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# Quiver matroids
Matroid morphisms, quiver Grassmannians,
their Euler characteristics and ${{\mathbb{F}}_{1}}$-points

Manoel Jarra  Manoel Jarra, University of Groningen, the Netherlands, and
IMPA, Rio de Janeiro, Brazil<EMAIL_ADDRESS>,  Oliver Lorscheid  Oliver
Lorscheid, University of Groningen, the Netherlands<EMAIL_ADDRESS>and
Eduardo Vital  Eduardo Vital, Bielefeld University, Germany<EMAIL_ADDRESS>bielefeld.de

###### Abstract.

In this paper, we introduce morphisms for matroids with coefficients (in the
sense of Baker and Bowler) and quiver matroids. We investigate their basic
properties, such as functoriality, duality, minors and cryptomorphic
characterizations in terms of vectors, circuits and bases (a.k.a. Grassmann-
Plücker functions). We generalize quiver matroids to quiver matroid bundles
and construct their moduli space, which is an ${{\mathbb{F}}_{1}}$-analogue of
a complex quiver Grassmannian. Eventually we introduce a suitable
interpretation of ${{\mathbb{F}}_{1}}$-points for these moduli spaces, so that
in ‘nice’ cases their number is equal to the Euler characteristic of the
associated complex quiver Grassmannian.

###### Contents

  1. 1 Background
  2. 2 Category of $F$-matroids
  3. 3 Quiver $F$-matroids
  4. 4 Bands and band schemes
  5. 5 Matroid bundles
  6. 6 The moduli space of quiver matroids
  7. 7 Euler characteristics of quiver Grassmannians

## Introduction

This text provides a general framework for a mélange of ideas around matroid
theory, quiver representations and ${{\mathbb{F}}_{1}}$-geometry. More
precisely, this concerns the following.

### Matroid quotients and flag matroids

A matroid quotient is a strong map that is the identity on the respective
underlying sets. Matroid quotients have been generalized to oriented matroids
in [LV75] (see also [BLVS+99, Def. 7.7.2]), to valuated matroids in and
[BEZ21] and to matroids with coefficients in idylls in [JL24].

Flag matroids have been introduced in [BGVW97] as a sequence of matroid
quotients, and they have been generalized to valuated flag matroids in [Haq12]
and [BEZ21] and to flag matroids with coefficients in idylls in [JL24],
including cryptomorphic descriptions in terms of circuits, vectors and
Grassmann-Plücker functions, duality and minors, and corresponding moduli
spaces, which are flag varieties over the regular partial field.

### Quiver representations and quiver Grassmannians

A quiver representation over a field is a collection of vector spaces, indexed
by the vertices of the quiver, together with a collection of linear maps,
indexed by the arrows of the quiver. Quiver representations have been widely
studied because of their central relevance for the representation theory of
finite dimensional algebras; for instance, see [ASS06].

The quiver Grassmannian of a quiver representation is the collection of all
subrepresentations with a given dimension vector. The Caldero-Chapoton formula
([CC06]) expresses the structure constants of cluster algebras in terms of the
Euler characteristics of complex quiver Grassmannians, which turns an
intricate combinatorial problem into a geometric problem. While the quiver
Grassmannians of a wild quiver assume every isomorphism type of a projective
scheme ([Rei13], [Rin18]), quiver Grassmannians of tame quivers are varieties
that have a Schubert decomposition into affine spaces ([LW19b]). In
particular, the Euler characteristic corresponds to the number of Schubert
cells in the latter case.

Flag varieties resurface as a particular type of quiver Grassmannians: the
underlying quiver is of (equi-oriented) type $A_{n}$ and the quiver
representation consists of a sequence of identity self-maps of a linear
$n$-space. A point of this flag variety corresponds to a flag of linear
subspaces, which is the same thing as a flag matroid over the base field.

###  ${{\mathbb{F}}_{1}}$-models

A general heuristic for ${{\mathbb{F}}_{1}}$-geometry is that the number of
${{\mathbb{F}}_{1}}$-points of an ${{\mathbb{F}}_{1}}$-variety $X$ should be
equal to the Euler characteristic of the base extension of $X$ to
${\mathbb{C}}$ (this goes back to [Sou04, Section 6.2]; for details see
[Lor16, Section 4]). This provides a new strategy to study the Euler
characteristic of a quiver Grassmannian in terms of
${{\mathbb{F}}_{1}}$-points of a suitable ${{\mathbb{F}}_{1}}$-model.111For
the purpose of this introduction, we use ${{\mathbb{F}}_{1}}$ as a place
holder for different theories that are used in the cited papers.

Quiver representations over ${{\mathbb{F}}_{1}}$ have been studied in [Szc12],
[JS21], [JS23], [JKS23], [Kle24]. As a particular example of a quiver
Grassmannian, flag varieties over ${{\mathbb{F}}_{1}}$ have been shown to be
the moduli space of flag matroids ([BL21], [JL24]). Models of quiver
Grassmannians over ${{\mathbb{F}}_{1}}$ have been considered in [Lor16] where
also the connection to Euler characteristics of the complex quiver
Grassmannian was studied in special cases. Tropical quiver Grassmannians have
been considered in [IS23].

### Outline of the contents of this text

We provide a common footing for all of the aforementioned concepts. This
includes the development of:

  * •

_morphisms of matroids with coefficients_ , as a common generalization of
strong maps of usual matroids (Relation to strong maps), quotients of oriented
matroids and $k$-linear maps where $k$ is a field;

  * •

_quiver matroids_ , which is a common generalization of quiver representations
and flag matroids;

  * •

_quiver Grassmannians over ${{\mathbb{F}}_{1}}$_, which generalize
Grassmannians and flag varieties over ${{\mathbb{F}}_{1}}$.

Our results include:

  * •

functoriality (Push-forward along an idyll morphism), duality (Duality), pre-
images (Pre-image along a matroid morphism) and minors (Minors) for matroid
morphisms;

  * •

cryptomorphic characterizations of matroid morphisms and quiver matroids in
terms of circuits, vectors and Grassmann-Plücker functions (Theorem A);

  * •

the quiver Grassmannian over ${{\mathbb{F}}_{1}}$ as the moduli space of
quiver matroid bundles (Theorem B);

  * •

the Euler characteristic of complex quiver Grassmannians as
${{\mathbb{F}}_{1}}$-points of their ${{\mathbb{F}}_{1}}$-models for ‘nice’
representations (Theorem C).

In the following, we explain these contents in more detail.

### Morphisms of matroids

For the purpose of this introduction, we assume that the reader is familiar
with Baker-Bowler theory (as in [BB19]); we refer to Section 1.3 for a
summary. An _idyll_ is a tract whose nullset is closed under addition. An
idyll is _perfect_ if vectors are orthogonal to covectors for all matroids
over the idyll, which is an important property for our purposes; in
particular, the notions of weak and strong matroids agree for perfect idylls
([BB19, Thm. 3.46]). Examples of perfect idylls are fields and partial fields,
the Krasner hyperfield ${\mathbb{K}}$, the tropical hyperfield ${\mathbb{T}}$,
the sign hyperfield ${\mathbb{S}}$ and, more generally, stringent hyperfields
([BP19]). More generally, every idyll can be completed to a perfect idyll
([BZ23]).

Let $F$ be a perfect idyll. A matrix $\Phi$ with entries $\Phi_{i,j}\in F$ is
_submonomial_ if every row and every column contains at most one nonzero
entry. If $\Phi$ is indexed by $T\times S$, then it defines a map
$\Phi:F^{S}\to F^{T}$ by the rule $(\Phi\cdot X)_{i}=\sum_{j\in
S}\Phi_{i,j}X_{j}$, which defines an element of $F$ since the sum contains at
most one nonzero term. Note that the composition of such maps is well-defined
since the product $\Psi\cdot\Phi$ of submonomial matrices $\Phi$ and $\Psi$ is
again submonomial.

Consider $F$-matroids $N$ and $M$ with respective underlying sets $S$ and $T$
and respective vector sets ${\mathcal{V}}_{N}\subseteq F^{S}$ and
${\mathcal{V}}_{M}\subseteq F^{T}$. A _morphism $\Phi:N\to M$ of $F$-matroids_
(or, for short, an _$F$ -morphism_) is a submonomial matrix $\Phi$ indexed by
$T\times S$ such that $\Phi\cdot{\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{M}$.
It is clear from the definition that morphisms of $F$-matroids are composable,
which yields the category $\operatorname{Mat}_{F}$ of $F$-matroids.

#### Push-forward along an idyll morphism

Let $f:F\to G$ be a morphism of perfect idylls and $\Phi$ a submonomial matrix
with coefficients in $F$. The _push-forward of $\Phi$ along $f$_ is the matrix
$f_{\ast}\Phi$ with coefficients $(f_{\ast}\Phi)_{i,j}=f(\Phi_{i,j})$, which
are elements of $G$. The following is Section 2.6.

######  Proposition  (functoriality).

The push-forward of an $F$-morphism $\Phi:N\to M$ along a morphism $f:F\to G$
of perfect idylls is a $G$-morphism $f_{\ast}\Phi:f_{\ast}N\to f_{\ast}M$.
This defines a functor
$f_{\ast}:\operatorname{Mat}_{F}\to\operatorname{Mat}_{G}$.

#### Duality

We denote the transpose of a submonomial matrix $\Phi$ by $\Phi^{t}$, which is
also a submonomial matrix.

######  Proposition  (duality).

The transpose of an $F$-morphism $\Phi:N\to M$ is an $F$-morphism
$\Phi^{t}:M^{\ast}\to N^{\ast}$. This defines a contravariant endofunctor on
$\operatorname{Mat}_{F}$ whose square is the identity.

#### Relation to strong maps

The _augmentation_ of a set $S$ is the pointed set
$\underline{S}=S\sqcup\\{0\\}$ with base point $0$. Let $\Phi$ be a
submonomial matrix with coefficients in $F$ and indexed by $T\times S$. The
_underlying map_ of $\Phi$ is the map
${\underline{\Phi}}:{\underline{S}}\to{\underline{T}}$ with
$\underline{\Phi}(j)=i$ if $\Phi_{i,j}\neq 0$ and $\underline{\Phi}(j)=0$ if
$j=0$ or if $\Phi_{i,j}=0$ for all $i$. Note that
${\underline{\Phi}}:{\underline{S}}\to{\underline{T}}$ is
_${{\mathbb{F}}_{1}}$ -linear_, in the sense that it preserves base points and
that for every $i\in T$, there is at most one $j\in S$ with
${\underline{\Phi}}(j)=i$.

We denote the underlying matroid of an $F$-matroid $M$ by $\underline{M}$.
Note that ${\underline{M}}$ has the same ground set as $M$, and that strong
maps of matroids are, by definition, maps between augmented ground sets.

######  Proposition  (relation to strong maps).

Let $N$ and $M$ be ${\mathbb{K}}$-matroids with respective underlying sets $S$
and $T$. A submonomial matrix $\Phi$ with entries in ${\mathbb{K}}$ is a
morphism $\Phi:N\to M$ of ${\mathbb{K}}$-matroids if and only if
${\underline{\Phi}}:{\underline{N}}\to{\underline{M}}$ is a strong map. In
particular, the underlying map
$\underline{\Phi}:\underline{S}\to\underline{T}$ of an $F$-morphism $\Phi:N\to
M$ (where $F$ is any perfect idyll) is a strong map
${\underline{\Phi}}:{\underline{N}}\to{\underline{M}}$ between the underlying
matroids.

######  Remark.

Since $1$ is the unique nonzero element of ${\mathbb{K}}$, a submonomial
matrix $\Phi$ with coefficients in ${\mathbb{K}}$ is uniquely determined by
its underlying map ${\underline{\Phi}}$. For the same reason, a
${\mathbb{K}}$-matroid $M$ is uniquely determined by its underlying matroid
${\underline{M}}$. This means that Relation to strong maps provides a faithful
embedding of $\operatorname{Mat}_{\mathbb{K}}$ into the category of usual
matroids together with strong maps.

While every matroid is the underlying matroid ${\underline{M}}$ of a (uniquely
determined) ${\mathbb{K}}$-matroid $M$, not every strong map comes from a
${\mathbb{K}}$-morphism. More precisely, a strong map
$\mysigma:{\underline{N}}\to{\underline{M}}$ comes from a
${\mathbb{K}}$-morphism if and only if it is ${{\mathbb{F}}_{1}}$-linear.

This means that isomorphisms, restrictions, contractions, quotients, and
compositions thereof, come from ${\mathbb{K}}$-morphism. Strong maps that
identify distinct parallel elements do not come from ${\mathbb{K}}$-morphisms.

#### Further examples

If $k$ is a field, then ${\mathcal{V}}_{N}$ is a linear subspace of $k^{S}$. A
$k$-morphism is a $k$-linear map $\Phi:k^{S}\to k^{T}$ that maps the subspace
${\mathcal{V}}_{N}$ to ${\mathcal{V}}_{M}$, that maps coordinate vectors of
$k^{S}$ to coordinate vectors of $k^{T}$ and whose kernel is spanned by
coordinate vectors of $k^{S}$.

A ${\mathbb{T}}$-matroid is a tropical linear space ${\mathcal{V}}_{N}$ in
${\mathbb{T}}^{S}$. A ${\mathbb{T}}$-morphism is a ${\mathbb{T}}$-linear map
$\Phi:{\mathbb{T}}^{S}\to{\mathbb{T}}^{T}$ that preserves the corresponding
tropical linear spaces ${\mathcal{V}}_{N}$ and ${\mathcal{V}}_{M}$, that maps
coordinate vectors to coordinate vectors and whose non-empty fibres all have
the same dimension; see also [IS23, Section 2].

Quotients of oriented matroids are ${\mathbb{S}}$-morphisms; see [BLVS+99,
Def. 7.7.2].

######  Warning.

There is a tension between linear algebra and matroid theory, which stems from
the fact for a field $k$, the vector set of a $k$-matroid represents the dual
of its underlying matroid. This culminates in the most confusing fact that a
(submonomial) linear map represents a strong map in the opposite direction.

To explain, the rank of a $k$-matroid $M$ is equal to the _codimension_ of its
vector set ${\mathcal{V}}_{M}\subseteq k^{T}$. More to the point,
${\mathcal{V}}_{M}$ represents the dual $\underline{M}^{\ast}$ of the
underlying matroid $\underline{M}$ of $M$. The _dimension_ of
${\mathcal{V}}_{M}$ equals the rank of $\underline{M}^{\ast}$.

A $k$-morphism $\Phi:N\to M$ between $k$-matroids $N$ and $M$ defines a strong
map $\underline{\Phi}:\underline{N}\to\underline{M}$ between the respective
underlying matroids _in the same direction_. The matroids _represented by
${\mathcal{V}}_{N}$ and ${\mathcal{V}}_{M}$_ are, however, the duals of
$\underline{N}$ and $\underline{M}$, respectively. Therefore $\Phi$ induces
the strong map
$\underline{\Phi}^{t}:\underline{M}^{\ast}\to\underline{N}^{\ast}$ between the
matroids represented by ${\mathcal{V}}_{N}$ and ${\mathcal{V}}_{M}$ _in the
opposite direction_.

#### Pre-image along a matroid morphism

Let $M$ be an $F$-matroid on $T$ and $\Phi$ a submonomial matrix with
coefficients in $F$ indexed by $T\times S$.

######  Proposition  (pre-image).

There is a unique $F$-matroid $N$ on $S$ with vector set

${\mathcal{V}}_{N}\ =\ \Phi^{-1}({\mathcal{V}}_{M})\ =\ \big{\\{}V\in
F^{S}\,\big{|}\,\Phi\cdot V\in{\mathcal{V}}_{M}\big{\\}}.$

We write $\Phi^{-1}(M)$ for $N$ and call it the _pre-image of $M$ along
$\Phi$_. See Section 2.4 for a characterization of the pre-image in terms of
Grassmann-Plücker functions.

#### Cryptomorphisms

Let $N$ and $M$ be $F$-matroids with respective underlying sets $S$ and $T$,
respective circuit sets ${\mathcal{C}}_{N}$ and ${\mathcal{C}}_{M}$, and
respective Grassmann-Plücker functions $\mynu$ and $\mymu$. We denote the
cocircuits of $M$ by ${\mathcal{C}}_{M}^{\ast}$. Let $w$ and $r$ be the
respective ranks of $N$ and $M$. The following summarizes Theorem 2.1 and
Theorem 2.3.

######  Theorem A (cryptomorphisms).

Let $\Phi$ be a submonomial $F$-matrix, indexed by $T\times S$. Then the
following are equivalent:

  1. (1)

$\Phi:N\to M$ is a morphism, i.e.
$\Phi\cdot{\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{M}$;

  2. (2)

$\Phi^{-1}(M)$ is a quotient of $N$, i.e.
${\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{\Phi^{-1}(M)}$;

  3. (3)

$\Phi\cdot{\mathcal{C}}_{N}\perp{\mathcal{C}}_{M}^{\ast}$;

  4. (4)

for all $y_{0},\dotsc,y_{w}\in S$ and $x_{2},\dotsc,x_{r}\in T$,

$\sum_{k=0}^{w}\
(-1)^{k}\cdot\mynu(y_{0},\dotsc,\widehat{y_{k}},\dotsc,y_{w})\cdot\Phi_{{\underline{\Phi}}(y_{k}),y_{k}}\cdot\mymu\big{(}\underline{\Phi}(y_{k}),x_{2},\dotsc,x_{r}\big{)}\quad\in\quad
N_{F},$

with the convention that $\Phi_{0,y_{k}}=\mymu(0,x_{2},\dotsc,x_{r}\big{)}=0$.

#### Minors

Let $N$ and $M$ be $F$-matroids with respective ground sets $S$ and $T$ and
$\Phi:N\to M$ an $F$-morphism. For subsets $A\subseteq S$ and $B\subseteq T$
with complements $A^{c}=S-A$ and $B^{c}=T-B$, we define the following
submatrices of $\Phi$:

$\Phi|(A\to B)\ =\ (\Phi_{i,j})_{(i,j)\in B\times
A}\quad\text{and}\quad\Phi/(A\to B)\ =\ (\Phi_{i,j})_{(i,j)\in B^{c}\times
A^{c}}.$

######  Proposition  (minors).

If ${\underline{\Phi}}({\underline{A}})\subseteq{\underline{B}}$, then

$\Phi|{(A\to B)}:N|A\to M|B\qquad\text{and}\qquad\Phi/{(A\to B)}:N/A\to M/B$

are $F$-morphisms (called the _restriction_ and _contraction_ of $\Phi$ to
$A\to B$, respectively). They satisfy $\Phi^{t}|(B^{c}\to
A^{c})=\big{(}\Phi/(A\to B)\big{)}^{t}$ and $f_{\ast}\big{(}\Phi|{(A\to
B)}\big{)}=(f_{\ast}\Phi)|{(A\to B)}$ for any morphism $f:F\to G$ of perfect
idylls.

### Quiver matroids

Let $F$ be a perfect idyll and $Q$ a quiver with vertex set $Q_{0}$ and arrow
set $Q_{1}$. A _$(Q,F)$ -matroid_ is a tuple $(M_{v})_{v\in Q_{0}}$ of
$F$-matroids $M_{v}$ together with a tuple $(M_{\myalpha})_{\myalpha\in
Q_{1}}$ of $F$-morphisms $M_{\myalpha}:M_{s(\myalpha)}\to M_{t(\myalpha)}$
where $s(\myalpha)$ is the source of $\myalpha$ and $t(\myalpha)$ is the
target of $\myalpha$. We write $M=\big{(}(M_{v}),(M_{\myalpha})\big{)}$ for
this $(Q,F)$-matroid.

#### Examples

Particular instances of quiver matroids include:

  * •

Flag $F$-matroids (in the sense of [JL24]) are $(Q,F)$-matroids $M$ for which
$Q$ is of the form

${1}$${2}$${\dotsb}$${s}$

and for which $M_{\myalpha}$ is the square identity matrix (for every
$\myalpha$); see Section 3.2.

  * •

Let $k$ be a field and $Q$ a quiver. A $(Q,k)$-matroid corresponds to a
subrepresentation of a $Q$-representation $\Lambda$ over $k$ together with a
choice of basis $E_{v}$ of $\Lambda_{v}$ (for every $v\in Q_{0}$) such that
$\Lambda_{\myalpha}$ is submonomial with respect to the bases
$E_{s(\myalpha)}$ and $E_{t(\myalpha)}$ (for every $\myalpha\in Q_{1}$); see
Section 2.2.

  * •

In the case of the tropical hyperfield ${\mathbb{T}}$, we gain the notion of
valuated quiver matroid, which is a tuple of tropical linear spaces together
with submonomial matrices that preserve them; see Section 2.2.

#### Duality

Let $M$ be an $(Q,F)$-matroid and $Q^{\ast}$ the _opposite quiver_ , which has
the same vertices and arrows, but with reversed direction. Then the _dual of
$M$_ is the $(Q^{\ast},F)$-matroid $M^{\ast}$ with
$(M^{\ast})_{v}:=(M_{v})^{\ast}$ for $v\in Q_{0}^{\ast}=Q_{0}$ and
$(M^{\ast})_{\myalpha}:=M_{\myalpha}^{t}$ for $\myalpha\in
Q_{1}^{\ast}=Q_{1}$.

