine the $q$-PM.

  * (d)

Part (c) shows another striking difference to $q$-matroids. If all spaces of a
$q$-matroid ${\mathcal{M}}=(E,\rho)$ are independent (i.e., $1$-independent),
then ${\mathcal{M}}$ is the uniform matroid ${\mathcal{U}}_{n}(E)$, where
$n=\dim E$. If furthermore ${\mathcal{M}}={\mathcal{M}}_{\rm
c}({\mathcal{C}})$ for some right ${\mathbb{F}}_{q^{m}}$-linear code
${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$, then
${\mathcal{C}}=\mbox{${\mathbb{F}}^{n\times m}$}$. This follows immediately
from 3.2 together with (2.1).

We continue with some basic facts.

######  Remark 3.5.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$.

  * (a)

The zero subspace of $E$ is $\mu$-independent.

  * (b)

Every dependent space $V$ contains a circuit: take any subspace $W$ of $V$ of
smallest dimension satisfying $\rho(W)<\dim W/\mu$ (which clearly exists).

  * (c)

Let $V\in{\mathcal{V}}(E)$ be a $\mu$-circuit. Then $\mu\rho(V)=\dim
V-1=\mu\rho(W)$ for all hyperplanes $W$ in $V$. Indeed, independence of $W$
along with (R2) tells us that

$\dim V-1=\dim W\leq\mu\rho(W)\leq\mu\rho(V)<\dim V.$

Thus we have equality since $\mu\rho$ takes integer values. 3.6 below shows
that not every subspace $V$ satisfying $\mu\rho(V)=\dim V-1=\mu\rho(W)$ for
all its hyperplanes $W$ is a circuit.

  * (d)

Part (c) implies that if $\dim V=1$, then $V$ is a $\mu$-loop iff $\rho(V)=0$.
As a consequence, loops do not depend on the choice of denominator. On the
other hand, if $\mu_{1}<\mu_{2}$ are distinct denominators of ${\mathcal{M}}$,
then the $\mu_{2}$-circuits with positive rank value are distinct from the
$\mu_{1}$-circuits. But every $\mu_{2}$-circuit is also $\mu_{1}$-dependent
and thus contains a $\mu_{1}$-circuit by (b).

  * (e)

Let $x_{1},\ldots,x_{t}\in E$ be linearly independent vectors such that
$\langle{x_{i}}\rangle$ is a $\mu$-loop for all $i\in[t]$. Then
$\rho(\mbox{$\langle{x_{1}}\rangle$}+\ldots+\mbox{$\langle{x_{t}}\rangle$})=0$.
For $t=2$ this is a consequence of the submodularity (R3) because
$\rho(\mbox{$\langle{x_{1}}\rangle$}+\mbox{$\langle{x_{2}}\rangle$})=\rho(\mbox{$\langle{x_{1}}\rangle$}+\mbox{$\langle{x_{2}}\rangle$})+\rho(\mbox{$\langle{x_{1}}\rangle$}\cap\mbox{$\langle{x_{2}}\rangle$})\leq\rho(\mbox{$\langle{x_{1}}\rangle$})+\rho(\mbox{$\langle{x_{2}}\rangle$})$
(see also [7, Lem. 11] for $q$-matroids), and the general case follows
similarly via induction.

######  Example 3.6.

Let ${\mathbb{F}}={\mathbb{F}}_{2}$. Consider the primitive polynomial
$f=x^{4}+x+1\in{\mathbb{F}}[x]$ and let $U=\Delta_{f}^{5}$, where $\Delta_{f}$
is defined as in (2.2). Thus $U\in{\rm GL}_{4}({\mathbb{F}})$. Let

$A_{1}=\begin{pmatrix}0&1&1&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ 1&0&0&1\\\ 0&0&1&1\\\
1&0&0&1\end{pmatrix},\quad A_{2}=\begin{pmatrix}1&0&0&0\\\ 0&0&0&0\\\
1&0&0&0\\\ 1&0&0&1\\\ 0&1&1&1\\\ 0&0&0&1\end{pmatrix},\quad
A_{3}=\begin{pmatrix}1&1&0&1\\\ 0&0&1&0\\\ 0&0&1&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
1&0&1&0\end{pmatrix}$

and define
${\mathcal{C}}=\mbox{$\langle{A_{1},\,A_{2},\,A_{3},\,A_{1}U,\,A_{2}U,\,A_{3}U}\rangle$}$.
Then ${\mathcal{C}}$ is a right ${\mathbb{F}}_{2^{2}}$-linear rank-metric code
of dimension $6$ and rank distance $3$ (see 2.13 for right
${\mathbb{F}}_{2^{2}}$-linearity). The principal denominator of
${\mathcal{M}}_{\rm c}({\mathcal{C}})$ is $\mu=2$. There exist 497
$\mu$-circuits, one of which has dimension $1$ and all others have dimension
$4$. An additional 169 spaces $V$ satisfy $\mu\rho_{\rm c}(V)=\dim V-1$ (all
of them have dimension $2,3$, or $4$), and 97 of them also satisfy
$\mu\rho_{\rm c}(W)=\dim V-1$ for all its hyperplanes $W$.

Independence behaves well under scaling-equivalence if the denominator is
taken into account.

######  Remark 3.7.

Let $\dim E_{1}=\dim E_{2}$ and ${\mathcal{M}}_{i}=(E_{i},\rho_{i}),i=1,2$, be
$q$-PMs with principal denominators $\mu_{i}$. Suppose ${\mathcal{M}}_{1}$ and
${\mathcal{M}}_{2}$ are scaling-equivalent, say
$\rho_{2}(\alpha(V))=a\rho_{1}(V)$ for all $V\in{\mathcal{V}}(E_{1})$, where
$a\in{\mathbb{Q}}_{>0}$ and $\alpha:E_{1}\longrightarrow E_{2}$ an
isomorphism. Then
$a^{-1}\mu_{1}\rho_{2}(\alpha(V))=\mu_{1}\rho_{1}(V)\in{\mathbb{N}}$ and thus
$a^{-1}\mu_{1}$ is a denominator of ${\mathcal{M}}_{2}$. Hence
$a^{-1}\mu_{1}=k\mu_{2}$ for some $k\in{\mathbb{N}}$; see Remark 2.2.
Similarly, $a\mu_{2}=\hat{k}\mu_{1}$ for some $\hat{k}\in{\mathbb{N}}$. Thus
$k=\hat{k}=1$, and hence $a\mu_{2}=\mu_{1}$. Now we have
$\mu_{2}\rho_{2}(\alpha(V))=\mu_{1}\rho_{1}(V)$ for all $V\in{\mathcal{V}}(E)$
and therefore

$\displaystyle V\text{ is $\mu_{1}$-independent in ${\mathcal{M}}_{1}$}$
$\displaystyle\Longleftrightarrow\mu_{1}\rho_{1}(W)\geq\dim W\text{ for all
subspaces }W\leq V$
$\displaystyle\Longleftrightarrow\mu_{2}\rho_{2}(\alpha(W))\geq\dim\alpha(W)\text{
for all subspaces }\alpha(W)\leq\alpha(V)$
$\displaystyle\Longleftrightarrow\alpha(V)\text{ is $\mu_{2}$-independent in
${\mathcal{M}}_{2}$}.$

Before we continue with our study of independent spaces, we briefly focus on
$q$-PMs induced by rank-metric codes and discuss the relation of
(in-)dependent spaces and code properties. Obviously, the relation depends on
the chosen denominator.