#### Minors

Let $M$ be an $(Q,F)$-matroid and $E_{v}$ the underlying set of $M_{v}$ for
$v\in Q_{0}$. Let $\Lambda_{v}\subseteq\underline{E_{v}}$ be a pointed subset
(for every $v\in Q_{0}$) such that
$\underline{M_{\myalpha}}(\Lambda_{s})\subseteq\Lambda_{t}$ for every
$\myalpha\in Q_{1}$ with source $s$ and target $t$. Define
$A_{v}:=\Lambda_{v}-\\{0\\}$ and $\Lambda:=(\Lambda_{v})_{v\in Q_{0}}$.

The _restriction of $M$ to $\Lambda$_ is the $(Q,F)$-matroid $M|\Lambda$ with
$(M|\Lambda)_{v}=M_{v}|A_{v}$ (for $v\in Q_{0}$) and
$(M|\Lambda)_{\myalpha}=M_{\myalpha}|(A_{s}\to A_{t})$ (for $\myalpha\in
Q_{1}$ with source $s$ and target $t$).

The _contraction of $M$ by $\Lambda$_ is the $(Q,F)$-matroid $M/\Lambda$ with
$(M/\Lambda)_{v}=M_{v}/A_{v}$ (for $v\in Q_{0}$) and
$(M/\Lambda)_{\myalpha}=M_{\myalpha}/(A_{s}\to A_{t})$ (for $\myalpha\in
Q_{1}$ with source $s$ and target $t$).

For more details, see Section 3.4.

#### Relation to ${{\mathbb{F}}_{1}}$-representations of quivers

An ${{\mathbb{F}}_{1}}$-representation of $Q$ is a tuple
$\Lambda=\big{(}({\Lambda}_{v})_{v\in Q_{0}},(\Lambda_{\myalpha})_{\myalpha\in
Q_{1}}\big{)}$ where $\Lambda_{v}$ is a finite pointed set with base point $0$
(for $v\in Q_{0}$) and
$\Lambda_{\myalpha}:{\Lambda}_{s(\myalpha)}\to{\Lambda}_{t(\myalpha)}$ (for
$\myalpha\in Q_{1}$) is an _${{\mathbb{F}}_{1}}$ -linear map_, i.e.
$\Lambda_{\myalpha}(0)=0$ and $\\#\Lambda_{\myalpha}^{-1}(e)\leqslant 1$ for
$e\neq 0$.

It is a general phenomenon that (postulated) objects in
${{\mathbb{F}}_{1}}$-geometry surface as certain types of matroids, which are
usually of a particularly simple shape. Quiver representations form a first
such instance (for this paper): an ${{\mathbb{F}}_{1}}$-representation
$\Lambda$ defines a $(Q,{\mathbb{K}})$-matroid $\mathsf{M}_{Q}(\Lambda)$ whose
matroids $\mathsf{M}_{Q}(\Lambda)_{v}$ (for $v\in Q_{0}$) are characterized by
their vector sets
${\mathcal{V}}_{\mathsf{M}_{Q}(\Lambda)_{v}}={\mathbb{K}}^{\Lambda_{v}-\\{0\\}}$
and whose submonomial matrices $\mathsf{M}_{Q}(\Lambda)_{\myalpha}$ (for
$\myalpha\in Q_{1}$) have coefficients

$\big{(}\mathsf{M}_{Q}(\Lambda_{\myalpha})\big{)}_{i,j}\ =\
\begin{cases}1&\text{if }\Lambda_{\myalpha}(j)=i,\\\ 0&\text{if
}\Lambda_{\myalpha}(j)\neq i.\end{cases}$

In other words, $\mathsf{M}_{Q}(\Lambda)_{\myalpha}$ is the unique submonomial
matrix with coefficients in ${\mathbb{K}}$ such that
$\underline{\mathsf{M}_{Q}(\Lambda)_{\myalpha}}=\Lambda_{\myalpha}$.

Conversely, every $(Q,{\mathbb{K}})$-matroid $M$ has an _underlying
${{\mathbb{F}}_{1}}$-representation_ $\mathsf{U}_{Q}(M)$ with pointed sets
$\mathsf{U}_{Q}(M)_{v}=\underline{E_{v}}$ (for $v\in Q_{0}$), where $E_{v}$ is
the ground set of $M_{v}$, and
$\mathsf{U}_{Q}(M)_{\myalpha}=\underline{M_{\myalpha}}$. We have
$\mathsf{U}_{Q}\circ\mathsf{M}_{Q}(\Lambda)=\Lambda$ for every
${{\mathbb{F}}_{1}}$-representation $\Lambda$ of $Q$, which, in fact, extends
to an adjunction between the category of ${{\mathbb{F}}_{1}}$-representations
of $Q$ and $(Q,{\mathbb{K}})$-matroids with respect to a suitable notion of
morphisms (see Section 3.3).

### Quiver Grassmannians

Let $\Lambda$ be an ${{\mathbb{F}}_{1}}$-representation of $Q$ and r a _rank
vector for $Q$_, which is a tuple $\textbf{r}=(r_{v})_{v\in Q_{0}}$ of
integers $r_{v}\geqslant 0$. The _rank_ of a $(Q,F)$-matroid $M$ is the tuple
$\underline{\operatorname{{rk}}}(M)=\big{(}\operatorname{{rk}}(M_{v})\big{)}_{v\in
Q_{0}}$.

As a point set over $F$, the quiver Grassmannian of $(\Lambda,F)$-matroids of
rank r is

$\operatorname{Gr}_{\textbf{r}}(\Lambda)(F)\ =\
\big{\\{}\text{$(Q,F)$-matroids $M$ of rank $\textbf{r}$ with
$\mathsf{U}_{Q}(M)=\Lambda$}\big{\\}}$

where $\mathsf{U}_{Q}(M)$ is the underlying
${{\mathbb{F}}_{1}}$-representation of $M$.

#### Relation to complex quiver Grassmannians

An ${{\mathbb{F}}_{1}}$-representation $\Lambda$ determines a complex
representation $\Lambda_{\mathbb{C}}$ of $Q$ with vector spaces
$\Lambda_{{\mathbb{C}},v}={\mathbb{C}}^{E_{v}}$ where
$E_{v}=\Lambda_{v}-\\{0\\}$ (for $v\in Q_{0}$) and complex matrices
$\Lambda_{{\mathbb{C}},\myalpha}$ (for $\myalpha\in Q_{1}$) with coefficients

$(\Lambda_{{\mathbb{C}},\myalpha})_{i,j}\ =\ \begin{cases}1&\text{if
}\Lambda_{\myalpha}(j)=i,\\\ 0&\text{if }\Lambda_{\myalpha}(j)\neq
i\end{cases}$

for $(i,j)\in E_{t(\myalpha)}\times E_{s(\myalpha)}$. The quiver Grassmannian
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{C}})$ in the above sense
agrees with the usual complex quiver Grassmannian

$\operatorname{Gr}_{{\textbf{r}}^{\ast}}(\Lambda_{\mathbb{C}})({\mathbb{C}})\
=\ \big{\\{}\text{subrepresentations $N$ of $\Lambda_{\mathbb{C}}$ with
$\underline{\dim}(N)={\textbf{r}}^{\ast}$}\big{\\}},$

where ${\textbf{r}}^{\ast}=(\\#E_{v}-r_{v})_{v\in Q_{0}}$. Note that we need
to pass to the dimension vector ${\textbf{r}}^{\ast}$ since the rank of a
${\mathbb{C}}$-matroid $M$ with ground set $E$ equals the codimension of its
vectors ${\mathcal{V}}_{M}$ as a subspace of ${\mathbb{C}}^{E}$.

Note further that we do not need to invoke subrepresentations in the
definition of $\operatorname{Gr}_{\textbf{r}}(\Lambda)(F)$ since (the vector
set of) a ${\mathbb{C}}$-matroid $M$ with ground set $E$ is by definition a
subspace ${\mathcal{V}}_{M}$ of ${\mathbb{C}}^{E}$, i.e. a
$(Q,{\mathbb{C}})$-matroid $N$ in
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{C}})$ is naturally a
_subrepresentation_ of $\Lambda_{\mathbb{C}}$, or a _quotient_ if we adhere to
the terminological conventions of matroid theory.

In a similar vein, $\operatorname{Gr}_{{\mathbf{r}}}(\Lambda)({\mathbb{T}})$
stays in a natural bijection with the quiver Dressian of [IS23].

#### The quiver Grassmannian as a moduli space

The push-forward along an perfect idyll morphism $f:F\to G$ defines a map

$f_{\ast}:\ \operatorname{Gr}_{\textbf{r}}(\Lambda)(F)\ \longrightarrow\
\operatorname{Gr}_{\textbf{r}}(\Lambda)(G),$

which turns $\operatorname{Gr}_{\textbf{r}}(\Lambda)$ into a functor from
$\operatorname{{Idylls}}$ to $\operatorname{Sets}$. This functor is
represented by an ${{\mathbb{F}}_{1}}$-variety in a suitable sense. For this,
we employ the theory of band schemes, as introduced in [BJL24]; see Section 6
for details.

More accurately, we first extend the functor
$\operatorname{Gr}_{\mathbf{r}}(\Lambda)$ to band schemes in the following
way. A _matroid bundle_ over a band scheme is a suitable generalization of a
subbundle of ${\mathcal{O}}_{X}^{n}$ in usual algebraic geometry, where $X$ is
a scheme and $n$ is the cardinality of the ground set $E$. This notion extends
to $\Lambda$-matroid bundles on a band scheme $X$ in the same way as
$F$-matroids generalize to $(\Lambda,F)$-matroids. This leads to the
definition of $\operatorname{Gr}_{\mathbf{r}}(\Lambda)(X)$ as the collection
of all $\Lambda$-matroid bundles on $X$ with rank vector ${\mathbf{r}}$.

######  Theorem B.

The functor $\operatorname{Gr}_{\mathbf{r}}(\Lambda)$ has the structure of a
band scheme. In other words, the band scheme
$\operatorname{Gr}_{\mathbf{r}}(\Lambda)$ is the fine moduli space of
$\Lambda$-matroid bundles.

### Euler characteristic and ${{\mathbb{F}}_{1}}$-rational points

The following heuristic reasoning leads to the expectation that the number of
${{\mathbb{F}}_{1}}$-points of a sufficiently nice
${{\mathbb{F}}_{1}}$-variety $X$ is equal to the (topological) Euler
characteristic of its ${\mathbb{C}}$-points $X({\mathbb{C}})$ (with respect to
the complex topology).

Namely, assume that $X$ is an ${{\mathbb{F}}_{1}}$-model of a smooth
projective scheme $X_{\mathbb{Z}}$ for which the number
$N(q)=\\#X_{\mathbb{Z}}({\mathbb{F}}_{q})$ of ${\mathbb{F}}_{q}$-rational
points (for prime powers $q$) is a polynomial in $q$. As a consequence of the
comparison result of singular cohomology with $l$-adic cohomology and of
Deligne’s proof of the Weil conjectures, this polynomial is

$N(q)\ =\ \sum_{i=0}^{\dim X}b_{2i}\cdot q^{i}$

where $b_{i}$ are the Betti numbers of the complex manifold $X({\mathbb{C}})$,
with $b_{2i+1}=0$ for odd indices (see [Lor16, Section 7] for more details).
Thus the number of ${{\mathbb{F}}_{1}}$-points of $X$ equals the Euler
characteristic of $X({\mathbb{C}})$:

$\mychi\big{(}X({\mathbb{C}})\big{)}\ =\ \sum_{i=0}^{2\dim X}(-1)^{i}\cdot
b_{i}\ =\ \sum_{i=0}^{\dim X}b_{2i}\cdot 1^{i}\ =\ N(1).$

This heuristic is reflected by the following expectation on the Grassmannian
over ${{\mathbb{F}}_{1}}$: an $r$-dimensional subspace of an
${{\mathbb{F}}_{1}}$-vector space $\Lambda$ (which is a pointed set) is a
pointed subset $N\subseteq\Lambda$ with $\\#(N-\\{0\\})=r$. In other words,
$N-\\{0\\}$ is an $r$-subset of $\Lambda-\\{0\\}$. This leads to the
expectation that

$\operatorname{Gr}(r,n)({{\mathbb{F}}_{1}})\ =\ \big{\\{}\text{$r$-subsets of
$\\{1,\dotsc,n\\}$}\big{\\}}$

has $\binom{n}{r}$ elements, which equals the Euler characteristic of
$\operatorname{Gr}(r,n)({\mathbb{C}})$.

These expectations are not met, however, if we interpret the notion of
${{\mathbb{F}}_{1}}$-points of $X$ in the usual way, i.e. as morphisms
$\operatorname{Spec}{{\mathbb{F}}_{1}}\to X$. For example,
$X={\mathbb{P}}^{n}_{{\mathbb{F}}_{1}}$ has $2^{n+1}-1$ such morphisms (say,
when considered as monoid schemes222Note that finite type monoid schemes form
a full subcategory of virtually any other approach to
${{\mathbb{F}}_{1}}$-geometry, so the number of ${{\mathbb{F}}_{1}}$-rational
points of ${\mathbb{P}}^{n}$ does not depend on the notion of
${{\mathbb{F}}_{1}}$-geometry that we use.), but the Euler characteristic of
${\mathbb{P}}^{n}({\mathbb{C}})$ is $n+1$.

In many instances, our expectations on ${{\mathbb{F}}_{1}}$-points are met by
certain ${\mathbb{K}}$-points of a suitable ${{\mathbb{F}}_{1}}$-model. To
explain, a band scheme $X$ comes with a set $X({\mathbb{C}})$ of
${\mathbb{C}}$-points and a set $X({\mathbb{K}})$ of ${\mathbb{K}}$-points,
which are both equipped with a natural topology. The expected
${{\mathbb{F}}_{1}}$-points agree with the closed ${\mathbb{K}}$-points in the
following examples (cf. [LT23], [BJL24, Section 3.4]):

  * •

${\mathbb{P}}^{n}$ has $n+1$ closed ${\mathbb{K}}$-points, which recovers the
Euler characteristic of ${\mathbb{P}}^{n}({\mathbb{C}})$.

  * •

The Weyl group $\mathfrak{S}_{n}$ of $\operatorname{{SL}}_{n}$ stays in
bijection with the closed points of $\operatorname{{SL}}_{n}({\mathbb{K}})$.

  * •

The closed ${\mathbb{K}}$-points of a flag variety correspond bijectively to
the simplices (of the corresponding type) of the Coxeter complex for
$\mathfrak{S}_{n}$, and their number agrees with the Euler characteristic of
the corresponding complex flag variety.

We call the set of closed ${\mathbb{K}}$-points of $X$ the _Tits space_ and
denote it by $X^{\textup{Tits}}$. The relation between the Euler
characteristic and the Tits space generalizes to all quiver Grassmannians of
sufficiently ‘nice’ ${{\mathbb{F}}_{1}}$-representations. This includes the
following cases (see Theorem 7.3 and Section 7.5).

######  Theorem C.

Let $Q$ be a finite quiver and $\Lambda$ an
${{\mathbb{F}}_{1}}$-representation. Assume that one of the following
conditions holds:

  1. (1)

The coefficient quiver $\Gamma$ of $\Lambda$ is a tree (see Section 7.5 for
the definition of $\Gamma$).

  2. (2)

$Q$ is a quiver of simply laced (extended) Dynkin type and
$\Lambda_{\mathbb{C}}$ is irreducible.

Then

$\mychi\big{(}\operatorname{Gr}_{{\mathbf{r}}}(\Lambda)({\mathbb{C}})\big{)}\
=\ \\#\operatorname{Gr}_{{\mathbf{r}}}(\Lambda)^{\textup{Tits}}.$

for every dimension vector ${{\mathbf{r}}}$ for $Q$.

#### Geometric sketch of proof

Every dimension ${\textbf{r}^{*}}$ subrepresentation of $\Lambda$, which is a
collection of pointed subsets $\Omega_{v}$ of $\Lambda_{v}$ (for $v\in Q_{0}$)
that is stable under the maps $\Lambda_{\myalpha}$ (for $\myalpha\in Q_{1}$),
determines a so-called _coordinate matroid_ $M$, which has
$B_{v}=\Lambda_{v}-\Omega_{v}$ as its unique basis at $v$. By Theorem B, $M$
corresponds to a point $x_{M}$ of
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{K}})$. Since $x_{M}$ has a
unique nonzero coordinate at $B_{v}$ for each $v$, it is closed in
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{K}})$, which establishes an
injection

$\big{\\{}\text{dimension ${\textbf{r}^{*}}$ subrepresentations of
$\Lambda$}\big{\\}}\ \longrightarrow\
\operatorname{Gr}_{\textbf{r}}(\Lambda)^{\textup{Tits}}.$

Under the assumptions of Theorem C, there is a sequence of torus actions that
allows us to establish an equality between these sets and between their
cardinalities and the Euler characteristic of
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{C}})$.

Namely, Jun and Sistko’s work [JS21] provides a _nice sequence of gradings of
$\Lambda$ that distinguishes elements_. These gradings determine the weights
of actions

$\mytheta_{i}:\ {\mathbb{G}}_{m}\times X_{i}\ \longrightarrow\ X_{i}$

(for $i=1,\dotsc,n$) of the multiplicative group scheme ${\mathbb{G}}_{m}$ on
$X_{1}=\operatorname{Gr}_{\textbf{r}}(\Lambda)$ and on the subspace $X_{i+1}$
of fixed points of $\mytheta_{i}$ (for $i=1,\dotsc,n$), with $X_{n+1}$ being
finite.

By a theorem of Białynicki-Birula ([BB73]), the Euler characteristic of
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{C}})$ is equal to the Euler
characteristic of the ${\mathbb{C}}^{\times}$-fixed points
$X_{2}({\mathbb{C}})$ of $\mytheta_{1,{\mathbb{C}}}$, and thus, after
repeating the argument $n$ times, equal to the Euler characteristic, or
cardinality, of the finite set $X_{n+1}({\mathbb{C}})$.

Since the torus actions are defined in terms of gradings of $\Lambda$, it
follows from first principles that $X_{n+1}({\mathbb{C}})$ corresponds to the
dimension ${\textbf{r}^{*}}$ subrepresentations of $\Lambda$.

In order to show that every point of
$\operatorname{Gr}_{\textbf{r}}(\Lambda)^{\textup{Tits}}$ stems from a
subrepresentation of $\Lambda$, we consider an arbitrary point $x$ in
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{K}})$. Since ${\mathbb{K}}$
is contained in the tropical hyperfield ${\mathbb{T}}$, we can consider $x$ as
an element of the tropical quiver Grassmannian
$\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{T}})$. If $x$ does not
correspond to a point in $X_{n+1}$, then there is an $i$ such that
${\mathbb{G}}_{m}({\mathbb{T}})$ acts via $\mytheta_{i,{\mathbb{T}}}$ non-
trivially on $x\in X_{i}({\mathbb{T}})$. Since
${\mathbb{G}}_{m}({\mathbb{T}})={\mathbb{R}}_{>0}$ (as multiplicative group),
we can consider the limit

$x_{0}\ =\ \lim_{t\to 0}\ \mytheta_{i,{\mathbb{T}}}(t,x),$

which is a point of $\operatorname{Gr}_{\textbf{r}}(\Lambda)({\mathbb{K}})$
with strictly more zero coefficients than $x$. Thus $x_{0}\neq x$ is in the
closure of $x$, which shows that $x$ is not closed and not in
$\operatorname{Gr}_{\textbf{r}}(\Lambda)^{\textup{Tits}}$. ∎

### Remark on generalizations

While the definitions and concepts that we develop in this paper are
sufficient to illustrate the interplay of matroid theory,
${{\mathbb{F}}_{1}}$-geometry and quiver representations, they lean themselves
to generalizations in different directions, which might be of independent
interest for a broader framework for matroid theory and quiver Grassmannians.
At this point, we restrict ourselves to the following brief comments.

#### Idylls with involutions

Baker and Bowler introduce $F$-matroids for idylls (and more generally,
tracts) $F$ together with an involution $i:F\to F$, which allows, for
instance, to take Hermitian complements of complex matroids, opposed to
orthogonal complements. All results of this paper generalize readily to
perfect idylls and tracts with involution; also cf. Section 2.7.

#### Morphisms of matroid bundles

The notion of morphisms of matroid bundles over band schemes has to be seen as
a provisional concept. The problem is that the composition of two such
morphisms are in general not a morphism in the sense of this text anymore.
This effect occurs already for non-perfect idylls; cf. Section 5.3.

The failure of composability can be fixed in terms of the so-called
_perfection_ of an idyll $F$, which is a perfect idyll that has the same class
of (weak) $F$-matroids, but more morphisms. In particular, the composition of
$F$-morphisms is defined over the perfection of $F$. A similar, but more
subtle, line of thought leads to a satisfactory notion of morphisms for
matroid bundles. All this is part of follow-up work.

#### Arbitrary matrices as morphisms

Even for perfect idylls $F$, there is a more ample class of morphisms of
$F$-matroids, which are given by $F$-matrices that are not necessarily
submonomial. This notion of morphism is required to preserve certain
$F^{+}$-semimodules ${\mathcal{V}}_{M}^{+}\subseteq(F^{+})^{E}$ of “additive”
vectors for an $F$-matroid $M$.

This larger class of morphisms contains all strong maps (and more) in the case
of the Krasner hyperfield $F={\mathbb{K}}$. In case of a field $F$, it is
equal to the class of all linear maps that preserve the given subspaces of
vectors. Also this is part of follow-up work.

#### Quiver Grassmannians for $F$-representations

In the definition of quiver Grassmannians, it is not necessary that the chosen
ambient representation $\Lambda$ is ${{\mathbb{F}}_{1}}$-linear. For a fixed
quiver $Q$ and a perfect idyll $F$, any choice of a rank vector
${\mathbf{r}}=(r_{v})_{v\in Q_{0}}$, of a collection $(\Lambda_{v})_{v\in
Q_{0}}$ of finite ground sets $\Lambda_{v}$ and of a collection
$(\Lambda_{\myalpha})_{\myalpha\in Q_{1}}$ of
$\Lambda_{t(\myalpha)}\times\Lambda_{s(\myalpha)}$-matrices
$\Lambda_{\myalpha}$ with coefficients in $F$ leads to the definition of a
quiver Grassmannian $\operatorname{Gr}_{\mathbf{r}}(\Lambda)$ in terms of the
quiver Plücker relations (cf. Section 6.1), which is a band scheme defined
over $F$.

If the matrices $\Lambda_{\myalpha}$ are submonomial, then for every perfect
idyll $P$ together with a structure map $F\to P$, the rational point set
$\operatorname{Gr}_{\mathbf{r}}(\Lambda)(P)$ consists of all $(Q,P)$-matroids
$M$ with rank vector $\underline{\operatorname{{rk}}}(M)={\mathbf{r}}$ and
with $M_{\myalpha}=\Lambda_{\myalpha}$ for all $\myalpha\in Q_{1}$.

Since there is a unique morphism $F\to{\mathbb{K}}$, this leads to the
definition of the _Tits space_
$\operatorname{Gr}_{\mathbf{r}}(\Lambda)^{\textup{Tits}}$ as the subspace of
closed points of $\operatorname{Gr}_{\mathbf{r}}(\Lambda)({\mathbb{K}})$,
which should be thought of as the ${{\mathbb{F}}_{1}}$-points of
$\operatorname{Gr}_{\mathbf{r}}(\Lambda)$.

This more ample notion of quiver Grassmannians over $F={\mathbb{C}}$ applies,
in particular, to irreducible representations $\Lambda_{\mathbb{C}}$ of tame
quiver (of extended Dynkin type $ADE$). A _basis for $\Lambda_{\mathbb{C}}$_
is a collection of bases $\Lambda_{v}$ for $\Lambda_{{\mathbb{C}},v}$ (for all
$v\in Q_{0}$). A basis for $\Lambda_{\mathbb{C}}$ identifies
$\Lambda_{{\mathbb{C}},\myalpha}$ with a matrix $\Lambda_{\myalpha}$ (for all
$\myalpha\in Q_{1}$). The verification of sample cases leads us to expect the
following generalization of Theorem C.

######  Conjecture .

Let $Q$ be a tame quiver, ${\mathbf{r}}$ a rank vector for $Q$ and
$\Lambda_{\mathbb{C}}$ an irreducible representation of $Q$. Then there exists
a basis for $\Lambda_{\mathbb{C}}$ such that

$\mychi\big{(}\operatorname{Gr}_{{\mathbf{r}}^{\ast}}(\Lambda_{\mathbb{C}})\big{)}\
=\ \\#\operatorname{Gr}_{\mathbf{r}}(\Lambda)^{\textup{Tits}}.$

### Acknowledgements

The authors thank Jaiung Jun and Alex Sistko for their explanations on their
technique to compute Euler characteristics of complex quiver Grassmannians.
They thank Matt Baker and Victoria Schleis for helpful conversations. E. Vital
was supported by the National Council for Scientific and Technological
Development - CNPq - Brazil, Proc. 200268/2022-8.

##  1\. Background

###  1.1. Vector spaces over ${\mathbb{F}}_{1}$

A _vector space over ${{\mathbb{F}}_{1}}$_ is a pointed set, whose base point
we denote typically by $0$. An _${{\mathbb{F}}_{1}}$ -linear map_ between
${{\mathbb{F}}_{1}}$-vector spaces $V$ and $W$ is a base point preserving map
$f:V\to W$ such that $\\#f^{-1}(w)\leqslant 1$ for all $w\neq 0$. This defines
the category $\operatorname{{Vect}}_{{\mathbb{F}}_{1}}$.

The _adjoint_ of an ${\mathbb{F}}_{1}$-linear map $f:V\to W$ is the
${\mathbb{F}}_{1}$-linear map $f^{t}:W\to V$ given by

$f^{t}(w)\ =\ \begin{cases}v&\text{if}\;\;w\neq 0\text{ and }f(v)=w\\\
0&\text{otherwise}.\end{cases}$

###  1.2. Strong maps of classical matroids

Let $M$ be a matroid with ground set $E$. We denote the sets of circuits of
$M$ by ${\mathcal{C}}_{M}$ and its set of cocircuits by
${\mathcal{C}}_{M}^{\ast}$.

######  Definition .

A _vector_ of $M$ is a union of circuits of $M$. A _covector_ of $M$ is a
vector of the dual matroid $M^{*}$. We denote the set of vectors of $M$ by
$\mathcal{V}_{M}$ and its set of covectors by $\mathcal{V}^{*}_{M}$.

Since a circuit is the same as the complement of a cohyperplane ([Oxl11, Prop.
2.1.6]), a vector is nothing else than the complement of a coflat. For our
comparison of usual matroid theory with linear algebra and Baker-Bowler theory
(cf. Section 1.3), it is more convenient to work with vectors than with
coflats.

An alternative characterization of vectors is as follows. Two subsets $A$ and
$B$ of $E$ are _orthogonal_ , denoted by $A\perp B$, if $\\#(A\cap B)\neq 1$.
Two collections $\mathcal{X}$ and $\mathcal{Y}$ of subsets of $E$ are
_orthogonal_ , denoted by $\mathcal{X}\perp\mathcal{Y}$, if $X\perp Y$ for all
$X\in\mathcal{X}$ and $Y\in\mathcal{Y}$. For $\mathcal{X}\subseteq 2^{E}$, we
define $\mathcal{X}^{\perp}:=\\{Z\subseteq E\mid X\perp Z\text{ for all
}X\in\mathcal{X}\\}$. Since circuits are orthogonal to cocircuits ([Oxl11,
Prop. 2.1.11]), we have that

$\mathcal{V}_{M}^{\ast}\ =\ \mathcal{C}_{M}^{\perp}\ =\
{\mathcal{V}}_{M}^{\perp}.$

######  Notation .

We denote the closure operator of $M$ by $\textup{cl}_{M}$ and define the
matroid $\widetilde{M}:=M\oplus U_{1}^{0}$ on the set $\underline{E}$. In
other words, $\widetilde{M}$ is obtained from $M$ by adding a loop at $0$.

Let $N$ and $M$ be classical matroids with respective underlying sets $S$ and
$T$. A _strong map_ $\mysigma$ from $N$ to $M$, denoted by
$\mysigma:N\rightarrow M$, is a base point preserving map
$\mysigma:\underline{S}\rightarrow\underline{T}$ such that $\mysigma^{-1}(F)$
is a flat of $\widetilde{N}$ for any flat $F$ of $\widetilde{M}$ (see [Kun86,
Def. 8.1.1]). If $S=T$ and $\mysigma=\textup{id}_{\underline{S}}$ is a strong
map, $M$ is called a _quotient_ of $N$. By [Kun86, Prop. 8.1.3], a base point
preserving map $\mysigma:\underline{S}\rightarrow\underline{T}$ is a strong
map if, and only if,
$\mysigma\big{(}\textup{cl}_{N}(A)\big{)}\subseteq\textup{cl}_{\widetilde{M}}\big{(}\mysigma(A)\big{)}$
for every $A\subseteq S$.

######  Definition .

An _${\mathbb{F}}_{1}$ -linear strong map_ $\mysigma:N\rightarrow M$ is a
strong map $\mysigma:\underline{S}\rightarrow\underline{T}$ that is
${\mathbb{F}}_{1}$-linear.

Examples of strong maps are isomorphisms of matroids, restrictions $M|A\to M$,
contractions $M\to M/A$, quotient maps and the identification of parallel
elements. Every strong map is the composition of these elementary types of
strong maps (see [Kun86, pp. 227-228]). Among these types of strong maps,
isomorphisms, restrictions, contractions and quotient maps are
${{\mathbb{F}}_{1}}$-linear. Nontrivial identifications of parallel elements
are not.

In fact, the factorization theorem for strong maps ([Kun86, Thm 8.2.7])
asserts that every quotient map is the composition of a restriction followed
by a contraction. This fact does not generalize to other idylls (see [RG93,
Cor. 3.5]), which means that quotient maps play a central role as a building
block of matroid morphisms; also cf. Section 1.2 and Theorem 2.3.

######  Definition  ([Kun86, p. 228]).

Let $\mysigma:\underline{S}\rightarrow\underline{T}$ be a base point
preserving map and $M$ a matroid on $T$. We define $\mysigma^{-1}(M)$ as the
matroid on $S$ whose rank function is given by

$\operatorname{{rk}}_{\mysigma^{-1}(M)}(A):=\operatorname{{rk}}_{\widetilde{M}}\big{(}\mysigma(A)\big{)}.$

######  Remark .

Note that $M|(\operatorname{im}(\mysigma)-\\{0\\})$ is obtained from
$\mysigma^{-1}(M)$ by identifying parallel elements, deleting loops and
relabeling elements.

######  Proposition .

Let $\mysigma:\underline{S}\rightarrow\underline{T}$ be a base point
preserving map. Then $\mysigma:N\rightarrow M$ is a strong map if, and only
if, $\mysigma^{-1}(M)$ is a quotient of $N$.

###### Proof.

Using the fact that

$\big{\\{}\text{flats of
}\widetilde{\big{(}\mysigma^{-1}(M)\big{)}}\big{\\}}=\big{\\{}\mysigma^{-1}(F)\;\big{|}\;F\text{
is a flat of }\widetilde{M}\big{\\}},$

the Proposition follows. ∎

######  Remark .

By Section 1.2, a strong map $\mysigma:N\rightarrow M$ is
${\mathbb{F}}_{1}$-linear if, and only if, it does not identify parallel
elements.

The following result generalizes [Kun86, Prop. 8.1.6] from matroid quotients
to all strong maps.

######  Proposition .

Let $\mysigma:\underline{S}\rightarrow\underline{T}$ be a base point
preserving map. Then $\mysigma:N\rightarrow M$ is a strong map if, and only
if, $\mysigma^{-1}:2^{T}\rightarrow 2^{S}$ sends cocircuits of $M$ to
covectors of $N$.

###### Proof.

Assume that $\mysigma:N\rightarrow M$ is a strong map. Let $Z$ be a cocircuit
of $M$. Next we show that $\mysigma^{-1}(Z)\in\mathcal{V}^{*}_{N}$. Suppose
that there exists $C\in\mathcal{C}_{N}$ such that
$C\cap\mysigma^{-1}(Z)=\\{x\\}$ is a singleton. As
$\mathcal{C}^{*}_{M}=\mathcal{C}^{*}_{\widetilde{M}}$, the set
$H:=\underline{T}-Z$ is a hyperplane of $\widetilde{M}$ that contains
$\mysigma(C-\\{x\\})$. Thus

$\mysigma(x)\in\mysigma(C)\subseteq\mysigma\big{(}\textup{cl}_{N}(C-\\{x\\})\big{)}\subseteq\textup{cl}_{\widetilde{M}}\big{(}\mysigma(C-\\{x\\})\big{)}\subseteq
H,$

which is a contradiction, because $\mysigma(x)\in Z$.

Assume $\mysigma^{-1}(\mathcal{C}^{*}_{M})\subseteq\mathcal{V}^{*}_{N}$. Let
$A\subseteq S$. Next we prove that
$\mysigma\big{(}\textup{cl}_{N}(A)\big{)}\subseteq\textup{cl}_{\widetilde{M}}\big{(}\mysigma(A)\big{)}$.
Suppose that there exists $x\in\textup{cl}_{N}(A)$ such that
$\mysigma(x)\in\underline{T}-\textup{cl}_{\widetilde{M}}\big{(}\mysigma(A)\big{)}$.
As $\textup{cl}_{\widetilde{M}}\big{(}\mysigma(A)\big{)}$ is the intersection
of all hyperplanes of $\widetilde{M}$ that contain $\mysigma(A)$, there exists
a hyperplane $H$ of $\widetilde{M}$ such that $\mysigma(A)\subseteq H$ and
$\mysigma(x)\notin H$. Thus $Z:=\underline{T}-H$ is a cocircuit of
$\widetilde{M}$ with $\mysigma(x)\in Z$ and $\mysigma(A)\cap Z=\emptyset$.

As $\mathcal{C}^{*}_{\widetilde{M}}=\mathcal{C}^{*}_{M}$, it follows that
$\mysigma^{-1}(Z)\in\mathcal{V}^{*}_{N}$, which implies that there exists a
cocircuit $Y$ of $N$ such that $x\in Y\subseteq\mysigma^{-1}(Z)$. Thus
$L:=S-Y$ is a hyperplane of $N$ with $x\notin L$. Because $A\subseteq L$, it
follows that $\textup{cl}_{N}(A)\subseteq L$ and, consequently,
$x\notin\textup{cl}_{N}(A)$, which is a contradiction. ∎

######  Corollary .

Let $\mysigma:\underline{S}\rightarrow\underline{T}$ be an
${\mathbb{F}}_{1}$-linear map. Then the following are equivalent:

  1. (1)

$\mysigma:N\rightarrow M$ is a strong map;

  2. (2)

$\mysigma^{t}:M^{*}\rightarrow N^{*}$ is a strong map;

  3. (3)

$\mysigma({\mathcal{C}}_{N})\perp{\mathcal{C}}^{*}_{M}$.

###### Proof.

As $\mysigma$ is ${\mathbb{F}}_{1}$-linear

$\\#\big{(}X\cap\mysigma^{-1}(Y)\big{)}=\\#\big{(}(\mysigma^{t})^{-1}(X)\cap
Y\big{)}$

for all $X\subseteq S$ and $Y\subseteq T$. In particular,
$\mysigma^{-1}(\mathcal{C}^{*}_{M})\perp\mathcal{C}_{N}$ if, and only if,
$\mathcal{C}_{M^{*}}\perp(\mysigma^{\mathfrak{t}})^{-1}(\mathcal{C}^{*}_{N^{*}})$.
Thus by Section 1.2, we conclude that (1) is equivalent to (2).

Now, note that $\mysigma(X)-\\{0_{T}\\}=(\mysigma^{\mathfrak{t}})^{-1}(X)$ for
each $X\subseteq S$. Therefore the equivalence between (2) and (3) follows
from Section 1.2. ∎

###  1.3. Baker-Bowler theory

We recall the theory of matroids with coefficients introduced by Baker and
Bowler in [BB19].

Idylls. A _pointed abelian group_ is a (multiplicatively written) commutative
monoid $F$ with a distinguished element $0$ such that $F^{\times}=F-\\{0\\}$
and $0\cdot a=0$ for all $a\in F$. An _idyll_ is a pointed abelian group $F$
together with a subset $N_{F}$ (called the _nullset_) of the group semiring
${\mathbb{N}}[F^{\times}]$ that satisfies the following axioms:

  1. (I1)

$N_{F}$ is a proper ideal of ${\mathbb{N}}[F^{\times}]$, i.e. $0\in N_{F}$,
$1\notin N_{F}$, $a+b\in N_{F}$ and $ac\in N_{F}$ for all $a,b\in N_{F}$ and
$c\in{\mathbb{N}}[F^{\times}]$.

  2. (I2)

There is a unique element $-1\in F^{\times}$ with $1+(-1)\in N_{F}$;

Note that these axioms imply that every element $a\in F$ has an _additive
inverse_ $-a=(-1)\cdot a$, which is the unique element with $a+(-a)\in N_{F}$.
This further implies that $(-1)^{2}=1$ and $a\in N_{F}$ only if $a=0$.

A _morphism of idylls_ is a multiplicative map $f:F\rightarrow G$ with
$f(0)=0$ and $f(1)=1$ such that $\sum f(a_{i})\in N_{G}$ for all $\sum
a_{i}\in N_{F}$. This defines the category $\operatorname{{Idylls}}$.

######  Example  (Fields as idylls).

A field $k$ is naturally an idyll if we replace its addition by the nullset

$\textstyle N_{k}:=\left\\{\sum a_{i}\in{\mathbb{N}}[k^{\times}]\mid\sum
a_{i}=0\text{ in }k\right\\}.$

By abuse of notation, we denote this idyll by the same symbol $k$. More to the
point, the category of fields embeds as a full subcategory into
$\operatorname{{Idylls}}$, which allows us to consider fields as particular
types of idylls.

######  Example .

The _regular partial field_ is the multiplicative pointed group

${\mathbb{F}}_{1}^{\pm}=\\{0,1,-1\\}\quad\text{ with nullset }\quad
N_{{\mathbb{F}}_{1}^{\pm}}=\\{m\cdot 1+n\cdot(-1)\mid m=n\\}.$

This idyll is the initial object of the category $\operatorname{{Idylls}}$.
Given an idyll $F$, the unique morphism ${\mathbb{F}}_{1}^{\pm}\rightarrow F$
is determined by $1_{{\mathbb{F}}_{1}^{\pm}}\mapsto 1_{F}$ and
$-1_{{\mathbb{F}}_{1}^{\pm}}\mapsto-1_{F}$.

######  Example  (Hyperfields as idylls).

Roughly speaking, a _hyperfield_ is a field where the standard addition is
replaced by a multivaluated _hyperaddition_ , see [DL07] for details. Given a
hyperfield $H$ with hyperaddition $\boxplus:H\times H\rightarrow 2^{H}$, we
consider $H$ as an idyll with respect to the nullset $N_{H}:=\big{\\{}\sum
a_{i}\in{\mathbb{N}}[H^{\times}]\mid
0\in\operatorname*{\,\raisebox{-2.2pt}{\larger[2]{$\boxplus$}}\,}a_{i}\big{\\}}$.

Some examples of interest are:

  * •

The _Krasner hyperfield_ is ${\mathbb{K}}=\\{0,1\\}$ with hyperaddition given
by $a\boxplus 0=\\{a\\}$ and $1\boxplus 1=\\{0,1\\}$. As an idyll,
$-1_{{\mathbb{K}}}=1$ and its nullset is ${\mathbb{N}}-\\{1\\}$. This
hyperfield is the final object in the category $\operatorname{{Idylls}}$.

  * •

The _tropical hyperfield_ is ${\mathbb{T}}={\mathbb{R}}_{\geqslant 0}$ with
the usual multiplication, whose hyperaddition is given by $a\boxplus
b=\textup{max}\\{a,b\\}$ for $b\neq a$, and $a\boxplus a=[0,a]$. As an idyll,
$-1_{{\mathbb{T}}}=1$ and its nullset is

$\textstyle N_{\mathbb{T}}=\big{\\{}\sum a_{i}\,\big{|}\,\text{ the maximum of
}\\{a_{i}\\}\text{ appears at least twice}\big{\\}}.$

  * •

The _phase hyperfield_ is ${\mathbb{P}}=S^{1}\cup\\{0\\}$, where
$S^{1}=\\{z\in{\mathbb{C}}\mid z\text{ has norm }1\\}$, with usual
multiplication and hyperaddition given by

$a\boxplus b=\left\\{\frac{\myalpha a+\mybeta b}{||\myalpha a+\mybeta
b||}\;\bigg{|}\;\myalpha,\mybeta\in{\mathbb{R}}_{>0}\right\\}$

for $b\neq-a$, and $a\boxplus-a=\\{-a,0,a\\}$. As an idyll,
$-1_{{\mathbb{P}}}=-1$ and its nullset is

$\textstyle N_{{\mathbb{P}}}=\left\\{\sum
x_{i}\in{\mathbb{N}}[S^{1}]\;\big{|}\;\sum\myalpha_{i}x_{i}=0\text{ in
}{\mathbb{C}},\text{ for some }\myalpha_{i}\text{ in
}{\mathbb{R}}_{>0}\right\\}.$

######  Remark .

Historically, the notion of _hypergroup_ was introduced by Marty in 1934
[Mar34]. Around 1940, more general aspects of _hyperstructure_ were
considered, when Krasner defined the notion of _hyperring_ and _hyperfield_.
For instance, Jun develop algebraic geometry over hyperrings in [Jun18], and
the second author studies tropical geometry over the tropical hyperfield in
[Lor22].

######  Notation .

For an idyll $F$, a set $E$, an integer $r$, a tuple ${\bf
x}=(x_{1},\dots,x_{r})\in E^{r}$ and a permutation
$\mysigma\in\mathfrak{S}_{r}$, we define:

  * •

$|{\bf x}|:=\\{x_{1},\dots,x_{r}\\}\subseteq E$;

  * •

the sign $\operatorname{{sign}}(\mysigma)\in\\{\pm 1\\}$, considered as an
element of $F$;

  * •

${\bf x}^{\mysigma}:=(x_{\mysigma(1)},\dots,x_{\mysigma(r)})$;

  * •

${\bf x}_{\widehat{k}}:=(x_{1},\dots,\widehat{x}_{k},\dots,x_{r})\in E^{r-1}$.