######  Remark 3.8.

Let ${\mathcal{C}}\leq\mbox{${\mathbb{F}}^{n\times m}$}$ be a rank-metric code
with column polymatroid ${\mathcal{M}}=({\mathbb{F}}^{n},\rho_{\rm c})$.

  * (a)

The most obvious information about the code is contained in 2.10: the distance
$d^{\perp}$ of the dual code is the largest integer $\ell$ for which all
$\ell$-dimensional subspaces $V$ satisfy $\rho_{\rm c}(V)=\dim V$. From 3.2 we
know that all these subspaces are $\mu$-independent for every denominator
$\mu$ of ${\mathcal{M}}$.

  * (b)

3.1 shows that the condition for independence relaxes with increasing
denominator. For this reason there may be few dependent spaces if the
principal denominator of ${\mathcal{M}}$ is $m$. To make this more precise,
let us consider the circuits for a given denominator $\mu$. From 3.5(c) we
know that if $V$ is a $\mu$-circuit of dimension $v$, then $\mu\rho_{\rm
c}(V)=v-1$. The definition of $\rho_{\rm c}$ thus implies

$V\text{ is a $\mu$-circuit}\Longrightarrow\dim{\mathcal{C}}(V^{\perp},{\rm
c})=\dim{\mathcal{C}}-\frac{mv}{\mu}+1$

(recall from 2.2 that $\mu$ is a divisor of $m$). The right hand side means
that ${\mathcal{C}}$ is equivalent to a code

$\tilde{{\mathcal{C}}}=\bigg{\langle}\begin{pmatrix}A_{1,1}\\\
0\end{pmatrix},\ldots,\begin{pmatrix}A_{1,k-\gamma+1}\\\
0\end{pmatrix},\begin{pmatrix}A_{1,k-\gamma}\\\
A_{2,k-\gamma}\end{pmatrix},\ldots,\begin{pmatrix}A_{1,k}\\\
A_{2,k}\end{pmatrix}\bigg{\rangle},$

where $k=\dim{\mathcal{C}},\,\gamma=mv/\mu$, and
$A_{1,j}\in{\mathbb{F}}^{(n-v)\times m},\,A_{2,j}\in{\mathbb{F}}^{v\times m}$.
In particular, if ${\mathcal{M}}$ contains an $m$-loop, then $\gamma=1$ and
${\mathcal{C}}$ is equivalent to a row degenerate code: the last row of all
matrices in $\tilde{{\mathcal{C}}}$ is zero. More generally, for any fixed
dimension $v$, the existence of a $v$-dimensional $\mu$-circuit becomes more
restrictive with increasing $\mu$.

We now return to general $q$-PMs. In order to derive the main result about the
collection of $\mu$-independent spaces, we will make use of an auxiliary
$q$-matroid. The following construction mimics the corresponding one in [8,
Prop. 11.1.7] for classical polymatroids.

######  Theorem 3.9.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$. Define the
map

$r_{\rho,\mu}:{\mathcal{V}}(E)\longrightarrow{\mathbb{N}}_{0},\quad
V\longmapsto\min\\{\mu\rho(W)+\dim V-\dim W\mid W\leq V\\}.$

Then ${\mathcal{Z}}:={\mathcal{Z}}_{{\mathcal{M}},\mu}:=(E,r_{\rho,\mu})$ is a
$q$-matroid, and the independent spaces of ${\mathcal{Z}}$ coincide with the
$\mu$-independent spaces of ${\mathcal{M}}$, i.e.,

${\mathcal{I}}_{\mu}({\mathcal{M}})={\mathcal{I}}({\mathcal{Z}})=\\{I\in{\mathcal{V}}(E)\mid
r_{\rho,\mu}(I)=\dim I\\}.$

###### Proof.

We make use of the induced integer $\rho$-function $\tau=\mu\rho$. Thus
$\tau(V)=\mu\rho(V)\leq\mu\dim V$ for all $V\in{\mathcal{V}}(E)$. Clearly the
map $r:=r_{\rho,\mu}$ takes integer values. We now verify (R1) – (R3) of 2.1
for $r$.
(R1) Clearly $r(V)\geq 0$ for all $V$. Furthermore,
$r(V)\leq\tau(0)+\dim(V)-\dim(0)=\dim V$.
(R2) Let $V\leq V^{\prime}$. Without loss of generality we may assume $\dim
V^{\prime}=\dim V+1$ and thus $V^{\prime}=V\oplus\mbox{$\langle{x}\rangle$}$
for some $x\in E$. Assume by contradiction that $r(V)>r(V^{\prime})$. Then
there exists $W^{\prime}\leq V^{\prime}$ such that

$\tau(W^{\prime})+\dim V^{\prime}-\dim W^{\prime}<\tau(W)+\dim V-\dim W\
\text{ for all }W\leq V.$ (3.2)

Clearly $W^{\prime}\not\leq V$ and thus we may write
$W^{\prime}=X\oplus\mbox{$\langle{y}\rangle$}$ for some $X\leq V$ and
$y\not\in V$. Then $\dim X=\dim V-\dim V^{\prime}+\dim W^{\prime}$ and (3.2)
leads to

$\tau(W)\\!-\\!\dim W>\tau(W^{\prime})\\!+\\!\dim V^{\prime}\\!-\\!\dim
W^{\prime}\\!-\\!\dim V=\tau(W^{\prime})\\!-\\!\dim X\text{ for all }W\leq V.$

Choosing $W=X$, we arrive at $\tau(X)>\tau(W^{\prime})$ and thus
$\rho(X)>\rho(W^{\prime})$. Since $X\leq W^{\prime}$ this contradicts that
$\rho$ is a rank function. All of this establishes (R2) for the map $r$.
(R3) Let $V,V^{\prime}\in{\mathcal{V}}(E)$. Choose $W\leq V,\,W^{\prime}\leq
V^{\prime}$ such that

$r(V)=\tau(W)+\dim V-\dim W\ \text{ and }\ r(V^{\prime})=\tau(W^{\prime})+\dim
V^{\prime}-\dim W^{\prime}.$

Then $W+W^{\prime}\leq V+V^{\prime}$ and $W\cap W^{\prime}\leq V\cap
V^{\prime}$ and therefore

$\displaystyle r(V\\!+\\!V^{\prime})+r(V\cap V^{\prime})$
$\displaystyle\leq\tau(W+W^{\prime})+\dim(V+V^{\prime})-\dim(W+W^{\prime})$
$\displaystyle\quad\ +\tau(W\cap W^{\prime})+\dim(V\cap V^{\prime})-\dim(W\cap
W^{\prime})$
$\displaystyle=\tau(W\\!+\\!W^{\prime})+\tau(W\\!\cap\\!W^{\prime})+\dim
V-\dim W+\dim V^{\prime}-\dim W^{\prime}$
$\displaystyle\leq\tau(W)+\tau(W^{\prime})+\dim V-\dim W+\dim V^{\prime}-\dim
W^{\prime}$ $\displaystyle=r(V)+r(V^{\prime}),$

where the second inequality follows from (R3) for $\rho$. This establishes
(R3) for the map $r$.
It remains to investigate the $\mu$-independent spaces. From 3.1, (R1), and
3.2 we obtain immediately

$V$ is $\mu$-independent $\displaystyle\Longleftrightarrow\tau(W)\geq\dim
W\text{ for all }W\leq V$ $\displaystyle\Longleftrightarrow\tau(W)+\dim V-\dim
W\geq\dim V\text{ for all }W\leq V$ $\displaystyle\Longleftrightarrow
r(V)\geq\dim V$ $\displaystyle\Longleftrightarrow r(V)=\dim V,$

and this establishes the stated result. ∎

As the next example shows, if ${\mathcal{M}}$ is a $q$-matroid, then it
coincides with its auxiliary $q$-matroid if we choose the principal
denominator $\mu=1$.