Matroids over idylls. Let $F$ be an idyll, $E$ a finite set and $r$ an
integer. A _Grassmann-Plücker function_ of rank $r$ on $E$ with coefficients
in $F$ is a nonzero function $\myvarphi:E^{r}\rightarrow F$ satisfying the
following two properties:

  1. (GP1)

$\myvarphi$ is _alternating_ , i.e. $\myvarphi({\bf
x})=\operatorname{{sign}}(\mysigma)\cdot\myvarphi({\bf x}^{\mysigma})$ for all
$\mysigma\in\mathfrak{S}_{r}$ and $\myvarphi({\bf x})=0$ whenever
$\\#|{\mathbf{x}}|<r$;

  2. (GP2)

for all $x_{2},\dotsc,x_{r},y_{0},\dotsc,y_{r}\in E$,

$\sum_{k=0}^{r}\
(-1)^{k}\cdot\myvarphi(y_{0},\dotsc,\widehat{y_{k}},\dotsc,y_{r})\cdot\myvarphi(y_{k},x_{2},\dotsc,x_{r})\
\in\ N_{F}.$

The relations in (GP2) are called _Plücker relations_. Two Grassmann-Plücker
functions $\myvarphi_{1}$ and $\myvarphi_{2}$ are _equivalent_ if
$\myvarphi_{1}=a\cdot\myvarphi_{2}$ for some $a$ in $F^{\times}$. An _$F$
-matroid_ of rank $r$ on $E$ over $F$ is an equivalence class $[\myvarphi]$ of
a Grassmann-Plücker function $\myvarphi:E^{r}\to F$.

######  Example .

Given a ${\mathbb{K}}$-matroid $M=[\myvarphi]$ of rank $r$ on the set $E$, the
collection

${\mathcal{B}}=\big{\\{}\\{x_{1},\dots,x_{r}\\}\subseteq
E\mid\myvarphi(x_{1},\dots,x_{r})\neq 0\big{\\}}$

is the set of bases of a classical matroid, called the _underlying matroid_ of
$M$ and denoted by $\underline{M}$. By [BB19, Remark 3.19], the map

$\begin{array}[]{ccc}\\{{\mathbb{K}}\text{-matroids on
}E\\}&\longrightarrow&\\{\text{classical matroids on }E\\}\\\
M&\longmapsto&\underline{M}\end{array}$

is a bijection. This shows that matroids over idylls are generalizations of
classical matroids.

Functoriality. Given a morphism of idylls $f:F\rightarrow G$ and a Grassmann-
Plücker function $\myvarphi:E^{r}\rightarrow F$ with coefficients in $F$, the
composition $f\circ\myvarphi:E^{r}\rightarrow G$ is a Grassmann-Plücker
function with coefficients in $G$. For an $F$-matroid $M=[\myvarphi]$, we
define the _push-forward_ of $M$ _along_ $f$ as the $G$-matroid
$f_{*}(M):=[f\circ\myvarphi]$.

######  Definition .

Given an $F$-matroid $M$, its _underlying matroid_ is the classical matroid
$\underline{M}:=\underline{f_{*}(M)}$, where $f:F\to{\mathbb{K}}$ the unique
idyll morphism into ${\mathbb{K}}$.

Duality. Let $F$ be an idyll and $M=[\myvarphi]$ an $F$-matroid of rank $r$ on
the set $E$. Fix a total order on $E$. For ${\bf y}\in E^{n}$ with
$|\mathbf{y}|=E$, let $\mysigma$ be the permutation for which
$y_{\mysigma(1)}<\cdots<y_{\mysigma(n)}$, and define
$\operatorname{{sign}}({\bf
y}):=\operatorname{{sign}}(\mysigma)\in\\{1,-1\\}$.

The _dual_ of $M$ is the $F$-matroid $M^{*}:=[\myvarphi^{*}]$, where
$\myvarphi^{*}:E^{n-r}\rightarrow F$ is the Grassmann-Plücker function given
by

$\myvarphi^{*}({\bf x})\ :=\ \begin{cases}0,\text{ if there exists }i\neq
j\text{ with }x_{i}=x_{j}\\\ \operatorname{{sign}}({\bf x},{\bf
x}^{\prime})\cdot\myvarphi({\bf x}^{\prime}),\text{ otherwise},\end{cases}$

where ${\bf x}^{\prime}\in E^{r}$ is such that $E=|{\bf x}|\sqcup|{\bf
x}^{\prime}|$. The dual of $M$ is independent of the order on $E$ and
satisfies $(M^{*})^{*}=M$.

Circuits and vectors. Let $X=(X_{e})_{e\in E}$ be an element of $F^{E}$. The
_support_ of $X$ is the set $\underline{X}:=\\{e\in E\mid X_{e}\neq 0\\}$.
More generally, for a subset ${\mathcal{S}}$ of $F^{E}$, its support is
$\underline{{\mathcal{S}}}:=\\{\underline{X}\mid X\in{\mathcal{S}}\\}$.

The _inner product_ of two elements $X$ and $Y$ of $F^{E}$ is defined as

$X\cdot Y\ =\ \sum X_{e}\cdot Y_{e}\in{\mathbb{N}}[F^{\times}].$

We say that $X$ is _orthogonal_ to $Y$, denoted by $X\perp Y$, if $X\cdot Y$
is in $N_{F}$. Two subsets ${\mathcal{X}}$ and ${\mathcal{Y}}$ of $F^{E}$ are
_orthogonal_ , written as ${\mathcal{X}}\perp{\mathcal{Y}}$, if $X\perp Y$ for
each $X$ in ${\mathcal{X}}$ and $Y$ in ${\mathcal{Y}}$. The _orthogonal
complement_ of a subset ${\mathcal{S}}\subseteq F^{E}$ is
${\mathcal{S}}^{\perp}:=\big{\\{}X\in
F^{E}\mid\\{X\\}\perp{\mathcal{S}}\big{\\}}$.

Let $M=[\myvarphi:E^{r}\to F]$ be an $F$-matroid and define $\Xi_{M}$ as the
collection of all ${\mathbf{y}}=(y_{0},\dotsc,y_{r})\in E^{r+1}$ for which
$\\#|{\mathbf{y}}|=r+1$ and for which
$\myvarphi(y_{0},\dotsc,\widehat{y_{k}},\dotsc,y_{r})\neq 0$ for some
$k=0,\dotsc,r$. We define the _fundamental circuit of ${\mathbf{y}}\in\Xi_{M}$
with respect to $\myvarphi$_ as

$X_{{\mathbf{y}}}(x)\ =\
\begin{cases}(-1)^{i}\cdot\myvarphi(y_{0},\dotsc,\widehat{y_{i}},\dotsc,y_{r})&\text{if
}x=y_{i},\\\ 0&\text{otherwise}.\end{cases}$

The set of _$F$ -circuits_ of $M$ is

${\mathcal{C}}_{M}\ =\ \big{\\{}a\cdot X_{{\mathbf{y}}}\,\big{|}\,a\in
F^{\times},\ {\mathbf{y}}\in\Xi_{M}\big{\\}}.$

The set of _$F$ -cocircuits_ of $M$ is
${\mathcal{C}}^{*}_{M}:={\mathcal{C}}_{M^{*}}$. We define the set of _$F$
-vectors_ and of _$F$ -covectors_ of $M$ as

${\mathcal{V}}_{M}:=({\mathcal{C}}^{*}_{M})^{\perp}\quad\text{and}\quad{\mathcal{V}}^{*}_{M}:={\mathcal{C}}_{M}^{\perp},$

respectively. The circuits of $\underline{M}$ are exactly the supports

$\underline{X}\ =\ \\{e\in E\mid X_{e}\neq 0\\}$

of the $F$-circuits $X\in{\mathcal{C}}_{M}$ of $M$, i.e. the elements of
$\underline{{\mathcal{C}}_{M}}$.

An idyll $F$ is _perfect_ if ${\mathcal{V}}_{M}\perp{\mathcal{V}}^{*}_{M}$ for
all $F$-matroids $M$. Examples of perfect idylls are: fields, the regular
partial field ${\mathbb{F}}^{\pm}_{1}$, and the hyperfields ${\mathbb{K}}$ and
${\mathbb{T}}$. The phase hyperfield ${\mathbb{P}}$ is not perfect. For more
details, see [BB19].

######  Example .

Let $k$ be a field, $E$ a finite set and $r$ an integer. By [BB19, Example
3.30], the following maps are bijections:

$\begin{array}[]{ccccc}\left\\{\begin{matrix}\text{subspaces of }k^{E}\\\
\text{ of dimension
}r\end{matrix}\right\\}&\longleftarrow&\left\\{\begin{matrix}k\text{-matroids
on }$E$\\\ \text{ of rank
}r\end{matrix}\right\\}&\longrightarrow&\left\\{\begin{matrix}\text{subspaces
of }k^{E}\text{ of}\\\ \text{dimension }\\#E-r\end{matrix}\right\\}\\\ &&\\\
\mathcal{V}^{*}_{M}=(\mathcal{V}_{M})^{\perp}&\longmapsfrom&M&\longmapsto&\mathcal{V}_{M}.\end{array}$

Minors. Let $M=[\myvarphi]$ be an $F$-matroid of rank $r$ on $E$ and
$A\subseteq E$ a subset.

  1. (1)

(Contraction) Let $\ell$ be the rank of $A$ in $\underline{M}$ and
$\\{a_{1},\dotsc,a_{\ell}\\}$ a maximal independent subset of $A$. We define
$\myvarphi/A:E^{r-\ell}\rightarrow F$ by

$(\myvarphi/A)(x_{1},\dotsc,x_{r-\ell})\ =\
\myvarphi(x_{1},\dotsc,x_{r-\ell},a_{1},\dotsc,a_{\ell}).$

  2. (2)

(Deletion) Let $k$ be the rank of $E\backslash A$ in $\underline{M}$ and
$\\{b_{1},\dotsc,b_{r-k}\\}\subseteq A$ a basis of $\underline{M}/(E\backslash
A)$. We define $\myvarphi\backslash A:(E\backslash A)^{k}\rightarrow F$ by

$(\myvarphi\backslash A)(y_{1},\dotsc,y_{k})\ =\
\myvarphi(y_{1},\dotsc,y_{k},b_{1},\dotsc,b_{r-k}).$

  3. (3)

(Restriction) Let $q$ be the rank of $A$ in $\underline{M}$. Define
$\myvarphi|A:=\myvarphi\backslash(E\backslash A):A^{q}\rightarrow F$.

By [BB19, Lemma 4.4], $\myvarphi/A$, $\myvarphi\backslash A$ and $\myvarphi|A$
are Grassmann-Plücker functions, and the $F$-matroids $M/A:=[\myvarphi/A]$
(_contraction_ of $M$ by $A$), $M\backslash A:=[\myvarphi\backslash A]$
(_deletion_ of $A$ from $M$) and $M|A:=[\myvarphi|A]$ (_restriction_ of $M$ to
$A$) do not depend on any choices. By [BB19, Thm. 3.29],
$(M/A)^{*}=M^{*}\backslash A$ and $(M\backslash A)^{*}=M^{*}/A$.

##  2\. Category of $F$-matroids

###  2.1. Submonomial matrices

Let $F$ be a perfect idyll and $T$ and $S$ finite sets.

######  Definition .

An $F$_-matrix_ (indexed by $T\times S$) is a matrix
$\Phi=(\Phi_{i,j})_{(i,j)\in T\times S}$ with entries $\Phi_{i,j}\in F$. An
$F$-matrix $\Phi$ is _submonomial_ if every row and column of $\Phi$ contains
at most one nonzero entry. We denote the _transpose_ of $\Phi$ by $\Phi^{t}$,
which has entries $\Phi^{t}_{i,j}:=\Phi_{j,i}$.

Note that the product $\Phi\cdot X$ of a submonomial $F$-matrix $\Phi$ indexed
by $T\times S$ with $X\in F^{S}$ is a well-defined element of $F^{T}$, since
the sum in the definition of $(\Phi\cdot X)_{i}=\sum_{j\in S}\Phi_{i,j}X_{j}$
contains at most one nonzero term. Due to this, we also consider a submonomial
$F$-matrix $\Phi$ as a map $\Phi:F^{S}\to F^{T}$. For a subset
$\mathcal{X}\subseteq F^{S}$, we use $\Phi\cdot\mathcal{X}$ to denote the
image $\\{\Phi\cdot X\mid X\in\mathcal{X}\\}\subseteq F^{T}$.

Let $\Phi$ and $\Psi$ be submonomial $F$-matrices indexed by $T\times S$ and
$W\times T$, respectively. The product $\Psi\cdot\Phi$ is well-defined and
corresponds to the composition $\Psi\circ\Phi:F^{S}\to F^{W}$ of the maps
$\Phi:F^{S}\to F^{T}$ and $\Psi:F^{T}\to F^{W}$. The transpose $\Phi^{t}$ of
$\Phi$ is also submonomial, and thus defines a map $\Phi^{t}:F^{T}\to F^{S}$.

######  Definition .

The _underlying map_ of a submonomial $F$-matrix $\Phi$ indexed by $T\times S$
is the ${\mathbb{F}}_{1}$-linear map $\uline{\Phi}:\uline{S}\to\uline{T}$
given by

$\uline{\Phi}(j)\ :=\ \begin{cases}i&\text{if $j\neq 0$ and $\Phi_{i,j}\neq
0$;}\\\ 0&\text{if $j=0$, or if $j\neq 0$ and $\Phi_{i,j}=0$ for all $i\in
T$.}\end{cases}$

which satisfies $(\underline{\Phi})^{t}=\underline{(\Phi^{t})}$ (see Section
1.1). For $s\in\underline{S}$, we define

$\Phi_{s}\ :=\ \begin{cases}\Phi_{\underline{\Phi}(s),s}&\text{if $s\neq 0$
and $\Phi_{\underline{\Phi}(s),s}\neq 0$;}\\\ 0&\text{if $s=0$, or if $s\neq
0$ and $\Phi_{i,s}=0$ for all $i\in T$.}\end{cases}$

######  Remark .

Note that since $\Phi$ is submonomial, there is at most one $i$ for every
given $j$ such that $\Phi_{i,j}\neq 0$. Note further that $(\Phi^{t}\cdot
Z)_{s}=\Phi_{s}\cdot Z_{\underline{\Phi}(s)}$ for $Z\in F^{T}$ and $s\in S$.

###  2.2. Morphisms of matroids with coefficients

######  Definition  (Category of $F$-matroids).

Let $N$ and $M$ be $F$-matroids with respective underlying sets $S$ and $T$. A
_morphism_ of $F$-matroids from $N$ to $M$ is a submonomial $F$-matrix $\Phi$
indexed by $T\times S$ such that
$\Phi\cdot{\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{M}$. We write $\Phi:N\to M$
for a morphism $\Phi$ from $N$ to $M$. This defines the category
$\operatorname{Mat}_{F}$ of $F$-matroids.

######  Example .

A morphism of ${\mathbb{K}}$-matroids is a particular type of strong map,
which includes isomorphisms, restrictions, contractions and quotients, but not
the identification of parallel elements. A rigorous discussion of this
relation can be found in Section 2.3.

In the case of a field $k$, a submonomial matrix $\Phi$ describes a linear map
$\Phi:k^{S}\rightarrow k^{T}$ that sends coordinate vectors of $k^{S}$ (i.e.
vectors with only one nonzero coefficient) to coordinate vectors of $k^{T}$ or
$0$. Therefore a morphism $\Phi:N\to M$ of $k$-matroids (with respective
ground sets $S$ and $T$) is a submonomial linear map $\Phi:k^{S}\rightarrow
k^{T}$ with $\Phi\cdot{\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{M}$, where
${\mathcal{V}}_{N}$ and ${\mathcal{V}}_{M}$ are the vector sets of $M$ and
$N$, respectively.

Similarly, a morphism $\Phi:N\to M$ of ${\mathbb{T}}$-matroids (with
respective ground sets $S$ and $T$) is a submonomial linear map
${\mathbb{T}}^{S}\to{\mathbb{T}}^{T}$ such that
$\Phi\cdot{\mathcal{V}}_{N}\subseteq{\mathcal{V}}_{M}$.

A quotient $M$ of an oriented matroid $N$ is the same thing as an
${\mathbb{S}}$-morphism $\Phi:N\to M$ that is given by a square identity
matrix $\Phi$. Examples of other ${\mathbb{S}}$-morphisms are

$\big{[}\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\big{]}:M\to
M^{\prime}\qquad\text{and}\qquad\big{[}\begin{smallmatrix}1&0\\\
0&0\end{smallmatrix}\big{]}:M\to N$

where $M$, $M^{\prime}$ and $N$ are the ${\mathbb{S}}$-matroids with
respective vector sets

$\textstyle{\mathcal{V}}_{M}\ =\ \big{\\{}\big{[}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}-1\\\
-1\end{smallmatrix}\big{]}\big{\\}},\quad{\mathcal{V}}_{M^{\prime}}\ =\
\big{\\{}\big{[}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
-1\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}-1\\\
1\end{smallmatrix}\big{]}\big{\\}},\quad{\mathcal{V}}_{N}\ =\
\big{\\{}\big{[}\begin{smallmatrix}0\\\
0\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]},\big{[}\begin{smallmatrix}-1\\\
0\end{smallmatrix}\big{]}\big{\\}}.$

Thus the underlying matroid of both $M$ and $M^{\prime}$ is $U^{1}_{2}$ and
the underlying matroid of $N$ is $U^{1}_{1}\oplus U^{0}_{1}$.

Recall that the inner product of two tuples $X,Y\in F^{E}$ is defined as the
element $X\cdot Y=\sum_{e\in E}X_{e}Y_{e}$ of ${\mathbb{N}}[F^{\times}]$.

######  Lemma .

Let $\Phi$ be a submonomial $F$-matrix indexed by $T\times S$. Let $X\in
F^{S}$ and $Y\in F^{T}$. Then $(\Phi\cdot X)\cdot Y=X\cdot(\Phi^{t}\cdot Y)$.

###### Proof.

This follows from a direct computation:

$(\Phi\cdot X)\cdot Y\ =\ \sum_{e\in T}\Phi_{e,\underline{\Phi}^{t}(e)}\cdot
X_{\underline{\Phi}^{t}(e)}\cdot Y_{e}\ =\ \sum_{e\in
S}X_{e}\cdot\Phi^{t}_{e,\underline{\Phi}(e)}\cdot Y_{\underline{\Phi}(e)}\ =\
X\cdot(\Phi^{t}\cdot Y).\qed$

######  Definition .

Given a function $\mymu:E^{r}\rightarrow F$ into a perfect idyll $F$, we
define

$\begin{array}[]{clll}\widetilde{\mymu}:&\underline{E}^{r}&\longrightarrow&\quad\;F\\\
&{\bf x}&\longmapsto&\begin{cases}\mymu({\bf x})&\text{if }x_{i}\neq 0\text{
for all }i\\\ 0&\text{otherwise}.\end{cases}\end{array}$

If $M=[\mymu]$ is an $F$-matroid on $E$, then
$\widetilde{M}:=[\widetilde{\mymu}]$ is an $F$-matroid on $\underline{E}$ with
$\underline{\widetilde{M}}=\underline{M}\oplus U_{1}^{0}$, i.e.
$\widetilde{M}$ is obtained from $M$ by adding a loop (cf. Section 1.2).

######  Theorem 2.1.

Let $N=[\mynu]$ be an $F$-matroid of rank $w$ on $S$. Let $M=[\mymu]$ be an
$F$-matroid of rank $r$ on $T$ and $\Phi$ a submonomial $F$-matrix indexed by
$T\times S$. Then the following are equivalent:

  1. (1)

$\Phi$ is a morphism from $N$ to $M$;

  2. (2)

$\Phi\cdot{\mathcal{C}}_{N}\perp{\mathcal{C}}_{M}^{\ast}$;

  3. (3)

for all $y_{0},\dotsc,y_{w}\in S$ and $x_{2},\dotsc,x_{r}\in T$,

$\sum_{k=0}^{w}\
(-1)^{k}\cdot\mynu(y_{0},\dotsc,\widehat{y_{k}},\dotsc,y_{w})\cdot{\Phi_{{\underline{\Phi}}(y_{k}),y_{k}}}\cdot\widetilde{\mymu}\big{(}\underline{\Phi}(y_{k}),x_{2},\dotsc,x_{r}\big{)}\
\in\ N_{F}.$

###### Proof.

We first prove the equivalence between (1) and (2). Assume (1). As
${\mathcal{C}}_{N}\subseteq{\mathcal{V}}_{N}$, it follows that
$\Phi\cdot{\mathcal{C}}_{N}\subseteq{\mathcal{V}}_{M}=({\mathcal{C}}_{M}^{*})^{\perp}$,
which implies (2).

Assume (2). By Section 2.2,
${\mathcal{C}}_{N}\perp\Phi^{t}\cdot{\mathcal{C}}_{M}^{*}$, which implies
${\mathcal{V}}_{N}^{*}=({\mathcal{C}}_{N})^{\perp}\supseteq\Phi^{t}\cdot{\mathcal{C}}_{M}^{*}$.
As $F$ is perfect, one has
${\mathcal{V}}_{N}=({\mathcal{V}}_{N}^{*})^{\perp}\subseteq(\Phi^{t}\cdot{\mathcal{C}}_{M}^{*})^{\perp}$,
i.e. ${\mathcal{V}}_{N}\perp\Phi^{t}\cdot{\mathcal{C}}_{M}^{*}$. Again by
Section 2.2, $\Phi\cdot{\mathcal{V}}_{N}\perp{\mathcal{C}}_{M}^{*}$, which
implies
$\Phi\cdot{\mathcal{V}}_{N}\subseteq({\mathcal{C}}_{M}^{*})^{\perp}={\mathcal{V}}_{M}$.
Thus (1) follows.

Next we prove the equivalence between (2) and (3). Assume (3). Let
$Z\in\mathcal{C}^{*}_{M}=\mathcal{C}_{M^{*}}$. By [BB19, p. 841], there exists
$z_{0}\in\underline{Z}$, a basis $\\{z_{1},\dotsc,z_{n-r}\\}$ of
$\underline{M}^{*}$ containing $\underline{Z}-z_{0}$, and $\myalpha\in
F^{\times}$ such that

$Z_{z_{i}}=\myalpha\cdot(-1)^{i}\cdot\mymu^{*}(z_{0},\dotsc,\widehat{z_{i}},\dotsc,z_{n-r})$

for all $i$. Let $\textbf{z}:=(z_{0},\dotsc,z_{n-r})\in T^{n-r+1}$ and
$\textbf{x}\in T^{r-1}$ such that $|{\bf z}|\sqcup|{\bf x}|=T$. Similarly, for
$Y\in\mathcal{C}_{N}$, there exist $y_{0}\in\underline{Y}$, a basis
$B=\\{y_{1},\dotsc,y_{w}\\}$ of $\underline{N}$ containing
$\underline{Y}-y_{0}$, and $\mychi\in F^{\times}$ such that

$Y_{y_{j}}=\mychi\cdot(-1)^{j}\cdot\mynu({\bf y}_{\widehat{j}})$

for all $j=0,\dotsc,w$, where $\textbf{y}:=(y_{0},\dotsc,y_{w})$. Let
$U:=|{\bf z}|\cap\underline{\Phi}\big{(}|{\bf y}|)$ and define

$I\ :=\ \big{\\{}i\in\\{0,\dotsc,n-r\\}\,\big{|}\,z_{i}\in
U\big{\\}}\qquad\text{and}\qquad J\ :=\
\big{\\{}j\in\\{0,\dotsc,w\\}\,\big{|}\,\underline{\Phi}(y_{j})\in
U\big{\\}}.$

As $\underline{\Phi}$ is ${\mathbb{F}}_{1}$-linear, it induces a bijection
$\mathfrak{b}:J\rightarrow I$ such that
$\underline{\Phi}(y_{j})=z_{\mathfrak{b}(j)}$, for all $j\in J$. By Section
2.2 and Section 2.1, we have (using the shorthand notation from Section 1.3
and Section 2.1):

$\displaystyle(\mychi{\myalpha})^{-1}\cdot\big{(}Z\cdot(\Phi\cdot
Y)\big{)}=(\mychi{\myalpha})^{-1}\cdot\big{(}Y\cdot(\Phi^{t}\cdot Z)\big{)}$
$\displaystyle=$ $\displaystyle(\mychi{\myalpha})^{-1}\cdot\underset{s\in
S}{\sum}Y_{s}\cdot(\Phi^{t}\cdot
Z)_{s}=(\mychi{\myalpha})^{-1}\cdot\underset{j\in J}{\sum}\
Y_{j}\cdot(\Phi^{t}\cdot Z)_{j}$ $\displaystyle=$
$\displaystyle(\mychi\myalpha)^{-1}\cdot\sum_{j\in
J}{\bigl{(}}{\mychi\cdot(-1)^{j}\mynu({\textbf{y}}_{\widehat{j}})}{\bigr{)}}\cdot{\bigl{(}}\Phi_{y_{j}}\cdot\myalpha\cdot(-1)^{{\mathfrak{b}}(j)}\mymu^{*}({\mathbf{z}}_{\widehat{{\mathfrak{b}}(j)}}){\bigr{)}}$
$\displaystyle=$ $\displaystyle\underset{j\in J}{\sum}\
(-1)^{\big{(}j+{\mathfrak{b}}(j)\big{)}}\cdot\mynu({\textbf{y}}_{\widehat{j}})\cdot{\Phi_{y_{j}}}\cdot\operatorname{{sign}}({\textbf{z}}_{\widehat{{\mathfrak{b}(j)}}},z_{\mathfrak{b}(j)},{\mathbf{x}})\cdot\mymu(z_{\mathfrak{b}(j)},{\mathbf{x}})$
$\displaystyle=$ $\displaystyle\underset{j\in J}{\sum}\
(-1)^{\big{(}j+\text{$\mathfrak{b}$}(j)\big{)}}\cdot\operatorname{{sign}}({\bf
z}_{\widehat{\text{$\mathfrak{b}$}(j)}},z_{\text{$\mathfrak{b}$}(j)},{\bf
x})\cdot\mynu({\bf
y}_{\widehat{j}})\cdot{\Phi_{y_{j}}}\cdot\mymu(\underline{\Phi}(y_{j}),{\bf
x}),$

because $Y_{s}\cdot(\Phi^{t}\cdot Z)_{s}=0$ for $s\notin J$. Furthermore, for
all $j\in J$ it follows that

$\displaystyle(-1)^{(n-r)}\cdot\operatorname{{sign}}({\bf z},{\bf x})$
$\displaystyle=(-1)^{2(n-r)-\mathfrak{b}(j)}\cdot\operatorname{{sign}}({\bf
z}_{\widehat{\mathfrak{b}(j)}},z_{\mathfrak{b}(j)},{\bf x})$
$\displaystyle=(-1)^{\mathfrak{b}(j)}\cdot\operatorname{{sign}}({\bf
z}_{\widehat{\mathfrak{b}(j)}},z_{\mathfrak{b}(j)},{\bf x}).$

Thus

$\displaystyle(-1)^{(n-r)}\cdot\operatorname{{sign}}({\bf z},{\bf
x})\cdot(\mychi{\myalpha})^{-1}\cdot\big{(}Y\cdot(\Phi^{t}\cdot Z)\big{)}$
$\displaystyle=$ $\displaystyle\underset{j\in J}{\sum}(-1)^{j}\cdot\mynu({\bf
y}_{\widehat{j}})\cdot{\Phi_{y_{j}}}\cdot\mymu\big{(}\underline{\Phi}(y_{j}),{\bf
x}\big{)}$ $\displaystyle=$
$\displaystyle\underset{j=0}{\overset{w}{\sum}}(-1)^{j}\cdot\mynu({\bf
y}_{\widehat{j}})\cdot{\Phi_{y_{j}}}\cdot\widetilde{\mymu}\big{(}\underline{\Phi}(y_{j}),{\bf
x}\big{)}.$

This implies that $Z\cdot(\Phi\cdot Y)=Y\cdot(\Phi^{t}\cdot Z)\in N_{F}$.
Therefore (2) follows.

Assume (2). Let ${\bf y}=(y_{0},\dotsc,y_{w})\in S^{w+1}$ and ${\bf
x}=(x_{1},\dotsc,x_{r-1})\in T^{r-1}$. There are two cases to analyze:

Case 1. If there is no $\ell$ in $\\{0,\dotsc,w\\}$ such that
$\big{\\{}\underline{\Phi}(y_{\ell})\big{\\}}\cup|{\bf x}|$ is a basis of
$\underline{M}$ and $|{\bf y}_{\widehat{\ell}}|$ is a basis of
$\underline{N}$, then $\mynu({\bf
y}_{\widehat{j}})\cdot\widetilde{\mymu}\big{(}\underline{\Phi}(y_{\ell}),{\bf
x}\big{)}=0$ for all $j\in\\{0,\dotsc,w\\}$. Thus

$\underset{k=0}{\overset{w}{\sum}}(-1)^{k}\leavevmode\nobreak\ \mynu({\bf
y}_{\widehat{k}})\leavevmode\nobreak\ {\Phi_{y_{k}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}\underline{\Phi}(y_{k}),{\bf x}\big{)}=0\ \in\ N_{F}.$

Case 2. If there is an $\ell$ in $\\{0,\dotsc,w\\}$ such that
$\big{\\{}\underline{\Phi}(y_{\ell})\big{\\}}\cup|{\bf x}|$ is a basis of
$\underline{M}$ and $|{\bf y}_{\widehat{\ell}}|$ is a basis of
$\underline{N}$, then
$\\{z_{1},\dotsc,z_{n-r}\\}:=T-\big{\\{}\\{\underline{\Phi}(y_{\ell})\\}\cup|{\bf
x}|\big{\\}}$ is a basis of $\underline{M}^{*}$. Define
$z_{0}:=\underline{\Phi}(y_{\ell})$ and ${\bf z}:=(z_{0},\dotsc,z_{n-r})\in
T^{n-r+1}$. Then

$H_{z}\ =\
\begin{cases}(-1)^{i}\cdot\mymu^{*}(\textbf{z}_{\widehat{i}})&\text{if}\
z=z_{i}\text{ for some}\ i\in\\{0,\dotsc,n-r\\}\\\
0&\text{otherwise},\end{cases}$

defines a circuit of $M^{*}$, and

$G_{y}\ =\ \begin{cases}(-1)^{j}\cdot\mynu({\bf y}_{\widehat{j}})&\text{if}\
y=y_{j}\text{ for some}\ j\in\\{0,\dotsc,w\\}\\\
0&\text{otherwise},\end{cases}$

defines a circuit of $N$.

Note that

$H_{z_{i}}=(-1)^{i}\cdot\operatorname{{sign}}(\textbf{z}_{\widehat{i}},z_{i},{\bf
x})\cdot\mymu(z_{i},{\bf x}),$

for all $i\in\\{0,\dotsc,n-r\\}$. Let
$U:=|\textbf{z}|\cap\underline{\Phi}(|{\bf y}|)$ and define

$I\ :=\ \big{\\{}i\in\\{0,\dotsc,n-r\\}\,\big{|}\,z_{i}\in
U\big{\\}}\qquad\text{and}\qquad J\ :=\
\big{\\{}j\in\\{0,\dotsc,w\\}\,\big{|}\,\underline{\Phi}(y_{j})\in
U\big{\\}}.$

There exists a bijection $\mathfrak{b}:J\rightarrow I$ such that
$\underline{\Phi}(y_{j})=z_{\mathfrak{b}(j)}$, for each $j\in J$. Because
$G\cdot(\Phi^{t}\cdot H)=H\cdot(\Phi\cdot G)\in N_{F}$ (see Section 2.2), it
follows that

$\displaystyle\underset{k=0}{\overset{w}{\sum}}(-1)^{k}\leavevmode\nobreak\
\mynu({{\mathbf{y}}}_{\widehat{k}})\leavevmode\nobreak\
{\Phi_{y_{j}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}\underline{\Phi}(y_{k}),{\mathbf{x}}\big{)}\;=$
$\displaystyle\underset{j\in J}{\sum}(-1)^{j}\leavevmode\nobreak\
\mynu(\textbf{y}_{\widehat{j}})\leavevmode\nobreak\
{\Phi_{y_{j}}}\leavevmode\nobreak\
\mymu(z_{{\mathfrak{b}}(j)},{{\mathbf{x}}})$ $\displaystyle=$
$\displaystyle(-1)^{(n-r)}\leavevmode\nobreak\
\operatorname{{sign}}(\textbf{z},\textbf{x})\leavevmode\nobreak\
\underset{j\in J}{\sum}G_{y_{j}}\leavevmode\nobreak\ (\Phi^{t}\cdot
H)_{z_{{\mathfrak{b}}(j)}}$ $\displaystyle=$
$\displaystyle(-1)^{(n-r)}\leavevmode\nobreak\
\operatorname{{sign}}(\mathbf{z},{\bf x})\leavevmode\nobreak\ \underset{s\in
S}{\sum}G_{s}\leavevmode\nobreak\ {(\Phi^{t}\cdot H)_{s}}$ $\displaystyle=$
$\displaystyle(-1)^{(n-r)}\leavevmode\nobreak\ \operatorname{{sign}}({\bf
z},{\bf x})\leavevmode\nobreak\ G\cdot(\Phi^{t}\cdot H)\ \in\ N_{F},$

which concludes the proof. ∎

######  Corollary .

Let $N$ be an $F$-matroid on $S$ and $M$ an $F$-matroid on $T$. Let $\Phi$ be
a submonomial $F$-matrix indexed by $T\times S$. Then the map $\Phi$ is a
morphism from $N$ to $M$ if, and only if, $\Phi^{t}$ is a morphism from
$M^{*}$ to $N^{*}$.

###### Proof.

Let $Z\in\mathcal{C}^{*}_{M}$ and $Y\in\mathcal{C}_{N}$. By Section 2.2, one
has ${Y\cdot(\Phi^{t}\cdot Z)}=Z\cdot(\Phi\cdot Y)$. In particular,
$Y\perp(\Phi^{t}\cdot Z)$ if, and only if, $Z\perp(\Phi\cdot Y)$. Thus the
result follows from Theorem 2.1. ∎

######  Example  (Quotients).

Let $N$ and $M$ be $F$-matroids on the same ground set $S=T$. If
$\Phi=\textup{id}_{S}$ is the identity matrix indexed by $S\times S$, then the
condition (3) of Theorem 2.1 correspond to the flag Plücker relations present
in [JL24, Def. 2.1]. Thus $\textup{id}_{S}$ is a morphism from $N$ to $M$ if,
and only if, $M$ is a quotient of $N$. In particular, this recovers quotients
of classical matroids for $F={\mathbb{K}}$, of oriented matroids for
$F={\mathbb{S}}$ and of valuated matroids for $F={\mathbb{T}}$.

######  Example  (Contractions and restrictions).

Let $M$ be an $F$-matroid on $E$ and $A\subseteq E$.

  1. (1)

(Contraction) Let $c_{A}:=(\mydelta_{ij})_{(i,j)\in(E-A)\times E}$, where
$\mydelta_{ij}=1$ for $i=j$, and $\mydelta_{ij}=0$ for $i\neq j$. Note that
$c_{A}$ is a submonomial $F$-matrix. Let $V=(v_{e})_{e\in
E}\in{\mathcal{V}}_{M}$. By [And19, Prop. 4.4], $c_{A}\cdot V=(v_{e})_{e\in
E-A}$ is an element of ${\mathcal{V}}_{M/A}$. Therefore $c_{A}:M\rightarrow
M/A$ is a morphism.

  2. (2)

(Restriction) Let $r_{A}:=(\mydelta_{ij})_{(i,j)\in E\times(E-A)}$. Then
$r_{A}$ is a submonomial matrix and a morphism $r_{A}:M|A\to M$, which follows
from Section 2.2 since $r_{A}=c_{A}^{t}$.

###  2.3. Morphisms of ${\mathbb{K}}$-matroids and strong maps

By Section 1.3, the map

$\begin{array}[]{ccc}\\{{\mathbb{K}}\text{-matroids}\\}&\longrightarrow&\\{\text{classical
matroids}\\}\\\ M&\longmapsto&\underline{M}\end{array}$

is a bijection. Next we show a similar bijection between morphisms of
${\mathbb{K}}$-matroids and ${\mathbb{F}}_{1}$-linear strong maps of classical
matroids.

######  Lemma .

Let $N$ and $M$ be ${\mathbb{K}}$-matroids on the sets $S$ and $T$,
respectively, and $\Phi:N\rightarrow M$ a morphism. Then the
${\mathbb{F}}_{1}$-linear map
$\underline{\Phi}:\underline{S}\rightarrow\underline{T}$ is a classical strong
map $\underline{\Phi}:\underline{N}\rightarrow\underline{M}$.

###### Proof.

Two tuples $X,Y$ in ${\mathbb{K}}^{T}$ are orthogonal if, and only if,
$\\#(\underline{X}\cap\underline{Y})\neq 1$, i.e. if, and only if,
$\underline{X}\perp\underline{Y}$ (cf. Section 1.2). As
$\underline{\mathcal{C}_{P}}=\mathcal{C}_{\underline{P}}$ for any
${\mathbb{K}}$-matroid $P$ and $\underline{\Phi\cdot
Z}=\underline{\Phi}(\underline{Z})\cap T$ for any $Z\in{\mathbb{K}}^{S}$, the
Lemma follows from Theorem 2.1 and Section 1.2. ∎

######  Theorem 2.2.

Let $N$ and $M$ be ${\mathbb{K}}$-matroids on the sets $S$ and $T$,
respectively. Then the map

$\begin{array}[]{rcl}\\{\text{morphisms }N\rightarrow
M\\}&\longrightarrow&\\{{\mathbb{F}}_{1}\text{-linear strong maps
}\underline{N}\rightarrow\underline{M}\\}\\\ &&\\\ \Phi:N\rightarrow
M&\longmapsto&\underline{\Phi}:\underline{N}\rightarrow\underline{M}\end{array}$

is a bijection.

###### Proof.

Note that for a morphism $\Phi:N\rightarrow M$,

$\Phi_{i,j}\ =\ \begin{cases}1&\text{if }\;\underline{\Phi}(j)=i\neq 0\\\
0&\text{otherwise.}\end{cases}$

Thus if $\Phi:N\rightarrow M$ and $\Phi^{\prime}:N\rightarrow M$ are morphisms
with $\underline{\Phi}=\underline{\Phi^{\prime}}$, then $\Phi=\Phi^{\prime}$.

Fix an ${\mathbb{F}}_{1}$-linear strong map
$\mysigma:\underline{N}\rightarrow\underline{M}$ and, for $(i,j)\in T\times
S$, let

$\Psi_{(i,j)}\ :=\ \begin{cases}1&\text{if }\mysigma(j)=i\neq 0\\\
0&\text{otherwise.}\end{cases}$

Note that $\Psi:=(\Psi_{i,j})_{(i,j)\in T\times S}$ is a submonomial
${\mathbb{K}}$-matrix that satisfies $\underline{\Psi\cdot
X}=\mysigma(\underline{X})\cap T$ for each $X$ in ${\mathbb{K}}^{S}$. As
$\underline{{\mathcal{C}}_{N}}={\mathcal{C}}_{\underline{N}}$ and
$\underline{{\mathcal{V}}_{M}}={\mathcal{V}}_{\underline{M}}$, by Section 1.2
and Theorem 2.1, $\Psi:N\rightarrow M$ is a morphism. As
$\underline{\Psi}=\mysigma$, the result follows. ∎

###  2.4. Factorization of morphisms through quotients

In this section we prove Theorem 2.3, which is the analogue of Section 1.2 for
morphisms of $F$-matroids.

######  Definition .

Let $\Phi=(\Phi_{i,j})_{(i,j)\in T\times S}$ be a submonomial $F$-matrix and
$\mymu$ a Grassmann-Plücker function on the set $T$ with coefficients in $F$.
Let $d$ be the rank of
$\underline{[\mymu]}|\big{(}\operatorname{im}(\underline{\Phi})\cap T\big{)}$
and define $\Phi^{-1}(\mymu):S^{d}\rightarrow F$ by

$\Phi^{-1}(\mymu)(z_{1},\dotsc,z_{r})\ :=\
\Big{(}\underset{\ell=1}{\overset{d}{\prod}}\Phi_{{\underline{\Phi}}(z_{\ell}),z_{\ell}}\Big{)}\cdot\widetilde{\big{(}\mymu|_{\operatorname{im}(\underline{\Phi})\cap
T}\big{)}}\big{(}\underline{\Phi}(z_{1}),\dots,\underline{\Phi}(z_{d})\big{)}.$

######  Proposition .

$\Phi^{-1}(\mymu)$ is a Grassmann-Plücker function on $S$.

###### Proof.

Note that $\Phi^{-1}(\mymu)$ is not identically zero when $d>0$ and is
alternating. Let
$\mypsi:=\widetilde{\big{(}\mymu|_{\operatorname{im}(\underline{\Phi})\cap
T}\big{)}}$, ${\bf y}\in S^{d+1}$ and ${\bf x}\in S^{d-1}$. For the purpose of
this proof, if ${\bf z}=(z_{1},\dotsc,z_{m})$ is a tuple of elements of $S$,
we write $\underline{\Phi}({\bf z})$ for
$\big{(}\underline{\Phi}(z_{1}),\dotsc,\underline{\Phi}(z_{m})\big{)}$. Then

$\displaystyle\leavevmode\nobreak\
\sum_{k=1}^{d+1}(-1)^{k}\leavevmode\nobreak\ \Phi^{-1}(\mymu)({\bf
y}_{\widehat{k}})\leavevmode\nobreak\ \Phi^{-1}(\mymu)(y_{k},{\bf
x})\leavevmode\nobreak\ $ $\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\sum_{k=1}^{d+1}(-1)^{k}\leavevmode\nobreak\ \Big{(}\underset{\ell\neq
k}{\prod}\Phi_{y_{\ell}}\Big{)}\leavevmode\nobreak\
\mypsi\big{(}\underline{\Phi}({\bf
y}_{\widehat{k}})\big{)}\leavevmode\nobreak\ \Phi_{y_{k}}\leavevmode\nobreak\
\Big{(}\underset{m=1}{\overset{d-1}{\prod}}\Phi_{x_{m}}\Big{)}\leavevmode\nobreak\
\mypsi\big{(}\underline{\Phi}(y_{k}),\underline{\Phi}({\bf x})\big{)}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\Big{(}\underset{\ell=1}{\overset{d+1}{\prod}}\Phi_{y_{\ell}}\Big{)}\leavevmode\nobreak\
\Big{(}\underset{m=1}{\overset{d-1}{\prod}}\Phi_{x_{m}}\Big{)}\leavevmode\nobreak\
\sum_{k=1}^{d+1}(-1)^{k}\leavevmode\nobreak\
\mypsi\big{(}\underline{\Phi}({\bf
y}_{\widehat{k}})\big{)}\leavevmode\nobreak\
\mypsi\big{(}\underline{\Phi}(y_{k}),\underline{\Phi}({\bf x})\big{)},$

which is in $N_{F}$, because $\mypsi$ is a Grassmann-Plücker function. ∎

######  Definition .

Let $M$ be the $F$-matroid $[\mymu]$ on $T$. The pre-image of $M$ under $\Phi$
is the $F$-matroid $\Phi^{-1}(M):=[\Phi^{-1}(\mymu)]$ on $S$.