######  Example 3.10.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-matroid, thus $\rho$ takes only integer
values.

  * (a)

Fix $\mu=1$. Then $r_{\rho,1}(V)=\min\\{\rho(W)+\dim V-\dim W\mid W\leq V\\}$
for $V\in{\mathcal{V}}(E)$. We show now that $r_{\rho,1}=\rho$. Fix
$V\in{\mathcal{V}}(E)$. Choosing $W=V$ we obtain $r_{\rho,1}(V)\leq\rho(V)$.
On the other hand, for every $W\leq V$ there exists $Z\leq V$ such that
$W\oplus Z=V$. Thus submodularity (R3) applied to $\rho$ yields

$\rho(V)=\rho(W+Z)\leq\rho(W)+\rho(Z)\leq\rho(W)+\dim Z=\rho(W)+\dim V-\dim
W.$

This shows $\rho(V)\leq r_{\rho,1}(V)$.

  * (b)

If we choose $\mu>1$, then there is in general no obvious relation between
$\rho$ and $r_{\rho,\mu}$. Consider for example the following $q$-matroid. Let
$n\geq 3$ and fix a $2$-dimensional subspace
$X\in{\mathcal{V}}({\mathbb{F}}^{n})$. Set $\rho(X)=1$ and
$\rho(V)=\min\\{\dim V,\,2\\}$ for $V\neq X$. One can check straightforwardly
that ${\mathcal{M}}=({\mathbb{F}}^{n},\rho)$ is a $q$-matroid (this also
follows from [5, Prop. 4.7]). Choosing $\mu=2$, one obtains
$r_{\rho,2}=\min\\{\dim V,4\\}$, and thus the $q$-matroids ${\mathcal{M}}$ and
${\mathcal{Z}}_{{\mathcal{M}},2}$ are not equivalent. Furthermore,
${\mathcal{Z}}_{{\mathcal{M}},2}={\mathcal{Z}}_{{\mathcal{U}},2}$, where
${\mathcal{U}}$ is the uniform $q$-matroid
${\mathcal{U}}_{2}({\mathbb{F}}^{n})$, and therefore the auxiliary $q$-matroid
${\mathcal{Z}}_{{\mathcal{M}},\mu}$ of a $q$-PM ${\mathcal{M}}$ does not
uniquely determine ${\mathcal{M}}$ (even if one specifies the denominator).

3.9 shows that the $\mu$-independent spaces of the $q$-PM ${\mathcal{M}}$
coincide with the independent spaces of the auxiliary $q$-matroid
${\mathcal{Z}}_{{\mathcal{M}},\mu}$. Therefore, all properties of independent
spaces of $q$-matroids that do not involve the value of the rank function hold
true for $q$-PMs as well. Such properties have been derived in [7, Thm. 8].
Before formulating our result we cast the following important notions.

######  Definition 3.11.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$. For
$V\in{\mathcal{V}}(E)$ we define

${\mathcal{I}}_{\mu}(V)=\\{I\in{\mathcal{I}}_{\mu}({\mathcal{M}})\mid I\leq
V\\}.$

A subspace $\hat{I}\in{\mathcal{I}}_{\mu}(V)$ is said to be a _$\mu$ -basis_
of $V$ if $\dim\hat{I}=\max\\{\dim I\mid I\in{\mathcal{I}}_{\mu}(V)\\}$. We
denote by ${\mathcal{B}}_{\mu}(V)$ the set of all $\mu$-bases of $V$. The
$\mu$-bases of $E$ are called the _$\mu$ -bases_ of ${\mathcal{M}}$.

A $\mu$-basis of $V$ is thus a maximal-dimensional $\mu$-independent subspace
of $V$. Their rank values will be discussed in the next section. Note that the
sets ${\mathcal{I}}_{\mu}(V)$ and ${\mathcal{B}}_{\mu}(V)$ are non-empty for
every $V\in{\mathcal{V}}(E)$ since clearly $\\{0\\}$ is $\mu$-independent.

We are now ready to present the following properties of the collection of
$\mu$-independent spaces of a $q$-PM. The result is an immediate consequence
of 3.9 together with [7, Thm. 8].

######  Corollary 3.12.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$ and set
${\mathcal{I}}_{\mu}:={\mathcal{I}}_{\mu}({\mathcal{M}})$. Then

  * (I1)

${\mathcal{I}}_{\mu}\neq\emptyset$, in fact $\\{0\\}\in{\mathcal{I}}_{\mu}$.

  * (I2)

If $I\in{\mathcal{I}}_{\mu}$ and $J\leq I$, then $J\in{\mathcal{I}}_{\mu}$.

  * (I3)

If $I,\,J\in{\mathcal{I}}_{\mu}$ and $\dim I<\dim J$, then there exists $x\in
J\setminus I$ such that
$I\oplus\mbox{$\langle{x}\rangle$}\in{\mathcal{I}}_{\mu}$.

  * (I4)

Let $V,\,W\in{\mathcal{V}}(E)$ and $I\leq V,\,J\leq W$ be $\mu$-bases of $V$
and $W$, respectively. Then there exists a $\mu$-basis of $V+W$ that is
contained in $I+J$.

Note that (I3) implies that for any $V\in{\mathcal{V}}(E)$ and any
$\hat{I}\in{\mathcal{I}}_{\mu}(V)$ we have

$\hat{I}\text{ is dimension-maximal in
}{\mathcal{I}}_{\mu}(V)\Longleftrightarrow\hat{I}\text{ is inclusion-maximal
in }{\mathcal{I}}_{\mu}(V).$

In other words, $\hat{I}$ is a $\mu$-basis of $V$ if and only if there exists
no $J\in{\mathcal{I}}_{\mu}(V)$ such that $\hat{I}\lneq J$. Thus, the
$\mu$-bases of $V$ are exactly the maximal elements of the poset
$({\mathcal{I}}_{\mu}(V),\leq)$.

Since the independent spaces of the auxiliary $q$-matroid
${\mathcal{Z}}_{{\mathcal{M}},\mu}$ coincide with those of the $q$-PM
${\mathcal{M}}$, the same is true for the dependent spaces, circuits, and
bases. As a consequence, any property about the collection of these spaces in
$q$-matroids holds true for $q$-PMs as well. Let us illustrate this for the
dependent spaces and bases. The following properties for $q$-matroids have
been established in [3, Thm. 63 and Lem. 66] and therefore apply to $q$-PMs as
well.

######  Corollary 3.13.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$. Let
${\mathcal{D}}_{\mu}$ and ${\mathcal{B}}_{\mu}$ be the collection of
$\mu$-dependent spaces and $\mu$-bases of ${\mathcal{M}}$, respectively. Then
${\mathcal{D}}_{\mu}$ and ${\mathcal{B}}_{\mu}$ satisfy

  * (D1)

$\\{0\\}\not\in{\mathcal{D}}_{\mu}$.

  * (D2)

If $D_{1}\in{\mathcal{D}}_{\mu}$ and $D_{2}\in{\mathcal{V}}(E)$ such that
$D_{1}\subseteq D_{2}$, then $D_{2}\in{\mathcal{D}}_{\mu}$.