######  Remark .

Note that $\Phi^{-1}(M)$ is well defined, because
$\myalpha\Phi^{-1}(\mymu)=\Phi^{-1}(\myalpha\mymu)$ for all $\myalpha\in
F^{\times}$.

######  Proposition .

The underlying matroids $\underline{\Phi^{-1}(M)}$ and
$\underline{\Phi}^{-1}(\underline{M})$ are equal.

###### Proof.

It follows from the fact that $\\{z_{1},\dotsc,z_{d}\\}$ is a basis of
$\underline{\Phi^{-1}(M)}$ if, and only if,
$\\{\underline{\Phi}(z_{1}),\dotsc,\underline{\Phi}(z_{d})\\}$ is a basis of
$\underline{M}|\big{(}\operatorname{im}(\underline{\Phi})\cap T\big{)}$. ∎

######  Theorem 2.3.

Let $\Phi=(\Phi_{i,j})_{(i,j)\in T\times S}$ be a submonomial $F$-matrix, and
$N$, $M$ two $F$-matroids with respective underlying sets $S$ and $T$. Then
$\Phi:N\rightarrow M$ is a morphism if, and only if, $\Phi^{-1}(M)$ is a
quotient of $N$.

###### Proof.

Let $w$ be the rank of $N$ and $d$ the rank of $\Phi^{-1}(M)$. Assume that
$\Phi:N\rightarrow M$ is a morphism. Let ${\bf y}\in S^{w+1}$ and ${\bf z}\in
S^{d-1}$. Thus

$\displaystyle\underset{k=1}{\overset{w+1}{\sum}}(-1)^{k}\leavevmode\nobreak\
\mynu({\bf y}_{\widehat{k}})\leavevmode\nobreak\ \Phi^{-1}(\mymu)(y_{k},{\bf
z})\leavevmode\nobreak\ =\leavevmode\nobreak\
\Big{(}\underset{\ell=1}{\overset{d-1}{\prod}}\Phi_{z_{\ell}}\Big{)}\leavevmode\nobreak\
\underset{k=1}{\overset{w+1}{\sum}}(-1)^{k}\leavevmode\nobreak\ \mynu({\bf
y}_{\widehat{k}})\leavevmode\nobreak\ \Phi_{y_{k}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}\underline{\Phi}(y_{k}),\underline{\Phi}({\bf
z})\big{)}$

is in $N_{F}$, which implies that $\Phi^{-1}(M)$ is a quotient of $N$.

Conversely, assume that $\Phi^{-1}(M)$ is a quotient of $N$. The proof is
divided into two cases.

Case 1: Assume $T\subseteq\operatorname{im}(\underline{\Phi})$. Let ${\bf
y}\in S^{w+1}$ and ${\bf x}\in T^{d-1}$. Note that
$\underline{\Phi^{t}}(\mathbf{x})\in S^{d-1}$ satisfies
$\underline{\Phi}\big{(}\underline{\Phi^{t}}(\mathbf{x})\big{)}=\mathbf{x}$.

$\displaystyle\underset{k=1}{\overset{w+1}{\sum}}(-1)^{k}\leavevmode\nobreak\
\mynu({\bf y}_{\widehat{k}})\leavevmode\nobreak\
\Phi_{y_{k}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}\underline{\Phi}(y_{k}),{\bf x}\big{)}$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak$
$\displaystyle\Big{(}\underset{\ell=1}{\overset{d-1}{\prod}}\Phi_{\underline{\Phi^{t}}(x_{\ell})}\Big{)}^{-1}\leavevmode\nobreak\
\underset{k=1}{\overset{w+1}{\sum}}(-1)^{k}\leavevmode\nobreak\ \mynu({\bf
y}_{\widehat{k}})\leavevmode\nobreak\
\Phi^{-1}(\mymu)\big{(}y_{k},\underline{\Phi^{t}}({\bf x})\big{)}$

is in $N_{F}$, which implies that $\Phi:N\rightarrow M$ is a morphism.

Case 2: Assume $T\not\subseteq\operatorname{im}(\underline{\Phi})$. Let
$\Psi:=\Phi|\big{(}S\rightarrow\operatorname{im}(\underline{\Phi})\cap
T\big{)}$. Using Case 1, we conclude that $\Psi:N\to
M|\big{(}\operatorname{im}(\underline{\Phi})\cap T\big{)}$ is a morphism. By
Section 2.2 (2), the map $r_{\operatorname{im}(\underline{\Phi})\cap
T}:M|\big{(}\operatorname{im}(\underline{\Phi})\cap T\big{)}\rightarrow M$ is
a morphism. Then $\Phi=r_{\operatorname{im}(\underline{\Phi})\cap
T}\cdot\Psi:N\rightarrow M$ is a morphism. ∎

Next result justifies the choice of the term “pre-image” for $\Phi^{-1}(M)$ in
Section 2.4.

######  Proposition .

Let $\Phi=(\Phi_{i,j})_{(i,j)\in T\times S}$ be a submonomial $F$-matrix and
$M$ an $F$-matrix on $T$. Then
$\mathcal{V}_{\Phi^{-1}(M)}=\Phi^{-1}(\mathcal{V}_{M})$.

###### Proof.

As any $F$-matroid is a quotient of itself, by Theorem 2.3, it follows that
the map $\Phi:\Phi^{-1}(M)\rightarrow M$ is a morphism. Thus
$\Phi\cdot\mathcal{V}_{\Phi^{-1}(M)}\subseteq\mathcal{V}_{M}$.

Let $X\in\Phi^{-1}(\mathcal{V}_{M})-\\{0\\}$. By [BB19, Thm. 3.12 and Thm.
3.21], there exists an $F$-matroid $N$ such that $\mathcal{C}_{N}=\\{\myalpha
X\mid\myalpha\in F^{\times}\\}$. As
$\Phi\cdot\mathcal{C}_{N}\subseteq\mathcal{V}_{M}$, which is equivalent to
$\Phi\cdot\mathcal{C}_{N}\perp\mathcal{C}^{*}_{M}$, we conclude, by Theorem
2.1, that $\Phi:N\rightarrow M$ is a morphism. Thus by Theorem 2.3,
$\Phi^{-1}(M)$ is a quotient of $N$. By [JL24, Rem. 2.12], we have
$\mathcal{C}_{N}\subseteq\mathcal{V}_{\Phi^{-1}(M)}$. ∎

###  2.5. Minors of morphisms

In this subsection we show how to construct minors of morphisms.

######  Definition .

Let $\Phi$ be a submonomial $F$-matrix indexed by $T\times S$, and $A\subseteq
S$ and $B\subseteq T$ subsets such that
$\underline{\Phi}(\underline{A})\subseteq\underline{B}$. We define:

$\begin{array}[]{lccl}\text{({Contraction} of }\Phi\text{ by }A\to
B\text{)}&\Phi/(A\to B)&:=&(\Phi_{i,j})_{(i,j)\in B^{c}\times A^{c}};\\\
\text{({Restriction} of }\Phi\text{ to }A\rightarrow B\text{)}&\Phi|(A\to
B)&:=&(\Phi_{i,j})_{(i,j)\in B\times A}.\\\ \end{array}$

######  Remark .

The deletion of a morphism is the restriction of its transpose to the
complementary subsets. More explicitly, consider a submonomial $F$-matrix
$\Phi$ indexed by $T\times S$ and subsets $A\subseteq S$ and $B\subseteq T$
such that $\underline{\Phi}(\underline{A})\subseteq\underline{B}$. Since
$\underline{\Phi}^{t}(\underline{B^{c}})\subseteq\underline{A^{c}}$ for the
respective complements $A^{c}=S-A$ and $B^{c}=T-B$ of $A$ and $B$, we define:

$\begin{array}[]{lccl}\text{({Deletion} of }A\to B\text{ from
}\Phi^{t}\text{)}&\Phi^{t}\backslash(A\to B)&:=&\Phi^{t}|(B^{c}\to
A^{c}).\end{array}$

Taking transposes yields the following alternative presentations of the
deletion

$\Phi^{t}\backslash(A\to B)\ =\ \big{(}\Phi|(A^{c}\to B^{c})\big{)}^{t}\ =\
\big{(}\Phi/(A\to B)\big{)}^{t}.$

######  Theorem 2.4.

Let $N$ and $M$ be two $F$-matroids on $S$ and $T$, respectively,
$\Phi:N\rightarrow M$ a morphism, and $A\subseteq S$ and $B\subseteq T$
subsets such that $\underline{\Phi}(\underline{A})\subseteq\underline{B}$.
Then

  1. (1)

$\Phi/(A\rightarrow B):N/A\rightarrow M/B$ is a morphism;

  2. (2)

$\Phi|(A\rightarrow B):N|A\rightarrow M|B$ is a morphism;

  3. (3)

$\Phi^{t}\backslash(A\rightarrow B):M^{*}\backslash B\rightarrow
N^{*}\backslash A$ is a morphism.

###### Proof.

We first prove (2). Let $V=(v_{a})_{a\in A}\in{\mathcal{V}}_{N|A}$. By [And19,
Prop. 4.5], there exists $X=(x_{s})_{s\in S}\in{\mathcal{V}}_{N}$ such that
$x_{e}=v_{e}$ for $e\in A$, and $x_{e}=0$ for $e\in S-A$. As $\Phi:N\to M$ is
a morphism, $\Phi\cdot V\in{\mathcal{V}}_{M}$. Let $Y=(y_{t})_{t\in
T}:=\Phi\cdot V$. As $\underline{\Phi}(\underline{A})\subseteq\underline{B}$,
one has that $y_{t}=0$ for $t\in T-B$. Again, by [And19, Prop. 4.5], we
conclude that $Z:=(y_{b})_{b\in B}\in{\mathcal{V}}_{M|B}$. As
$Z=\big{(}\Phi|(A\to B)\big{)}\cdot V$, the result follows.

Item (3) follows from Section 2.2 and (2).

Next we prove (1). By Section 2.2, $\Phi^{t}:M^{*}\to N^{*}$ is a morphism. As
$\underline{\Phi^{t}}(\underline{B^{c}})\subseteq\underline{A^{c}}$, it
follows that $\Phi^{t}|(B^{c}\to A^{c}):M^{*}|B^{c}\to N^{*}|A^{c}$ is a
morphism, by (2). Thus the result follows from Section 2.5 and another
application of Section 2.2. ∎

###  2.6. Functoriality

The next Proposition shows that morphisms of matroids with coefficients are
well behaved with respect to idyll morphisms.

Let $f:F\rightarrow G$ be a morphism of perfect idylls and $\Phi$ a
submonomial $F$-matrix indexed by $T\times S$.

######  Definition .

The push-forward of $\Phi$ by $f$ is the submonomial $G$-matrix

$f_{*}(\Phi):=\big{(}f(\Phi_{i,j})\big{)}_{(i,j)\in T\times S}.$

######  Proposition .

Let $N$ and $M$ be $F$-matroids with respective underlying sets $S$ and $T$.
If $\Phi:N\rightarrow M$ is a morphism of $F$-matroids, then
$f_{*}(\Phi):f_{*}(N)\rightarrow f_{*}(M)$ is a morphism of $G$-matroids.
Therefore $f_{*}$ defines a functor from $\mathsf{Mat}_{F}$ to
$\mathsf{Mat}_{G}$.

###### Proof.

Let $f^{+}:{\mathbb{N}}[F^{\times}]\to{\mathbb{N}}[G^{\times}]$ be the
morphism of semirings induced by $f$. For a set $E$, we also use $f_{*}$ to
denote the map $F^{E}\to G^{E}$ given by $f_{*}\big{(}(x_{e})_{e\in
E}\big{)}:=\big{(}f(x_{e})\big{)}_{e\in E}$.

Note that

$f^{+}\big{(}Z\cdot(\Phi\cdot
Y)\big{)}\;=\;f_{*}(Z)\cdot\big{(}f_{*}(\Phi)\cdot f_{*}(Y)\big{)}$

for all $Z\in F^{T}$ and $Y\in F^{S}$. In particular, if $Z\perp\Phi\cdot Y$,
then $f_{*}(Z)\perp f_{*}(\Phi)\cdot f_{*}(Y)$. Therefore the result follows
from Theorem 2.1 and [BB19, Lemma 3.39]. ∎

######  Corollary .

Let $M$ and $N$ be $F$-matroids on the sets $T$ and $S$, respectively, and
$\Phi:N\rightarrow M$ a morphism. Then
$\underline{\Phi}:\underline{S}\rightarrow\underline{T}$ is an
${\mathbb{F}}_{1}$-linear strong map
$\underline{\Phi}:\underline{N}\rightarrow\underline{M}$.

###### Proof.

Let $f:F\rightarrow{\mathbb{K}}$ be the unique idyll morphism from $F$ to
${\mathbb{K}}$. By Section 2.6, $f_{*}(\Phi):f_{*}(N)\rightarrow f_{*}(M)$ is
a morphism. As $\underline{\Phi}=\underline{f_{*}(\Phi)}$ and
$\underline{f_{*}(P)}=\underline{P}$ for any $F$-matroid $P$, the Corollary
follows from Section 2.3. ∎

###  2.7. Remark on morphisms for idylls with involution

Baker and Bowler formulate their theory of matroids with coefficients for
idylls $F$ with an involution $x\mapsto\overline{x}$ (which is an automorphism
$F\to F$ of order $2$). The most prominent example for such an involution are
the complex numbers ${\mathbb{C}}$ together with the complex conjugate
function $z\mapsto\overline{z}$. In this case, the dual of a
${\mathbb{C}}$-matroid $M$ with vectors
${\mathcal{V}}_{M}\subseteq{\mathbb{C}}^{T}$ is the ${\mathbb{C}}$-matroid
$M^{\ast}$ whose vectors ${\mathcal{V}}_{M^{\ast}}$ is the Hermitian
complement of ${\mathcal{V}}_{M}$ in ${\mathbb{C}}^{T}$.

The presence of an involution changes the notion of orthogonality and
therefore the notion of the dual of an $F$-matroid. This extends to morphisms
of $F$-matroids: the dual of a morphism $\Phi:N\to M$ of $F$-matroids (with
respect to an involution $x\mapsto\overline{x}$) is described by the
_conjugate transpose_ $\overline{\Phi}^{t}:M^{\ast}\to N^{\ast}$ of $\Phi$.

It is readily verified that the conjugate transpose is indeed a morphism of
$F$-matroids, and that the cryptomorphic descriptions of morphisms and minor
constructions extend to matroids over idylls with involution. We omit further
details.

##  3\. Quiver $F$-matroids

###  3.1. Quiver representations

A _quiver_ is a directed graph which allows loops and multiples arrows. More
precisely, a quiver $Q$ is a tuple $(Q_{0},Q_{1},s,t)$ where $Q_{0}$ is the
set of _vertices_ , $Q_{1}$ is the set of _arrows_ , and $s,t:Q_{1}\rightarrow
Q_{0}$ are functions mapping an arrow to its _source_ and _target_ ,
respectively. We denote an arrow of $Q$ by
$s(\myalpha)\xrightarrow{\;\;\myalpha\;\;}t(\myalpha)$.

Given a quiver $Q$, let $\mathsf{C}(Q)$ be the free category generated by $Q$.
A _representation_ of $Q$ over a category $\mathsf{C}$ is a (covariant)
functor

$\Gamma:\mathsf{C}(Q)\longrightarrow\mathsf{C}.$

Let $\Lambda:\mathsf{C}(Q)\rightarrow\mathsf{C}$ and
$\Gamma:\mathsf{C}(Q)\rightarrow\mathsf{C}$ be two representations of $Q$. A
_morphism_ of representations is a natural transformation
$\Lambda\rightarrow\Gamma$. A _subrepresentation_ of $\Gamma$ is an
isomorphism class of subfunctors of $\Gamma$ 333A subfunctor of $\Gamma$ is a
pair $(\Gamma^{\prime},\mytau)$, where $\Gamma^{\prime}$ is a functor and
$\mytau:\Gamma^{\prime}\rightarrow\Gamma$ a natural transformation such that
$\mytau_{v}:\Gamma^{\prime}v\rightarrow\Gamma v$ is a monomorphism for all
$v\in\textup{ob}\big{(}\mathsf{C}(Q)\big{)}=Q_{0}$. Two subfunctors
$(\Gamma^{\prime}_{1},\mytau_{1})$ and $(\Gamma^{\prime}_{2},\mytau_{2})$ are
isomorphic if there exists a natural isomorphism
$T:\Gamma^{\prime}_{1}\rightarrow\Gamma^{\prime}_{2}$ such that
$\mytau_{1}=\mytau_{2}\circ T$..

We denote by $\mathsf{Rep}_{\mathsf{C}}(Q)$ the category of representations of
$Q$ over $\mathsf{C}$.

######  Example  (Representations of quivers over vector spaces).

Let $k$ be a field, $\operatorname{{Vect}}_{k}$ the category of $k$-vector
spaces, and $Q$ a quiver. A representation of $Q$ over
$\operatorname{{Vect}}_{k}$ corresponds to the data
$\Gamma=\big{(}(V_{v})_{v\in Q_{0}},(\myvarphi_{\myalpha})_{\myalpha\in
Q_{1}}\big{)}$, where $V_{v}$ is a $k$-vector space for each $v\in Q_{0}$ and
$\myvarphi_{\myalpha}:V_{s(\myalpha)}\rightarrow V_{t(\myalpha)}$ is a
$k$-linear map for each $\myalpha\in Q_{1}$. The _dimension vector_ of
$\Gamma$ is the tuple $\underline{\dim}(\Gamma):=(\dim V_{v})_{v\in Q_{0}}$.

Let $\Lambda=(V^{\prime}_{v},\myvarphi^{\prime}_{\myalpha})$ and
$\Gamma=(V_{v},\myvarphi_{\myalpha})$ be two representations of $Q$ over
$\operatorname{{Vect}}_{k}$. A morphism of representations
$f:\Lambda\rightarrow\Gamma$ is a collection $(f_{v}:V^{\prime}_{v}\rightarrow
V_{v})_{v\in Q_{0}}$ of $k$-linear maps, such that
$f_{t(\myalpha)}\circ\myvarphi^{\prime}_{\myalpha}=\myvarphi_{\myalpha}\circ
f_{s(\myalpha)}$ for each arrow $\myalpha$ in $Q_{1}$. The morphism
$f=(f_{v})$, of representations, is a monomorphism, an epimorphism or an
isomorphism if each $f_{v}$ is, respectively.

Each subrepresentation of $\Gamma=(V_{v},\myvarphi_{\myalpha})$ is isomorphic
to a unique representation
$\Gamma^{\prime}=(V^{\prime}_{v},\myvarphi^{\prime}_{\myalpha})$ for which
$V_{v}^{\prime}\subseteq V_{v}$ is a linear subspace for all $v\in Q_{0}$ and
for which $\myvarphi^{\prime}_{\myalpha}$ is the restriction of
$\myvarphi_{\myalpha}$ to a map $V_{s(\myalpha)}^{\prime}\rightarrow
V_{t(\myalpha)}^{\prime}$ for all $\myalpha\in Q_{1}$. For more details on
representations of quivers over vector spaces, we refer the reader to [Kir16]
and [Sch14].

###  3.2. Quiver $F$-matroids

######  Definition  (Quiver $F$-Matroid).

Let $F$ be a perfect idyll and $Q=(Q_{0},Q_{1},s,t)$ a quiver. A _quiver
$F$-matroid of type $Q$_, or _$(Q,F)$ -matroid_ for short, is a representation
of $Q$ in $\mathsf{Mat}_{F}$. More explicitly, this is a tuple

$M=\big{(}(M_{v})_{v\in Q_{0}},(M_{\myalpha})_{\myalpha\in Q_{1}}),$

where $M_{v}$ is an $F$-matroid for all $v\in Q_{0}$, and
$M_{\myalpha}:M_{s(\myalpha)}\rightarrow M_{t(\myalpha)}$ is a morphism of
$F$-matroids for all $\myalpha\in Q_{1}$. The _rank vector_ of $M$ is the
tuple
$\underline{\operatorname{{rk}}}(M):=\big{(}\operatorname{{rk}}(M_{v})\big{)}_{v\in
Q_{0}}$.

######  Notation .

For $v\in Q_{0}$, we use $\mymu_{v}:E_{v}^{r_{v}}\rightarrow F$ to denote a
Grassmann-Plücker function such that $M_{v}=[\mymu_{v}]$. In particular,
$E_{v}$ is the underlying set of $M_{v}$ and $r_{v}$ is the rank of $M_{v}$.