  * (D2)

Let $D_{1},\,D_{2}\in{\mathcal{D}}_{\mu}$ be such that $D_{1}\cap
D_{2}\not\in{\mathcal{D}}_{\mu}$. Then every subspace of $D_{1}+D_{2}$ of
codimension $1$ is in ${\mathcal{D}}_{\mu}$.

  * (B1)

${\mathcal{B}}_{\mu}\neq\emptyset$.

  * (B2)

Let $B_{1},\,B_{2}\in{\mathcal{B}}_{\mu}$ be such that $B_{1}\leq B_{2}$. Then
$B_{1}=B_{2}$.

  * (B3)

Let $B_{1},\,B_{2}\in{\mathcal{B}}_{\mu}$ and $A$ be a subspace of $B_{1}$ of
codimension $1$ such that $B_{1}\cap B_{2}\leq A$. Then there exists a
$1$-dimensional subspace $Y$ of $B_{2}$ such that $A+Y\in{\mathcal{B}}_{\mu}$.

  * (B4)

Let $A_{1},\,A_{2}\in{\mathcal{V}}(E)$ and $I_{1},\,I_{2}$ be maximal
dimensional intersections of some members of ${\mathcal{B}}_{\mu}$ with
$A_{1}$ and $A_{2}$, respectively. Then there exist a maximal dimensional
intersection of a member of ${\mathcal{B}}_{\mu}$ with $A_{1}+A_{2}$ that is
contained in $I_{1}+I_{2}$.

In [3, Cor. 65] and [7, Thm. 37] it has been shown that any collection of
subspaces satisfying (D1)–(D3) (resp. (B1)–(B4)) is the collection of
dependent spaces (resp. bases) of a unique $q$-matroid. Similar statements
hold true for circuits in $q$-matroids (see [3, Cor. 68]). The following
examples illustrate that none of these properties extends to $q$-PMs – even if
we take the rank values into account.

######  Example 3.14.

  * (a)

Consider the rank-metric codes ${\mathcal{C}}_{1},\,{\mathcal{C}}_{2}$ in 2.16
and the associated column polymatroids ${\mathcal{M}}_{\rm
c}({\mathcal{C}}_{1})$ and ${\mathcal{M}}_{\rm c}({\mathcal{C}}_{2})$. Both
have principal denominator $2$, and in both cases every subspace of
${\mathbb{F}}^{5}$ is $2$-independent. Thus the only $2$-basis of
${\mathcal{M}}_{\rm c}({\mathcal{C}}_{i})$ is ${\mathbb{F}}^{5}$ for $i=1,2$,
and in both $q$-PMs it has rank value $5/2$. Yet, the two $q$-PMs are not
equivalent. This shows that the bases of a $q$-PM along with their rank values
do not uniquely determine the $q$-PM. Trivially, this example also shows that
the circuits and dependent spaces along with their rank values do not
determine the $q$-PM. For later purposes we also note that in both
${\mathcal{M}}_{\rm c}({\mathcal{C}}_{1})$ and ${\mathcal{M}}_{\rm
c}({\mathcal{C}}_{2})$ all 4-dimensional subspaces and plenty of 3-dimensional
subspaces have rank value $5/2$ as well.

  * (b)

Let ${\mathbb{F}}={\mathbb{F}}_{2}$ and consider the codes
${\mathcal{C}}=\mbox{$\langle{A_{1},A_{2},A_{3}}\rangle$},\,{\mathcal{C}}^{\prime}=\mbox{$\langle{A_{1},A_{2},A_{3}^{\prime}}\rangle$}\leq{\mathbb{F}}^{4\times
3}$, where

$A_{1}=\begin{pmatrix}0&0&0\\\ 1&0&0\\\ 0&1&1\\\ 0&1&0\end{pmatrix},\quad
A_{2}=\begin{pmatrix}1&0&1\\\ 1&0&0\\\ 0&0&0\\\ 1&1&1\end{pmatrix},\quad
A_{3}=\begin{pmatrix}0&1&0\\\ 0&1&1\\\ 0&1&0\\\ 0&1&1\end{pmatrix},\quad
A_{3}^{\prime}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\ 1&0&1\\\ 1&0&1\end{pmatrix}.$

Consider the associated polymatroids ${\mathcal{M}}={\mathcal{M}}_{\rm
c}({\mathcal{C}})=({\mathbb{F}}^{4},\rho_{\rm c})$ and
${\mathcal{M}}^{\prime}={\mathcal{M}}_{\rm
c}({\mathcal{C}}^{\prime})=({\mathbb{F}}^{4},\rho_{\rm c}^{\prime})$. Both
have principal denominator $3$, and in both $q$-PMs the space
${\mathbb{F}}^{4}$ is the only dependent space. Hence ${\mathcal{M}}$ and
${\mathcal{M}}^{\prime}$ share the same bases, namely all $3$-dimensional
spaces. Moreover, $\rho_{\rm c}(V)=1=\rho_{\rm c}^{\prime}(V)$ for all bases
$V$. Yet, ${\mathcal{M}}$ and ${\mathcal{M}}^{\prime}$ are not equivalent: in
${\mathcal{M}}$ the rank value $1$ is assumed by 33 subspaces of dimension
$2$, whereas in ${\mathcal{M}}^{\prime}$ it is assumed by 32 subspaces of
dimension $2$ (in both $q$-PMs 4 subspaces of dimension $1$ have rank value
$1$ as well).

On the positive side, in the next section we will show that we can fully
recover a $q$-PM from its independent spaces and their rank values. Recall
from 3.4(c) that the independent spaces alone (without their rank values) do
not uniquely determine the $q$-PM.

##  4 The Rank Function on Independent Spaces

We begin by showing that for a $q$-PM the rank function is fully determined by
its values on the independent spaces. We then go on to prove that all bases of
a given subspace have the same rank value, and this value coincides with the
rank value of the subspace. This result allows us to investigate whether a
collection of spaces satisfying (I1)–(I4) from 3.12 gives rise to a $q$-PM
whose collection of independent spaces is exactly the initial collection.
Since the rank value of independent spaces in a $q$-PM is not as rigid as in a
$q$-matroid, we also need to specify a meaningful rank function on the
collection of spaces. All of this results in Theorems 4.4 and 4.6.

######  Theorem 4.1.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$. Then

$\rho(V)=\max\\{\rho(I)\mid I\in{\mathcal{I}}_{\mu}(V)\\}\text{ for all
}V\in{\mathcal{V}}(E).$

###### Proof.

Let $V\in{\mathcal{V}}(E)$. Set $\rho^{\prime}(V)=\max\\{\rho(I)\mid
I\in{\mathcal{I}}_{\mu}(V)\\}$. Thanks to (R2), $\rho^{\prime}(V)\leq\rho(V)$,
and it remains to establish $\rho(V)\leq\rho^{\prime}(V)$. Let
$\hat{I}\in{\mathcal{I}}_{\mu}(V)$ be of maximal possible dimension such that
$\rho(\hat{I})=\rho^{\prime}(V)$. If $V$ is $\mu$-independent, then
$\hat{I}=V$ and we are done. Thus let $V$ be $\mu$-dependent.
Case 1: $\dim\hat{I}=\dim V-1$.
Then $V=\hat{I}\oplus\mbox{$\langle{x}\rangle$}$ for any $x\in
V\setminus\hat{I}$ and submodularity of $\rho$ implies
$\rho(V)\leq\rho(\hat{I})+\rho(\mbox{$\langle{x}\rangle$})$. As before, we use
the integer $\rho$-function $\tau=\mu\rho$. Let $s$ be minimal such that there
exists an $s$-dimensional $\mu$-dependent subspace of $V$, say $W$. Such space
exists by $\mu$-dependence of $V$. Then 3.1 implies that $\tau(W)<\dim W$. By
(I2) $W$ is not contained in $\hat{I}$ and thus
$W=(W\cap\hat{I})\oplus\mbox{$\langle{z}\rangle$}$ for some $z\in V\setminus
0$. Then $W\cap\hat{I}$ is $\mu$-independent and

$\dim W-1=\dim(W\cap\hat{I})\leq\tau(W\cap\hat{I})\leq\tau(W)<\dim W,$

and hence $\tau(W\cap\hat{I})=\tau(W)$ because $\tau$ takes integer values.
Using that $V=W+\hat{I}$, we obtain by submodularity of $\tau$

$\tau(V)\leq\tau(W)+\tau(\hat{I})-\tau(W\cap\hat{I})=\tau(\hat{I})=\mu\rho^{\prime}(V).$

All of this shows that $\rho(V)=\rho^{\prime}(V)$, as desired.
Case 2: $\dim\hat{I}<\dim V-1$.
Let $x\in V\setminus\hat{I}$. Using that
$\rho^{\prime}(W)\leq\rho^{\prime}(Z)$ for any subspaces $W,Z$ such that
$W\leq Z$, we obtain

$\rho(\hat{I})=\rho^{\prime}(\hat{I})\leq\rho^{\prime}(\hat{I}\oplus\mbox{$\langle{x}\rangle$})\leq\rho^{\prime}(V)=\rho(\hat{I}),$

and hence $\rho(\hat{I})=\rho^{\prime}(W)$, where
$W:=\hat{I}\oplus\mbox{$\langle{x}\rangle$}$. Note that $W$ is $\mu$-dependent
thanks to the maximality of $\hat{I}$. Furthermore, $\dim\hat{I}=\dim W-1$.
Therefore Case 1 yields $\rho^{\prime}(W)=\rho(W)$. Now we arrived at
$\rho(\hat{I})=\rho(\hat{I}+\mbox{$\langle{x}\rangle$})$ for all $x\in V$, and
2.3(a) tells us that $\rho(\hat{I})=\rho(V)$. Since
$\rho(\hat{I})=\rho^{\prime}(V)$, this concludes the proof. ∎

3.12 and 4.1 generalize one direction of [7, Thm. 8] where the same properties
are proven for the independent spaces of $q$-matroids. Our next goal is to
generalize the other direction of [7, Thm. 8], namely to characterize the
collections of spaces along with given rank values that give rise to a $q$-PM
having those spaces as independent spaces. The following result will be
crucial. It shows that the rank value of any $\mu$-basis of a subspace $V$
equals the rank value of $V$. Recall from 3.14 that the converse is not true:
not every $I\in{\mathcal{I}}_{\mu}(V)$ such that $\rho(I)=\rho(V)$ is a
$\mu$-basis of $V$.

######  Theorem 4.2.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$. Let
$V\in{\mathcal{V}}(E)$. Then

$\rho(I)=\rho(V)\text{ for all }I\in{\mathcal{B}}_{\mu}(V).$

In particular, all $\mu$-bases of $V$ have the same rank value.

###### Proof.

Throughout the proof we will omit the subscript $\mu$. The result is clearly
true if $V$ is independent. Thus, let $V$ be dependent. Set $t=\dim V$. In
order to avoid denominators we use again the integer $\rho$-function
$\tau:=\mu\rho$. First of all, there exists

$J\in{\mathcal{B}}(V)\text{ such that }\tau(J)=\tau(V).$ (4.1)

Indeed, by 4.1 there exists $J\in{\mathcal{I}}(V)$ such that
$\tau(J)=\tau(V)$, and by Property (I2) along with the monotonicity of $\tau$
we may assume that $J\in{\mathcal{B}}(V)$. Note that by 3.11 all spaces in
${\mathcal{B}}(V)$ have the same dimension, which we denote by $s$.
Case 1: $s=t-1$. Let $I\in{\mathcal{B}}(V)$. We want to show that
$\tau(I)=\tau(V)$. Choose a circuit, say $C$, in $V$. Then $\tau(C)=\dim C-1$
(see 3.5(c)). Clearly, $C\not\subseteq I$ by Property (I2) and thus $C+I=V$
thanks to $\dim I=\dim V-1$. Furthermore, $C\cap I$ is independent, being a
subspace of $I$, and thus $\tau(C\cap I)\geq\dim(C\cap I)$. Using
submodularity, we obtain

$\displaystyle\tau(V)=\tau(C+I)$ $\displaystyle\leq\tau(C)+\tau(I)-\tau(C\cap
I)$ $\displaystyle\leq\dim C-1+\tau(I)-\dim(C\cap I)$
$\displaystyle=\tau(I)+\dim(C+I)-(\dim I+1)$ $\displaystyle=\tau(I),$

where the last step follows from $C+I=V$ and $\dim I+1=\dim V$. All of this
implies $\tau(I)=\tau(V)$, and thus all bases of $V$ have the same rank value.
Case 2: $s<t-1$. We will show that

$\tau(I)=\tau(J)\text{ for all }I\in{\mathcal{B}}(V),$ (4.2)

where $J$ is as in (4.1). We induct on the codimension of $I\cap J$ in $I$.
Let $\dim(I\cap J)=s-r$, thus $0\leq r\leq s$. The case $r=0$ is trivial.
i) Let $r=1$. Then $I=(I\cap J)\oplus\mbox{$\langle{x}\rangle$}$ for some
$x\in I\setminus J$. Set $W=J\oplus\mbox{$\langle{x}\rangle$}$. Then $W\leq V$
and $\dim W=\dim J+1$. Thus $W$ is dependent by maximality of $J$. Hence $I$
and $J$ are elements of ${\mathcal{B}}(W)$, and Case 1 implies
$\tau(I)=\tau(J)$.
ii) Assume now $\tau(I)=\tau(J)$ for all $I\in{\mathcal{B}}(V)$ such that
$\dim(I\cap J)\geq s-(r-1)$ for some $r\geq 2$. Let $I\in{\mathcal{B}}(V)$ be
such that $\dim(I\cap J)=s-r$. Choose $K\leq I$ and $x\in I\setminus J$ such
that $I=(I\cap J)\oplus K\oplus\mbox{$\langle{x}\rangle$}$ and set
$I_{1}=(I\cap J)\oplus K$. Then $I_{1}$ is independent and $\dim I_{1}=\dim
I-1=\dim J-1$. Thanks to Property (I3) there exists $y\in J\setminus I_{1}$
such that

$I^{\prime}:=I_{1}\oplus\mbox{$\langle{y}\rangle$}\in{\mathcal{B}}(V).$

Now we have three bases, $I^{\prime},\,I,\,J$, of $V$. We show first
$\tau(I)=\tau(I^{\prime})$. Since $y\not\in I$ we have the subspace
$W:=I\oplus\mbox{$\langle{y}\rangle$}$ of $V$, which must be dependent due to
maximality of $I$. Furthermore, $I,\,I^{\prime}\leq W$ and $\dim
I^{\prime}=\dim I=\dim W-1$, and therefore $\tau(I)=\tau(I^{\prime})$ thanks
to Case 1. Next, we show $\tau(I^{\prime})=\tau(J)$. In order to do so, note
that $I^{\prime}=(I\cap J)\oplus K\oplus\mbox{$\langle{y}\rangle$}$, where
$y\in J$. Thus $\dim(I^{\prime}\cap J)\geq s-(r-1)$ and the induction
hypothesis yields $\tau(I^{\prime})=\tau(J)$. All of this establishes (4.2)
and concludes the proof. ∎

######  Remark 4.3.

In a $q$-matroid ${\mathcal{M}}=(E,\rho)$ a subspace $V\in{\mathcal{V}}(E)$
satisfies

$\text{$V$is independent and }\rho(V)=\rho(E)\Longleftrightarrow V\text{ is a
basis of }{\mathcal{M}}.$

The forward direction is in fact the definition of independence in [7, Def.
2]. Thanks to 4.2 the implication “$\Longleftarrow$” holds true for $q$-PMs as
well. However, “$\Longrightarrow$” is not true as the $q$-PMs in 3.14 show.