By Theorem 2.1, the term

$\sum_{k=0}^{r_{s(\myalpha)}}(-1)^{k}\cdot\mymu_{s(\myalpha)}(y_{0},\dotsc,\widehat{y_{k}},\dotsc,y_{r_{s(\myalpha)}})\cdot{M_{\myalpha,\underline{M_{\myalpha}}(y_{k}),y_{k}}}\cdot\widetilde{\mymu_{t(\myalpha)}}\big{(}\underline{M_{\myalpha}}(y_{k}),x_{2},\dotsc,x_{r_{t(\myalpha)}}\big{)}$

is contained in $N_{F}$ for all $\myalpha\in Q_{1}$, all
$y_{0},\dotsc,y_{r_{s(\myalpha)}}\in E_{s(\myalpha)}$ and all
$x_{2},\dotsc,x_{r_{t(\myalpha)}}\in E_{t(\myalpha)}$.

######  Example  (Flag $F$-matroids as quiver $F$-matroids).

Let $F$ be a perfect idyll and

$M=(M_{1},\dots,M_{s})$

a sequence of $F$-matroids on the same set $E$. The sequence $M$ is a _flag
$F$-matroid_ if $M_{i}$ is a quotient of $M_{j}$ for all $i<j$. By [JL24, Thm.
2.17] and Section 2.2, $M$ is a flag $F$-matroid if, and only if,

${M_{1}}$${M_{2}}$${\cdots}$${M_{s-1}}$${M_{s},}$$\scriptstyle{\textup{id}_{E}}$$\scriptstyle{\textup{id}_{E}}$$\scriptstyle{\textup{id}_{E}}$$\scriptstyle{\textup{id}_{E}}$

is a quiver $F$-matroid.

###  3.3. Relation to quiver representations over ${{\mathbb{F}}_{1}}$

Let $F$ be a perfect idyll. In the following, we denote the underlying set of
an $F$-matroid $M$ by $E_{M}$.

######  Definition .

The assignment

  * •

$M\mapsto\underline{E_{M}}$ for $F$-matroids

  * •

$(\Phi:N\rightarrow
M)\mapsto(\underline{\Phi}:\underline{E_{N}}\rightarrow\underline{E_{M}})$ for
morphisms

defines the _underlying set functor_
$\mathsf{U}_{F}:\mathsf{Mat}_{F}\rightarrow\operatorname{{Vect}}_{{\mathbb{F}}_{1}}$.

The _underlying ${{\mathbb{F}}_{1}}$-representation_ of a $(Q,F)$-matroid $M$
is the representation of $Q$ over $\mathsf{Vect}_{{\mathbb{F}}_{1}}$ given by
$\mathsf{U}_{F}(M):=\big{(}(\underline{E_{M_{v}}}),(\underline{M_{\myalpha}})\big{)}$.

Let $V$ be a pointed finite set, $E=V-\\{0\\}$ and $n=\\#E$. Let
$\mymu_{V}:E^{n}\rightarrow{\mathbb{K}}$ be the unique Grassmann-Plücker
function of rank $n$ on $E$, which sends ${\bf x}$ to $1$ if $|{\bf x}|=E$,
and to $0$ otherwise. Given an ${\mathbb{F}}_{1}$-linear map $g:V\to W$, let
$g_{\mathbb{K}}$ be the unique submonomial ${\mathbb{K}}$-matrix such that
$\underline{g_{\mathbb{K}}}=g$, which is indexed by
$(W-\\{0\\})\times(V-\\{0\\})$ and whose entries are

$g_{{\mathbb{K}},w,v}\ :=\ \begin{cases}1&\text{if }g(v)=w,\\\ 0&\text{if
}g(v)\neq w.\end{cases}$

######  Lemma .

Let $g:V\to W$ be an ${\mathbb{F}}_{1}$-linear map and $M=[\mymu]$ a
${\mathbb{K}}$-matroid on $W-\\{0\\}$. Then
$g_{\mathbb{K}}:[\mymu_{V}]\rightarrow M$ is a morphism of
${\mathbb{K}}$-matroids.

###### Proof.

Let $r$ be the rank of $M$. Let $\mathbf{y}\in(V-\\{0\\})^{\\#V}$ and
$\mathbf{x}\in(W-\\{0\\})^{r-1}$. Note that there exist $\ell\neq m$ such that
$y_{\ell}=y_{m}$, as $\mymu_{V}(\mathbf{y}_{\widehat{k}})=0$ for
$k\neq\ell,m$, it follows that

$\displaystyle\;\underset{k=1}{\overset{\\#V}{\sum}}\mymu_{V}(\mathbf{y}_{\widehat{k}})\leavevmode\nobreak\
{g_{{\mathbb{K}},y_{k}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}g(y_{k}),\mathbf{x}\big{)}$ $\displaystyle=$
$\displaystyle\;\mymu_{V}(\mathbf{y}_{\widehat{\ell}})\leavevmode\nobreak\
{g_{{\mathbb{K}},y_{\ell}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}g(y_{\ell}),\mathbf{x}\big{)}\;+\;\mymu_{V}(\mathbf{y}_{\widehat{m}})\leavevmode\nobreak\
{g_{{\mathbb{K}},y_{m}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}g(y_{m}),\mathbf{x}\big{)}$ $\displaystyle=$
$\displaystyle\;2\cdot\;\mymu_{V}(\mathbf{y}_{\widehat{\ell}})\leavevmode\nobreak\
{g_{{\mathbb{K}},y_{\ell}}}\leavevmode\nobreak\
\widetilde{\mymu}\big{(}g(y_{\ell}),\mathbf{x}\big{)},$

which is in $N_{\mathbb{K}}={\mathbb{N}}-\\{1\\}$. Thus the result follows. ∎

######  Definition .

The assignment

  * •

$V\mapsto[\mymu_{V}]$ for finite pointed sets

  * •

$(g:V\to W)\mapsto(g_{\mathbb{K}}:[\mymu_{V}]\rightarrow[\mymu_{W}])$ for
${\mathbb{F}}_{1}$-linear maps

defines the functor
$\mathsf{M}:\operatorname{{Vect}}_{{\mathbb{F}}_{1}}\rightarrow\mathsf{Mat}_{\mathbb{K}}$.

######  Theorem 3.1.

The functor $\mathsf{M}$ is left adjoint to $\mathsf{U}_{\mathbb{K}}$.

###### Proof.

Let $M$ be a ${\mathbb{K}}$-matroid and $V$ a finite pointed set. We define
the maps

${\textup{Hom}\big{(}\mathsf{M}(V),M\big{)}}$${\textup{Hom}\big{(}V,\mathsf{U}_{\mathbb{K}}(M)\big{)}}$$\scriptstyle{\myalpha}$$\scriptstyle{\mygamma}$

by

(1)
$\myalpha(\Phi):=\underline{\Phi}:V\rightarrow\underline{E_{M}}=\mathsf{U}_{\mathbb{K}}(M)\quad\text{and}\quad\mygamma(g):=g_{\mathbb{K}}:\mathsf{M}(V)\rightarrow
M.$

As $\myalpha$ and $\mygamma$ are mutually inverse, the result follows. ∎

######  Remark .

The functor $\mathsf{U}_{\mathbb{K}}$ also has a right adjoint
$\mathsf{M}_{0}$, which sends a finite pointed set $V$ to the unique
${\mathbb{K}}$-matroid of rank $0$ on $V-\\{0\\}$. Both $\mathsf{M}$ and
$\mathsf{M}_{0}$ are right inverses to $\mathsf{U}_{\mathbb{K}}$.

Let $Q$ be a quiver. Then the functors $\mathsf{U}_{\mathbb{K}}$ and
$\mathsf{M}$ induce functors

${\mathsf{U}_{Q}:\operatorname{Rep}_{\mathsf{Mat}_{\mathbb{K}}}(Q)}$${\operatorname{Rep}_{\mathsf{Vect}_{{\mathbb{F}}_{1}}}(Q)}$${\Gamma}$${\mathsf{U}_{\mathbb{K}}\circ\Gamma}$
and
${\mathsf{M}_{Q}:\operatorname{Rep}_{\mathsf{Vect}_{{\mathbb{F}}_{1}}}(Q)}$${\operatorname{Rep}_{\mathsf{Mat}_{\mathbb{K}}}(Q)}$${\Lambda}$${\mathsf{M}\circ\Lambda.}$

The functors $\mathsf{U}_{Q}$ and $\mathsf{M}_{Q}$ satisfy the following
corollary.

######  Corollary .

The functor $\mathsf{M}_{Q}$ is left adjoint to $\mathsf{U}_{Q}$.

###### Proof.

Let $\Lambda$ be a representation of $Q$ over
$\mathsf{Vect}_{{\mathbb{F}}_{1}}$ and $\Gamma$ a $(Q,{\mathbb{K}})$-matroid.
We define

${\textup{Hom}\big{(}\mathsf{M}_{Q}(\Lambda),\Gamma\big{)}}$${\textup{Hom}\big{(}\Lambda,\mathsf{U}_{Q}(\Gamma)\big{)}}$$\scriptstyle{\myalpha_{Q}}$$\scriptstyle{\mygamma_{Q}}$

by $\myalpha_{Q}=\big{(}\myalpha_{v})_{v\in Q_{0}}$ and
$\mygamma_{Q}=\big{(}\mygamma_{v})_{v\in Q_{0}}$, with $\myalpha_{v}$ and
$\mygamma_{v}$ defined as in (1), for each $v$. As
$\mygamma_{v}\circ\myalpha_{v}$ and $\myalpha_{v}\circ\mygamma_{v}$ are
identity maps, it follows that $\myalpha_{Q}\circ\mygamma_{Q}$ and
$\mygamma_{Q}\circ\myalpha_{Q}$ are identity maps as well, which finishes the
proof. ∎

######  Example  (Quiver matroids of type $D_{4}$).

Let $Q$ be a $D_{4}$-quiver and $\Lambda$ the
${{\mathbb{F}}_{1}}$-representation of $Q$ as illustrated in Figure 1: the
points vertically above $v_{i}$ (for $i=0,1,2,3$) correspond the nonzero
elements of $\Lambda_{v_{i}}$; the arrows above $\myalpha_{i}$ (for $i=1,2,3$)
illustrate the map $\Lambda_{\myalpha_{i}}:\Lambda_{v_{0}}\to\Lambda_{v_{i}}$;
the elements of $\Lambda_{v_{0}}$ without an emerging arrow above
$\myalpha_{i}$ are mapped to $0$. More to the point, Figure 1 illustrates the
coefficient quiver of $\Lambda$; cf. Section 7.5.

$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$v_{0}$$v_{1}$$v_{2}$$v_{3}$$\myalpha_{1}$$\myalpha_{2}$$\myalpha_{3}$$\Lambda$$Q$$\Lambda_{v_{0}}$$\Lambda_{v_{1}}$$\Lambda_{v_{2}}$$\Lambda_{v_{3}}$$\Lambda_{\myalpha_{1}}$$\Lambda_{\myalpha_{2}}$$\Lambda_{\myalpha_{3}}$
Figure 1. The ${{\mathbb{F}}_{1}}$-representation $\Lambda$ of type $D_{4}$

Next we determine all $(\Lambda,{\mathbb{K}})$-matroids $M$ with rank vector
${\mathbf{r}}=(r_{0},r_{1},r_{2},r_{3})=(2,1,1,1)$. Thus $M_{v_{i}}$ has
corank $1$ for every $i=0,1,2,3$ and therefore a unique circuit, which is the
unique nonzero vector of $M_{v_{i}}$. In the following, we list the different
choices for nonzero vectors (written as row vectors), ordered by the
isomorphism type of $M_{v_{i}}$ (for $i\neq 0$ and $i=0$, respectively):

type of $M_{v_{i}}$ | nonzero vectors |  | type of $M_{v_{0}}$ | nonzero vectors
---|---|---|---|---
$U_{1}^{1}\oplus U_{1}^{0}$ | $(1,0),\quad(0,1)$ |  | $U_{2}^{2}\oplus U_{1}^{0}$ | $(1,0,0),\quad(0,1,0),\quad(0,0,1)$
$U_{2}^{1}$ | $(1,1)$ |  | $U_{2}^{1}\oplus U_{1}^{1}$ | $(1,1,0),\quad(1,0,1),\quad(0,1,1)$
|  |  | $U_{3}^{2}$ | $(1,1,1)$

Let $\Phi_{i}$ be the unique submonomial ${\mathbb{K}}$-matrix such that
$\underline{\Phi_{i}}=\Lambda_{\myalpha_{i}}$ (for $i=1,2,3$), i.e.

$\Phi_{1}\ =\ \big{[}\begin{smallmatrix}1&0&0\\\
0&1&0\end{smallmatrix}\big{]},\qquad\Phi_{2}\ =\
\big{[}\begin{smallmatrix}1&0&0\\\
0&0&1\end{smallmatrix}\big{]},\qquad\Phi_{3}\ =\
\big{[}\begin{smallmatrix}0&1&0\\\ 0&0&1\end{smallmatrix}\big{]}.$

A quadruple of corank $1$ matroids $(M_{v_{0}},M_{v_{1}},M_{v_{2}},M_{v_{3}})$
with respective nonzero vectors $(w_{0},w_{1},w_{2},w_{3})$ as above is a
$(\Lambda,{\mathbb{K}})$-matroid if and only if $\Phi_{i}\cdot
w_{0}\subseteq{\mathcal{V}}_{M_{v_{i}}}=\\{0,w_{i}\\}$. We illustrate such
quadruples in Figure 2 as follows: each vertex $w$ within the colored area
labeled by $v_{i}$ represents a corank $1$ matroid $M_{w}$ on
$\Lambda_{v_{i}}-\\{0\\}$ whose label $w$ is the unique nonzero vector of
$M_{w}$. An arrow from a vertex with label $w$ in the green area (labeled by
$v_{0}$) to a vertex with label $w^{\prime}$ in a blue area (labeled by some
$v_{i}$) indicates that $\Phi_{i}:M_{w}\to M_{w^{\prime}}$ is a morphism.
Therefore $(\Lambda,{\mathbb{K}})$-matroids of rank r correspond to connected
full subquivers of $Q^{\prime}$ with exactly one vertex per colored area.

$v_{0}$$v_{2}$$v_{1}$$v_{3}$$\Big{[}\begin{smallmatrix}1\\\ 1\\\
1\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}0\\\ 1\\\
0\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}0\\\ 1\\\
1\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}0\\\ 0\\\
1\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}1\\\ 0\\\
1\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}1\\\ 0\\\
0\end{smallmatrix}\Big{]}$$\Big{[}\begin{smallmatrix}1\\\ 1\\\
0\end{smallmatrix}\Big{]}$$\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}0\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}1\\\
1\end{smallmatrix}\big{]}$$\big{[}\begin{smallmatrix}1\\\
0\end{smallmatrix}\big{]}$ Figure 2. The quiver $Q^{\prime}$, which
illustrates all $(\Lambda,{\mathbb{K}})$-matroids of rank ${\mathbf{r}}$

Note that each red vertex $w$ of $Q^{\prime}$ determines a unique
$(\Lambda,{\mathbb{K}})$-matroid $M$ of rank r with $M_{v_{0}}=M_{w}$, while
for each yellow vertex $w^{\prime}$ there exist $3$
$(\Lambda,{\mathbb{K}})$-matroids $M$ of rank r with
$M_{v_{0}}=M_{w^{\prime}}$. In total, there are $13$
$(\Lambda,{\mathbb{K}})$-matroids of rank r.

###  3.4. Duality and minors of quiver matroids

Let $Q=(Q_{0},Q_{1},s,t)$ be a quiver. The _opposite quiver of $Q$_ is the
quiver $Q^{\ast}$ with

$Q_{0}^{*}:=Q_{0},\qquad Q_{1}^{*}:=Q_{1},\qquad s^{*}:=t,\qquad t^{*}:=s.$

Let $F$ be a perfect idyll and $M$ a $(Q,F)$-matroid. By Section 2.2, the
transpose $\Phi_{\myalpha}^{t}:M_{t(\myalpha)}^{*}\rightarrow
M_{s(\myalpha)}^{*}$ is a morphism for all $\myalpha\in Q_{1}$.

######  Definition .

The _dual_ of $M$ is $M^{*}:=\big{(}(M_{v}^{*})_{v\in
Q_{0}^{*}},(M_{\myalpha}^{t})_{\myalpha\in Q_{1}^{*}}\big{)}$, which is a
quiver $F$-matroid of type $Q^{*}$.

Let
$\mathsf{U}_{\mathbb{K}}(M)=\big{(}(\underline{E_{v}}),(\underline{M_{\myalpha}})\big{)}$
be the underlying representation of $M$ and $\Lambda$ a subrepresentation of
$\mathsf{U}_{\mathbb{K}}(M)$ where $\Lambda_{v}\subseteq\underline{E_{v}}$ for
all $v\in Q_{0}$. Define $A_{v}:=\Lambda_{v}-\\{0\\}$. By Theorem 2.4, for any
$\myalpha\in Q_{1}$, the following are morphisms:

$\displaystyle M_{\myalpha}/(A_{s(\myalpha)}\rightarrow A_{t(\myalpha)})$
$\displaystyle:M_{s(\myalpha)}/A_{s(\myalpha)}\rightarrow
M_{t(\myalpha)}/A_{t(\myalpha)},$ $\displaystyle
M_{\myalpha}|(A_{s(\myalpha)}\rightarrow A_{t(\myalpha)})$
$\displaystyle:M_{s(\myalpha)}|A_{s(\myalpha)}\rightarrow
M_{t(\myalpha)}|A_{t(\myalpha)},$ $\displaystyle
M_{\myalpha}^{t}\backslash(A_{s(\myalpha)}\rightarrow A_{t(\myalpha)})$
$\displaystyle:M_{t(\myalpha)}^{*}\backslash A_{t(\myalpha)}\rightarrow
M_{s(\myalpha)}^{*}\backslash A_{s(\myalpha)}.$

######  Definition .

We define the following quiver $F$-matroids:

$\begin{array}[]{lccc}\text{(\emph{Contraction} of $M$ by
$\Lambda$)}&M/\Lambda&:=&\big{(}(M_{v}/A_{v})\;,\;\big{(}M_{\myalpha}/(A_{s(\myalpha)}\rightarrow
A_{t(\myalpha)})\big{)}\big{)},\\\ \text{(\emph{Restriction} of $M$ to
$\Lambda$)}&M|\Lambda&:=&\big{(}(M_{v}|A_{v})\;,\;\big{(}M_{\myalpha}|(A_{s(\myalpha)}\rightarrow
A_{t(\myalpha)})\big{)}\big{)},\\\ \text{(\emph{Deletion} of $\Lambda$ from
$M^{*}$)}&M^{*}\backslash\Lambda&:=&\big{(}(M_{v}^{*}\backslash
A_{v})\;,\;\big{(}M_{\myalpha}^{t}\backslash(A_{t(\myalpha)}\rightarrow
A_{s(\myalpha)})\big{)}\big{)}.\end{array}$

######  Remark .

Note that $M/\Lambda$ and $M|\Lambda$ are of type $Q$, and
$M^{*}\backslash\Lambda$ is of type $Q^{*}$.

##  4\. Bands and band schemes

The vessel of our approach to moduli spaces of quiver matroids are band
schemes, which form a streamlined approach to ${{\mathbb{F}}_{1}}$-geometry
that is developed in [BJL24]. We review this theory in brevity in this
section.

In particular, we tailor certain results about ordered blueprints to bands,
which is possible since bands are ordered blueprints of a particularly well-
behaved type (see Section 4.1.7)—morally speaking, a band is the part of an
ordered blueprint that is sensitive to matroid theory. For further details, we
refer the reader to [BJL24].

###  4.1. Bands

A band is a generalization of a ring, in the same way that an idyll is a
generalization of a field.

A _pointed monoid_ $B$ is a commutative monoid $B$ together with an element
$0$ such that $0\cdot a=0$ for all $a\in B$. The _ambient semiring_ of a
pointed monoid $B$ is the semiring

$\textstyle B^{+}\ :=\ {\mathbb{N}}[B]\,/\,\langle 0\rangle\ =\ \big{\\{}\sum
a_{i}\,\big{|}\,a_{i}\in B-\\{0\\}\big{\\}}.$

Note that $B$ embeds as a submonoid of $B^{+}$. An _ideal_ of $B^{+}$ is a
subset $I$ such that $0\in I$, $I+I=I$ and $B\cdot I=I$.

A _band_ is a pointed monoid $B$ together with an ideal $N_{B}\subseteq B^{+}$
(the _null set of $B$_) such that for every $a\in B$, there exists a unique
element $b\in B$ such that $a+b\in N_{B}$, which we call the _additive inverse
of $a$_ and which we denote by $-a$. We write $a-b$ for $a+(-b)$. A _band
morphism_ is a multiplicative map $f:B\to C$ of bands that preserves $0$ and
$1$ such that $\sum a_{i}\in N_{B}$ implies $\sum f(a_{i})\in N_{C}$. We
denote the category of bands by $\operatorname{Bands}$.

The null set $N_{B}$ of a band $B$ is _generated by a subset $S$ of $B^{+}$_
if it is generated by $S$ and $1-1$ as an ideal of $B^{+}$. We write
$N_{B}=\langle S\rangle_{B^{+}}$ in this case.

######  Example .

The main examples of interest to us are the following:

  1. (1)

Every idyll is naturally a band. More to the point, an idyll is the same as a
band $F$ with $F^{\times}=F-\\{0\\}$.