We are now ready to provide a characterization of the pairs
$({\mathcal{I}},\tilde{\rho})$ of collections ${\mathcal{I}}$ of subspaces and
rank functions $\tilde{\rho}$ on ${\mathcal{I}}$ that give rise to a $q$-PM
whose collection of independent spaces is ${\mathcal{I}}$ and whose rank
function restricts to $\tilde{\rho}$. Clearly, ${\mathcal{I}}$ has to satisfy
(I1)–(I4) from 3.12, and $\tilde{\rho}$ must satisfy (R1)–(R3). However, for
independence we also need the rank condition from 3.1. This leads to (R1′) in
4.4 below. Furthermore, since the sum of independent spaces need not be
independent, we have to adjust (R3) and replace $\tilde{\rho}(I+J)$ by
$\max\\{\tilde{\rho}(K)\mid K\in{\mathcal{I}},\,K\leq I+J\\}$, thereby
accounting for 4.1. This results in the submodularity condition (R3′) below.
Since one can easily find examples showing that (R1′)–(R3′) are not sufficient
to guarantee submodularity of the extended rank function (defined in (4.3)),
we also have to enforce 4.2. This leads to condition (R4′), which states that
for any space $V$ all maximal subspaces that are contained in ${\mathcal{I}}$
have the same rank value. As we will see, all these conditions together
guarantee submodularity of the extended rank function, and the spaces in
${\mathcal{I}}$ are independent in the resulting $q$-PM. However, the $q$-PM
may have additional independent subspaces; see 4.5 below. In order to prevent
this, we need a natural closure property. This will be spelled out in 4.6.

######  Theorem 4.4.

Let ${\mathcal{I}}$ be a subset of ${\mathcal{V}}(E)$. For
$V\in{\mathcal{V}}(E)$ set ${\mathcal{I}}(V)=\\{I\in{\mathcal{I}}\mid I\leq
V\\}$ and denote by $\mbox{${\mathcal{I}}_{\text{max}}$}(V)$ the set of
subspaces in ${\mathcal{I}}(V)$ of maximal dimension. Suppose ${\mathcal{I}}$
satisfies the following.

  * (I1)

$\\{0\\}\in{\mathcal{I}}$.

  * (I2)

If $I\in{\mathcal{I}}$ and $J\leq I$, then $J\in{\mathcal{I}}$.

  * (I3)

If $I,\,J\in{\mathcal{I}}$ and $\dim I<\dim J$, then there exists $x\in
J\setminus I$ such that $I\oplus\mbox{$\langle{x}\rangle$}\in{\mathcal{I}}$.

  * (I4)

Let $V,\,W\in{\mathcal{V}}(E)$ and
$I\in\mbox{${\mathcal{I}}_{\text{max}}$}(V),\,J\in\mbox{${\mathcal{I}}_{\text{max}}$}(W)$.
Then there exists a space $K\in\mbox{${\mathcal{I}}_{\text{max}}$}(V+W)$ that
is contained in $I+J$.

Furthermore, let $\tilde{\rho}:{\mathcal{I}}\longrightarrow{\mathbb{Q}}$ and
$\mu\in{\mathbb{Q}}_{>0}$ such that $\mu\tilde{\rho}(I)\in{\mathbb{Z}}$ for
all $I\in{\mathcal{I}}$. Suppose $\tilde{\rho}$ satisfies the following.

  * (R1′)

$0\leq\mu^{-1}\dim I\leq\tilde{\rho}(I)\leq\dim I$ for all
$I\in{\mathcal{I}}$.

  * (R2′)

If $I,J\in{\mathcal{I}}$ such that $I\leq J$, then
$\tilde{\rho}(I)\leq\tilde{\rho}(J)$.

  * (R3′)

For all $I,J\in{\mathcal{I}}$ we have $\max\\{\tilde{\rho}(K)\mid
K\in{\mathcal{I}}(I+J)\\}+\tilde{\rho}(I\cap
J)\leq\tilde{\rho}(I)+\tilde{\rho}(J)$.

  * (R4′)

For all $V\in{\mathcal{V}}(E)$ and
$I,J\in\mbox{${\mathcal{I}}_{\text{max}}$}(V)$ we have
$\tilde{\rho}(I)=\tilde{\rho}(J)$.

Define the map

$\rho:{\mathcal{V}}(E)\longrightarrow{\mathbb{Q}},\quad
V\longmapsto\max\\{\tilde{\rho}(I)\mid I\in{\mathcal{I}}(V)\\}.$ (4.3)

Then ${\mathcal{M}}=(E,\rho)$ is a $q$-PM with denominator $\mu$, and
${\mathcal{I}}\subseteq{\mathcal{I}}_{\mu}({\mathcal{M}})$.

Note that thanks to (I3) the set $\mbox{${\mathcal{I}}_{\text{max}}$}(V)$ is
the set of maximal elements in the poset $({\mathcal{I}}(V),\leq)$.
Furthermore, by (R2′) and (R4′) we have for all $V\in{\mathcal{V}}(E)$

$\rho(V)=\tilde{\rho}(I)\ \text{ for any }\
I\in\mbox{${\mathcal{I}}_{\text{max}}$}(V).$

###### Proof.

It is clear that $\mu$ is a denominator of $\rho$. We have to show that $\rho$
satisfies (R1)–(R3) from 2.1.
(R1) Let $V\in{\mathcal{V}}(E)$ and $I\in{\mathcal{I}}$ such that $I\leq V$
and $\tilde{\rho}(I)=\rho(V)$. Then $0\leq\tilde{\rho}(I)\leq\dim I\leq\dim
V$, which establishes (R1).
(R2) Let $V,W\in{\mathcal{V}}(E)$ be such that $V\leq W$. Let
$I\in{\mathcal{I}}$ be such that $I\leq V$ and $\tilde{\rho}(I)=\rho(V)$. Then
$I\leq W$ and the definition of $\rho$ implies
$\rho(W)\geq\tilde{\rho}(I)=\rho(V)$, as desired.
(R3) Let $V,W\in{\mathcal{V}}(E)$. Choose
$K\in\mbox{${\mathcal{I}}_{\text{max}}$}(V\cap W)$. Applying (I3) repeatedly,
we can find $I\in\mbox{${\mathcal{I}}_{\text{max}}$}(V)$ and
$J\in\mbox{${\mathcal{I}}_{\text{max}}$}(W)$ such that $K\leq I$ and $K\leq
J$. By (I4) there exists $H\in\mbox{${\mathcal{I}}_{\text{max}}$}(V+W)$ such
that $H\leq I+J$. Now (R4′) implies

$\tilde{\rho}(I)=\rho(V),\quad\tilde{\rho}(J)=\rho(W),\quad\tilde{\rho}(H)=\rho(I+J)=\rho(V+W),\quad\tilde{\rho}(K)=\tilde{\rho}(I\cap
J)=\rho(V\cap W).$

From (R3′) we obtain $\rho(I+J)+\tilde{\rho}(I\cap
J)\leq\tilde{\rho}(I)+\tilde{\rho}(J)$, and we finally arrive at

$\rho(V+W)+\rho(V\cap
W)=\tilde{\rho}(H)+\tilde{\rho}(K)=\rho(I+J)+\tilde{\rho}(I\cap
J)\leq\tilde{\rho}(I)+\tilde{\rho}(J)=\rho(V)+\rho(W),$

as desired. Finally, (R1′) shows that the spaces in ${\mathcal{I}}$ are
$\mu$-independent, thus
${\mathcal{I}}\subseteq{\mathcal{I}}_{\mu}({\mathcal{M}})$. ∎

The following example shows that in general the $q$-PM ${\mathcal{M}}$ from
4.4 has more independent spaces than ${\mathcal{I}}$.

######  Example 4.5.

Consider the $q$-PM ${\mathcal{M}}=({\mathbb{F}}^{3},\rho_{\rm c})$ from
3.3(a). We have seen already that
${\mathcal{I}}_{3}({\mathcal{M}})={\mathcal{V}}({\mathbb{F}}^{3})$. Define the
set ${\mathcal{I}}=\\{V\in{\mathcal{V}}({\mathbb{F}}^{3})\mid
V\neq\mbox{$\langle{e_{1}+e_{2},e_{3}}\rangle$}\text{ and
}V\neq{\mathbb{F}}^{3}\\}$ and let $\tilde{\rho}=\rho_{\rm
c}|_{{\mathcal{I}}}$. One easily verifies that $({\mathcal{I}},\tilde{\rho})$
satisfies (I1)–(I4) and (R1′)–(R4′). Furthermore, the extension $\rho$ defined
in 4.4 equals $\rho_{\rm c}$ and thus the induced $q$-PM
$({\mathbb{F}}^{3},\rho)$ equals ${\mathcal{M}}$. Now we have
${\mathcal{I}}\subsetneq{\mathcal{I}}_{3}({\mathcal{M}})$.

We can easily force equality
${\mathcal{I}}={\mathcal{I}}_{\mu}({\mathcal{M}})$ by adding the following
natural closure property.

######  Theorem 4.6.

Let the pair $({\mathcal{I}},\tilde{\rho})$ be as in 4.4. Suppose
$({\mathcal{I}},\tilde{\rho})$ satisfies (I1)–(I4) and (R1′)–(R4′) as well as
the following closure property:

  * (C)

If $V\in{\mathcal{V}}(E)$ is such that

    * (i)

all proper subspaces of $V$ are in ${\mathcal{I}}$,

    * (ii)

$\max\\{\tilde{\rho}(I)\mid I\in{\mathcal{I}}(V)\\}\geq\mu^{-1}\dim V$,

then $V$ is in ${\mathcal{I}}$.

Then ${\mathcal{I}}={\mathcal{I}}_{\mu}({\mathcal{M}})$ for the $q$-PM
${\mathcal{M}}$ from 4.4.

Note that by (I2) and (R1′), any subspace $V\in{\mathcal{I}}$ satisfies the
properties in (i) and (ii).

###### Proof.

Thanks to 4.4 it remains to show that any
$V\in{\mathcal{I}}_{\mu}({\mathcal{M}})$ is in ${\mathcal{I}}$. Recall that
$\rho(V)=\max\\{\tilde{\rho}(I)\mid I\in{\mathcal{I}}(V)\\}$. We induct on
$\dim V$.
1) Let $\dim V=1$. Then $\rho(V)\geq\mu^{-1}\dim V$ holds true by the
definition of $\mu$-independence, hence (ii) is satisfied. Property (i) is
trivially satisfied by (I1). Now (C) implies $V\in{\mathcal{I}}$.
2) Let $\dim V=r$ and assume that all subspaces
$V\in{\mathcal{I}}_{\mu}({\mathcal{M}})$ of dimension at most $r-1$ are in
${\mathcal{I}}$. Since $V\in{\mathcal{I}}_{\mu}({\mathcal{M}})$, the same is
true for all its subspaces. Hence all proper subspaces are in ${\mathcal{I}}$
by induction hypothesis. Again, $\rho(V)\geq\mu^{-1}\dim V$ is true by
$\mu$-independence and thus Property (C) implies that $V\in{\mathcal{I}}$. ∎

##  5 Minimal Spanning Spaces and Maximal Independent Spaces

By definition, the maximal independent spaces in a $q$-PM are the bases. In
this section, we introduce spanning spaces and show that – differently from
$q$-matroids – bases are not the same as minimal spanning spaces. However,
spanning spaces turn out to be the dual notion to strongly independent spaces,
which we also define in this section. This result may be regarded as the
generalization of the duality result for bases in $q$-matroids. The latter
states that for a $q$-matroid ${\mathcal{M}}$ a space $B$ is a basis of
${\mathcal{M}}$ if and only if $B^{\perp}$ is a basis of ${\mathcal{M}}^{*}$.
We show that in fact this equivalence characterizes $q$-matroids within the
class of $q$-PMs.

######  Definition 5.1.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM. A subspace $V\in{\mathcal{V}}(E)$ is
called a _spanning space_ if $\rho(V)=\rho(E)$. Furthermore, $V$ is a _minimal
spanning space_ if it is a spanning space and no proper subspace is a spanning
space.

For $q$-matroids the minimal spanning spaces are exactly the bases.

######  Proposition 5.2.

Suppose ${\mathcal{M}}=(E,\rho)$ is a $q$-matroid. A subspace is a basis if
and only if it is a minimal spanning space. As a consequence, all minimal
spanning spaces of ${\mathcal{M}}$ have the same dimension.

###### Proof.

“$\Rightarrow$” Let $V$ be a basis of ${\mathcal{M}}$. Then $\dim
V=\rho(V)=\rho(E)$. For every proper subspace $W\lneq V$ we have
$\rho(W)\leq\dim W<\dim V=\rho(V)=\rho(E)$, hence $W$ is not a spanning space.
This proves minimality of $V$. “$\Leftarrow$” Let now $V$ be a minimal
spanning space. By definition $\rho(V)=\rho(E)$, and thus it remains to show
that $V$ is independent (see 4.3). Suppose $V$ is dependent. Then there exists
a maximal independent subspace $W$ of $V$, and thanks to 4.2 we have
$\rho(W)=\rho(V)=\rho(E)$. This contradicts minimality of $V$. ∎

The last result is not true for $q$-PMs.

######  Example 5.3.

  * (a)

The $q$-PM ${\mathcal{M}}_{\rm c}({\mathcal{C}}_{1})$ from 3.14(a) has 126
minimal spanning spaces and they all have dimension $3$, whereas the bases are
the $4$-dimensional spaces.

  * (b)

The $q$-PM ${\mathcal{M}}$ from 3.14(b) has $4$ (resp. $10$) minimal spanning
spaces of dimension $1$ (resp. $2$), whereas the bases are the $3$-dimensional
subspaces.

There exist $q$-PMs that are not $q$-matroids and yet the bases coincide with
the minimal spanning spaces (for instance the $q$-PM ${\mathcal{M}}_{\rm
c}({\mathcal{C}})$ in [5, Ex. 4.2]). Thus the equivalence in 5.2 does not
characterize $q$-matroids.

The following describes the relation between bases and minimal spanning spaces
in a $q$-PM.

######  Proposition 5.4.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM with denominator $\mu$.

  * (a)

A minimal spanning space is $\mu$-independent.

  * (b)

Every $\mu$-basis of ${\mathcal{M}}$ contains a minimal spanning space and
every minimal spanning space is contained in a $\mu$-basis.

###### Proof.

(a) Let $V$ be a minimal spanning space. If $V$ is $\mu$-dependent, then $V$
contains a maximal $\mu$-independent subspace $W$, and 4.2 implies
$\rho(W)=\rho(V)=\rho(E)$. This contradicts minimality of $V$.
(b) is clear. ∎

Recall the dual $q$-PM from 2.5. Our next result shows that bases are
compatible with duality in “the expected way” if and only if the $q$-PM is a
$q$-matroid. Part (a) has been established in [7].

######  Proposition 5.5.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM. Fix a non-degenerate symmetric
bilinear form $\langle{\cdot}\\!\mid\\!{\cdot}\rangle$ on $E$ and let
${\mathcal{M}}^{*}=(E,\rho^{*})$ be the dual of ${\mathcal{M}}$ w.r.t.
$\langle{\cdot}\\!\mid\\!{\cdot}\rangle$.

  * (a)

If ${\mathcal{M}}$ is a $q$-matroid, then for every basis $B$ of
${\mathcal{M}}$ the orthogonal space $B^{\perp}$ is a basis of
${\mathcal{M}}^{*}$.

  * (b)

Let $\mu$ be a denominator of ${\mathcal{M}}$. Suppose there exists a
$\mu$-basis $B$ of ${\mathcal{M}}$ such that the orthogonal space $B^{\perp}$
is a $\mu$-basis of ${\mathcal{M}}^{*}$. Then ${\mathcal{M}}$ is a
$q$-matroid.

###### Proof.

(a) has been proven in [7, Thm. 45].
(b) Let $B$ be a $\mu$-basis of ${\mathcal{M}}$ and $B^{\perp}$ be a
$\mu$-basis of ${\mathcal{M}}^{*}$. Then $\rho(B)=\rho(E)$ and thus
$\rho^{*}(B^{\perp})=\dim B^{\perp}+\rho(B)-\rho(E)=\dim B^{\perp}$. 4.2
implies that every basis $\hat{B}$ of ${\mathcal{M}}^{*}$ satisfies
$\rho^{*}(\hat{B})=\rho^{*}(B^{\perp})=\dim B^{\perp}=\dim\hat{B}$. Now (3.1)
yields $\rho^{*}(I)=\dim I$ for all $\mu$-independent spaces $I$ of
${\mathcal{M}}^{*}$. Hence the dual rank function $\rho^{*}$ is integer-valued
on the $\mu$-independent spaces. But then the entire rank function $\rho^{*}$
is integer-valued thanks to 4.1. Now $\rho=\rho^{**}$ is also integer-valued,
which means that ${\mathcal{M}}$ is a $q$-matroid. ∎

The above result has an interesting consequence. Recall from 3.9 the auxiliary
$q$-matroid ${\mathcal{Z}}_{{\mathcal{M}},\mu}$ of a $q$-PM ${\mathcal{M}}$
with denominator $\mu$. Part (b) above implies that if ${\mathcal{M}}$ is a
$q$-PM that is not a $q$-matroid, then
${\mathcal{Z}}_{{\mathcal{M}}^{*},\mu}\not\approx{\mathcal{Z}}_{{\mathcal{M}},\mu}^{*}$.
Indeed, from 3.9 we know that a subspace $B\in{\mathcal{V}}(E)$ is a
$\mu$-basis in ${\mathcal{M}}$ if and only if it is a basis in
${\mathcal{Z}}_{{\mathcal{M}},\mu}$. Thanks to 5.5(a) the latter is equivalent
to $B^{\perp}$ being in basis in ${\mathcal{Z}}_{{\mathcal{M}},\mu}^{*}$. But
by 5.5(a) $B^{\perp}$ is not a basis of ${\mathcal{M}}^{*}$, and thus not of
${\mathcal{Z}}_{{\mathcal{M}}^{*},\mu}$.

As we will show next, Part (a) can be generalized to $q$-PMs if one replaces
bases by minimal spanning spaces in ${\mathcal{M}}$ and by maximally strongly
independent spaces in ${\mathcal{M}}^{*}$, where the latter are defined as
follows.

######  Definition 5.6.

Let ${\mathcal{M}}=(E,\rho)$ be a $q$-PM. A subspace $V\in{\mathcal{V}}(E)$ is
_strongly independent_ if $\rho(V)=\dim V$. A subspace $V\in{\mathcal{V}}(E)$
is _maximally strongly independent_ if it is strongly independent and not
properly contained in a strongly independent subspace.

From 3.2 we know that strongly independent subspaces are $\mu$-independent for
every denominator $\mu$ of ${\mathcal{M}}$. Furthermore, in $q$-matroids
strong independence coincides with independence. We remark that strongly
independent subspaces play a crucial role in [1] for the construction of
subspace designs.

Now we have the following simple result. It shows that spanning spaces and
strongly independent spaces are mutually dual. This may be regarded a
generalization of [3, Prop. 83] and [7, Thm. 45], where the same results have
been established for $q$-matroids.

######  Proposition 5.7.

Let ${\mathcal{M}}$ and ${\mathcal{M}}^{*}$ be as in 5.5 and let
$V\in{\mathcal{V}}(E)$. Then $V$ is a (minimal) spanning space in
${\mathcal{M}}$ if and only if $V^{\perp}$ is (maximally) strongly independent
in ${\mathcal{M}}^{*}$.

###### Proof.

This follows immediately from $\rho^{*}(V^{\perp})=\dim
V^{\perp}+\rho(V)-\rho(E)$.∎

In summary, for $q$-matroids the notions ‘minimal spanning space’, ‘maximally
strongly independent space’, and ‘basis’ coincide, whereas these are distinct
concepts for $q$-PMs. We close the paper with a few remarks on the properties
– or rather lack thereof – of strongly independent spaces and spanning spaces
in $q$-PMs. In particular, neither the maximally strongly independent spaces
nor the minimal spanning spaces behave as well as bases. This is not
surprising since neither collection consists of subspaces of constant
dimension (see 5.3(b) along with duality).

######  Remark 5.8.

  * (a)

Let ${\mathcal{M}}$ be a $q$-PM and $\tilde{{\mathcal{I}}}$ be its collection
of strongly independent subspaces. The duality in 5.7 implies that
$\tilde{{\mathcal{I}}}$ is invariant under taking subspaces, i.e., it
satisfies (I2) of 3.12 (see also 3.2 and [1, Lem. 6]). It is not hard to find
examples showing that $\tilde{{\mathcal{I}}}$ does not satisfy (I3) and (I4).

  * (b)

In 3.13 we listed conditions (B1)–(B4) that are satisfied for bases in a
$q$-PM. As discussed earlier, they give rise to a cryptomorphic definition of
$q$-matroids, but not for $q$-PMs (see 3.14). For $q$-PMs neither the
maximally strongly independent subspaces nor the minimal spanning spaces
satisfy (B3) or (B4).

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