  2. (2)

A ring $R$ is a band with null set $N_{R}=\\{\sum a_{i}\mid\sum a_{i}=0\text{
in }R\\}$.

####  4.1.1. Free algebras

Let $k$ be a band. A _$k$ -algebra_ is a band $B$ together with a band
morphism $\myalpha_{B}:k\to B$ (called the _structure map_). A _$k$ -linear
morphism of $k$-algebras_ from $B$ to $C$ is a band morphism $f:B\to C$ such
that $\myalpha_{C}=f\circ\myalpha_{B}$.

The _free $k$-algebra in $T_{1},\dotsc,T_{n}$_ is the multiplicative submonoid

$k[T_{1},\dotsc,T_{n}]\ =\ \big{\\{}a\cdot T_{1}^{e_{1}}\dotsb
T_{n}^{e_{n}}\,\big{|}\,a\in k,\,e_{1},\dotsc,e_{n}\in{\mathbb{N}}\big{\\}}$

of the polynomial semiring $k[T_{1},\dotsc,T_{n}]^{+}$ over $k^{+}$. It
contains $k$ as a submonoid and gains the structure of a band with respect to
the null set generated by $N_{k}$. The free algebra $k[T_{1},\dotsc,T_{n}]$ in
$T_{1},\dotsc,T_{n}$ satisfies the expected universal property: every map
$f_{0}:\\{T_{1},\dotsc,T_{n}\\}\to B$ into a $k$-algebra $B$ extends uniquely
to a $k$-linear morphism $f:k[T_{1},\dotsc,T_{n}]\to B$.

####  4.1.2. Quotients

A _null ideal_ of a band $B$ is an ideal $I$ of the ambient semiring $B^{+}$
that contains the null set $N_{B}$ and satisfies the substitution rule:

  1. (SR)

if $a-c\in I$ and $c+\sum b_{j}\in I$, then $a+\sum b_{j}\in I$.

Given a subset $S$ of $B^{+}$, we denote the smallest null ideal that contains
$S$ by $\langle S\rangle$.

The _quotient of $B$ by a null ideal $I$_ is the following band
$B\\!\sslash\\!I$: as a set, $B\\!\sslash\\!I$ is the quotient $B/\sim$ by the
equivalence relation given by $a\sim b$ if and only if $a-b\in I$. It is a
pointed monoid with product $[a]\cdot[b]=[ab]$. The null set of
$B\\!\sslash\\!I$ is defined as

$\textstyle N_{B\\!\sslash\\!I}\ =\ \big{\\{}\sum[a_{i}]\,\big{|}\,\sum
a_{i}\in I\big{\\}}.$

The association $a\mapsto[a]$ defines a band morphism $\mypi_{I}:B\to
B\\!\sslash\\!I$, which satisfies the following universal property: every band
morphism $f:B\to C$ with $\sum f(a_{i})\in N_{C}$ for every $\sum a_{i}\in I$
factors into $f=\bar{f}\circ\mypi_{I}$ for a uniquely determined morphism
$\bar{f}:B\\!\sslash\\!I\to C$.

####  4.1.3. Localizations

A _multiplicative subset_ of a band $B$ is a subset $S$ that contains $1$ and
is closed under multiplication. Given such $S$, we define the _localization of
$B$ at $S$_ as

$S^{-1}B\ =\ (S\times B)\,/\,\sim$

where $(s,a)\sim(s^{\prime},a^{\prime})$ if and only if
$tsa^{\prime}=ts^{\prime}a$ for some $t\in S$. We denote by $\frac{a}{s}$ the
class of $(s,a)$ in $S^{-1}B$. The product
$\frac{a}{s}\cdot\frac{b}{t}=\frac{ab}{st}$ turns $S^{-1}B$ into a pointed
monoid, and the null set

$\textstyle N_{S^{-1}B}\ =\ \big{\\{}\sum\frac{a_{i}}{s}\,\big{|}\,s\in S,\
\sum a_{i}\in N_{B}\big{\\}}$

endows $S^{-1}B$ with the structure of a band.

The association $a\mapsto\frac{a}{1}$ defines a band morphism
$\myiota_{S}:B\to S^{-1}B$, which satisfies the following universal property:
every band morphism $f:B\to C$ with $f(S)\subseteq C^{\times}$ factors into
$f=f_{S}\circ\myiota_{S}$ for a uniquely determined morphism $f_{S}:S^{-1}B\to
C$.

A _finite localization of $B$_ is a localization of the form
$B[h^{-1}]=S^{-1}B$ where $h\in B$ and $S=\\{h^{i}\\}_{i\in{\mathbb{N}}}$.

####  4.1.4. Ideals

An _$m$ -ideal_ of a band $B$ is a subset $I$ of $B$ that contains $0$ such
that $IB=I$. An $m$-ideal ${\mathfrak{p}}$ is _prime_ if $S=B-{\mathfrak{p}}$
is a multiplicative subset of $B$. The _localization of $B$ at
${\mathfrak{p}}$_ is $B_{\mathfrak{p}}=S^{-1}B$.

Every nonzero band $B$ has a unique maximal $m$-ideal, which is
${\mathfrak{m}}_{B}=B-B^{\times}$ and which is prime.

####  4.1.5. Limits and colimits

The category of a bands contains all limits and colimits. The regular partial
field ${{\mathbb{F}}_{1}^{\pm}}$ is initial and the zero band
$\mathbf{0}=\\{0\\}$ is terminal in $\operatorname{Bands}$. The product of two
bands $B$ and $C$ is the Cartesian product $B\times C$ together with the
inherited structure as a band. The tensor product $B\otimes_{k}C$ of two
$k$-algebras $B$ and $C$ is defined as the colimit of $B\leftarrow k\to C$.

####  4.1.6. The universal ring

The _universal ring_ of a band $B$ is the ring

$B^{+}_{\mathbb{Z}}\ :=\ {\mathbb{Z}}[B]/\langle N_{B}\rangle$

where $\langle N_{B}\rangle$ is the ideal of ${\mathbb{Z}}[B]$ generated by
all expression in $N_{B}$, interpreted as a subset of the monoid algebra
${\mathbb{Z}}[B]$. If $R$ is a ring with associated band $B$, then
$B^{+}_{\mathbb{Z}}=R$.

The universal ring comes together with a map $B\to B^{+}_{\mathbb{Z}}$, which
is universal among all maps $f:B\to R$ into a ring $R$ with the property that
$\sum f(a_{i})=0$ for all $\sum a_{i}\in N_{B}$. In other words,
$\operatorname{Rings}$ form a reflective subcategory of
$\operatorname{Bands}$.

Note that the canonical map $B\to B^{+}_{\mathbb{Z}}$ is in general not
injective. For example, ${\mathbb{K}}^{+}_{\mathbb{Z}}$ is the trivial ring
$\mathbf{0}=\\{0\\}$.

####  4.1.7. Bands as ordered blueprints

For the sake of this section, we assume that the reader is familiar with the
notion of an ordered blueprint. We employ the notation from [BL21] and
[BJL24].

A band $B$ can be viewed as the ordered blueprint $B^{\textup{oblpr}}$ with
underlying monoid $B$, ambient semiring $B^{+}$ and partial order generated by
the relations $0\leqslant\sum a_{i}$ for which $\sum a_{i}\in N_{B}$. This
association defines a fully faithful embedding
$\operatorname{Bands}\to\operatorname{{OBlpr}}$. In fact, this embedding is a
coreflection: every ordered blueprint $B=(B^{\bullet},B^{+},\leqslant)$ has an
associated band $B^{\textup{band}}=B^{\bullet}$ whose nullset is

$\textstyle N_{B^{\textup{band}}}\ =\ \big{\\{}\sum
a_{i}\,\big{|}\,0\leqslant\sum a_{i}\text{ in }B\big{\\}},$

and the canonical map $(B^{\textup{band}})^{\textup{oblpr}}\to B$ establishes
an adjunction

$\operatorname{Hom}_{\operatorname{Bands}}(B,\
C^{\textup{band}})\quad\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\quad\operatorname{Hom}_{\operatorname{{OBlpr}}}(B^{\textup{oblpr}},\
C).$

This tight connection between bands and ordered blueprints makes it possible
to specialize and transfer many results from ordered blueprints to bands.

###  4.2. Band schemes

The spectrum of a band is a topological space that comes with a structure
sheaf of coordinate bands. A band scheme is a space that locally looks like
the spectrum of a band. We review these definitions in the following.

####  4.2.1. Band spaces

A _band space_ is a topological space $X$ together with a sheaf
${\mathcal{O}}_{X}$ in $\operatorname{Bands}$. Every point $x\in X$ comes with
a _stalk_ , which is
${\mathcal{O}}_{X,x}=\operatorname{colim}{\mathcal{O}}_{X}(U)$ where $U$
ranges over all open neighbourhoods $U$ of $x$. A _morphism of band spaces_ is
a continuous map $\myvarphi:X\to Y$ together with a morphism
$\myvarphi^{\\#}:{\mathcal{O}}_{Y}\to\myvarphi_{\ast}{\mathcal{O}}_{X}$ of
sheaves that is _local_ in the sense that for every $x\in X$ and
$y=\myvarphi(x)$, the induced morphism of stalks
$\myvarphi^{\\#}_{x}:{\mathcal{O}}_{Y,y}\to{\mathcal{O}}_{X,x}$ maps the
maximal ideal ${\mathfrak{m}}_{y}$ of ${\mathcal{O}}_{Y,y}$ to the maximal
ideal ${\mathfrak{m}}_{x}$ of ${\mathcal{O}}_{X,x}$. This defines the category
$\operatorname{BandSpaces}$ of band spaces.

Every open subset $U$ of a band space $X$ is naturally a band space with
respect to the restriction of ${\mathcal{O}}_{X}$ to $U$.

####  4.2.2. The spectrum

Let $B$ be a band. The _spectrum of $B$_ is the set $X=\operatorname{Spec}B$
of all prime $m$-ideals ${\mathfrak{p}}$ of $B$, endowed with the topology
generated by the _principal opens_

$U_{h}\ =\ \\{{\mathfrak{p}}\in X\mid h\notin{\mathfrak{p}}\\}$

for $h\in B$ and with the sheaf ${\mathcal{O}}_{X}$ with values
${\mathcal{O}}_{X}(U_{h})=B[h^{-1}]$ and stalks
${\mathcal{O}}_{X,{\mathfrak{p}}}=B_{\mathfrak{p}}$.

Note that ${\mathcal{O}}_{X}$ is uniquely determined by its values on
principal opens since the $U_{h}$ form a basis for the topology of
$\operatorname{Spec}B$. The proof of the existence of ${\mathcal{O}}_{X}$ is
easier than in usual algebraic geometry since every non-empty spectrum
$\operatorname{Spec}B$ contains a unique closed point, which is the maximal
ideal ${\mathfrak{m}}_{B}=B-B^{\times}$ of $B$.

A band morphism $f:B\to C$ induces a morphism
$\myvarphi=f^{\ast}:\operatorname{Spec}C\to\operatorname{Spec}B$ with
$f^{\ast}({\mathfrak{q}})=f^{-1}({\mathfrak{q}})$ and the sheaf morphism
$\myvarphi^{\\#}$ determined by

$\begin{array}[]{cccc}\myvarphi^{\\#}(U_{h}):&B[h^{-1}]&\longrightarrow&C[f(h)^{-1}].\\\\[5.0pt]
&\frac{a}{h^{i}}&\longmapsto&\frac{f(a)}{f(h)^{i}}\end{array}$

This defines a contravariant functor
$\operatorname{Spec}:\operatorname{Bands}\to\operatorname{BandSpaces}$, which
is fully faithful.

######  Example .

The following are first examples of spectra:

  1. (1)

An idyll $F$ has a unique prime $m$-ideal, which is $\myeta=\\{0\\}$. Thus
$\operatorname{Spec}F=\\{\myeta\\}$ is a singleton.

  2. (2)

The _affine space_ over a band $B$ is
${\mathbb{A}}^{n}=\operatorname{Spec}B[T_{1},\dotsc,T_{n}]$. For an idyll $F$,
the prime $m$-ideals of ${\mathfrak{p}}$ of $F[T_{1},\dotsc,T_{n}]$ are of the
form ${\mathfrak{p}}_{I}=\langle T_{i}\mid i\in I\rangle$ for subsets $I$ of
$E=\\{1,\dotsc,n\\}$. Thus ${\mathbb{A}}^{n}_{F}=\\{{\mathfrak{p}}_{I}\mid
I\subseteq E\\}$ as a set, and ${\mathfrak{p}}_{I}$ is in the closure of
${\mathfrak{p}}_{J}$ if and only if $J\subseteq I$.

####  4.2.3. Band schemes

An _affine band scheme_ is a band space that is isomorphic to the spectrum of
a band. A _band scheme_ is a band space that has an open covering by _affine
opens_ , which are open subsets that are affine band schemes. This defines the
categories $\operatorname{BAff}$ of affine band schemes and
$\operatorname{BSch}$ of band schemes.

Taking global sections defines a (contravariant) functor
$\Gamma:\operatorname{BSch}\to\operatorname{Bands}$ that is adjoint to
$\operatorname{Spec}$, i.e. $\operatorname{Hom}(B,\Gamma
X)=\operatorname{Hom}(X,\operatorname{Spec}B)$ for all bands $B$ and band
schemes $X$. The restrictions
$\Gamma:\operatorname{BAff}\to\operatorname{Bands}$ and
$\operatorname{Spec}:\operatorname{Bands}\to\operatorname{BAff}$ are mutually
inverse anti-equivalences of categories.

######  Remark  (Local nature of a band scheme).

Since every band has a unique maximal ideal, every affine band scheme has a
unique closed point. This fact has strong consequences:

  1. (1)

Every closed point $x$ of a band scheme $X$ is contained in a unique affine
open neighbourhood, which is $U_{x}=\operatorname{Spec}{\mathcal{O}}_{X,x}$.
In particular, if $X$ is finite, then the affine opens of the form $U_{x}$
(with $x\in X$ closed) form the unique minimal affine open covering of $X$.

  2. (2)

If a band morphism $f:B\to C$ induces an open immersion
$f^{\ast}:\operatorname{Spec}C\to\operatorname{Spec}B$ of affine band scheme,
then $f$ is a finite localization, i.e. $C=B[h^{-1}]$ for some $h\in B$.

  3. (3)

Let $\myvarphi:X\to Y$ be a morphism of band schemes and $X=\bigcup_{i\in
I}U_{i}$ and $Y=\bigcup_{j\in J}V_{j}$ affine open coverings. Then there is a
map $\myvarphi_{0}:I\to J$ such that $\myvarphi(U_{i})\subseteq
V_{\myvarphi_{0}(i)}$ for every $i\in I$.

####  4.2.4. Base extension to ${\mathbb{Z}}$

A band scheme $X$ comes with a universal scheme $X^{+}$ and a canonical
inclusion of $R$-rational point sets $X(R)\subseteq X^{+}(R)$ for every ring
$R$, which is a bijection if $R$ is local.

The universal scheme $X^{+}$ is defined as follows. If
$X=\operatorname{Spec}B$ is affine, we define
$X^{+}_{\mathbb{Z}}:=\operatorname{Spec}B^{+}_{\mathbb{Z}}$. For arbitrary
$X$, we let ${\mathcal{U}}$ be the diagram of all affine opens $U$ of $X$
together with all inclusions $\myiota_{V,U}:V\hookrightarrow U$ as subsets of
$X$, which are open immersions of affine band schemes. Since
$(-)^{+}_{\mathbb{Z}}$ is functorial, this yields a diagram
${\mathcal{U}}^{+}_{\mathbb{Z}}$ of affine schemes. We define

$X^{+}_{\mathbb{Z}}\ =\ \operatorname{colim}{\mathcal{U}}^{+}_{\mathbb{Z}}.$

As a matter of fact, all morphisms of ${\mathcal{U}}^{+}_{\mathbb{Z}}$ are
principal open immersions and the canonical inclusions $U^{+}_{\mathbb{Z}}\to
X^{+}_{\mathbb{Z}}$ are open immersions for $U\in{\mathcal{U}}$.

The functoriality of this construction defines an injection

$X(R)\ =\ \operatorname{Hom}(\operatorname{Spec}R,X)\ \longrightarrow\
\operatorname{Hom}(\operatorname{Spec}R,X^{+}_{\mathbb{Z}})\ =\
X^{+}_{\mathbb{Z}}(R)$

for every ring $R$, which is bijective if $R$ is local since in this case
every morphism $\operatorname{Spec}R\to X^{+}_{\mathbb{Z}}$ factors through
some affine open of the form $U^{+}_{\mathbb{Z}}$ with $U\in{\mathcal{U}}$.

####  4.2.5. Fibre products

The categorical fibre product of two morphisms $\myvarphi_{X}:X\to Z$ and
$\myvarphi_{Y}:Y\to Z$ is represented by a band scheme $X\times_{Z}Y$, which
comes with two projections $\mypi_{X}:X\times_{Z}Y\to X$ and
$\mypi_{Y}:X\times_{Z}Y\to Y$. In the affine case $X=\operatorname{Spec}B$,
$Y=\operatorname{Spec}C$ and $Z=\operatorname{Spec}k$, we have
$X\times_{Z}Y=\operatorname{Spec}(B\otimes_{k}C)$. In general, the fibre
product $X\times_{Z}Y$ is covered by affine opens of this form.

####  4.2.6. Open and closed subschemes

An _open subscheme_ of a band scheme $X$ is an open subset $U$ together with
the restriction of the structure sheaf ${\mathcal{O}}_{X}$ of $X$ to $U$. An
_open immersion_ is a morphism of band schemes $\myvarphi:Y\to X$ that factors
through an isomorphism $Y\to U$ with $U$ an open subscheme of $X$.

An _affine morphism_ is a morphism $\myvarphi:Y\to X$ of band schemes such
that for every affine open $U$ of $X$, the inverse image
$\myvarphi^{-1}(U)=Y\times_{X}U$ is affine. A _closed immersion_ is an affine
morphism $\myvarphi:Y\to X$ such that for every affine open $U$ of $X$ and
$V=\myvarphi^{-1}(U)$, the map $\myvarphi^{\\#}(U):\Gamma U\to\Gamma V$ is
surjective. A _closed subscheme of $X$_ is an isomorphism class of closed
immersions $\myvarphi:Y\to X$.

###  4.3. Projective geometry

Similar to usual schemes, projective geometry for band schemes is governed by
homogeneous coordinate algebras and the $\operatorname{Proj}$-construction. We
introduce all relevant notions in the following.

####  4.3.1. Projective space

Let $k$ be a band. The free algebra $k[T_{0},\dotsc,T_{n}]$ is graded by $\deg
T_{i}=1$ for each $i$, and this grading is inherited by all of its
localizations at elements444Note that all elements of $k[T_{0},\dotsc,T_{n}]$
are homogeneous since they are monomials in $T_{0},\dotsc,T_{n}$. $h\in
k[T_{0},\dotsc,T_{n}]$.

The _projective $n$-space over $k$_ is the band scheme ${\mathbb{P}}^{n}_{k}$
that consists of all prime $m$-ideals of $k[T_{0},\dotsc,T_{n}]$ that do not
contain all of $T_{0},\dotsc,T_{n}$, endowed with the topology generated by
the _principal opens_

$U_{h}\ =\ \\{{\mathfrak{p}}\in{\mathbb{P}}^{n}_{k}\mid
h\notin{\mathfrak{p}}\\}$

for $h\in k[T_{0},\dotsc,T_{n}]$, and whose structure sheaf is characterized
by the values

${\mathcal{O}}_{{\mathbb{P}}^{n}_{k}}(U_{h})\ =\
\big{(}k[T_{0},\dotsc,T_{n}][h^{-1}]\big{)}_{0},$

on principal opens, where $(-)_{0}$ stands for the degree $0$ part. The
projective $n$-space is covered by the canonical affine opens

$U_{i}\ =\ U_{T_{i}}\ =\ \\{{\mathfrak{p}}\in{\mathbb{P}}^{n}_{k}\mid
T_{i}\notin{\mathfrak{p}}\\},$

which are isomorphic to the affine space ${\mathbb{A}}^{n}_{k}$ via the
morphism associated to the band isomorphism

$\begin{array}[]{ccccc}\Gamma({\mathbb{A}}^{n}_{k})&=&k[T_{0}/T_{i},\dotsc,T_{n}/T_{i}]&\longrightarrow&\big{(}k[T_{0},\dotsc,T_{n}][T_{i}^{-1}]\big{)}_{0}\\\
&&T_{j}/T_{i}&\longmapsto&T_{j}/T_{i}.\end{array}$

####  4.3.2. Projective band schemes

A _homogeneous element of $B=k[T_{0},\dotsc,T_{n}]^{+}$ of degree $d$_ is a
polynomial $\sum a_{\underline{e}}T_{1}^{e_{1}}\dotsb T_{n}^{e_{n}}$ such that
$e_{1}+\dotsb+e_{n}=d$ for all indices
$\underline{e}=(e_{i})\in{\mathbb{N}}^{n}$ with $a_{\underline{e}}\neq 0$. A
_homogeneous null ideal of $B$_ is a null ideal $I$ that is generated by
homogeneous elements (of possibly different degrees).

A homogeneous null ideal $I$ of $B$ defines the closed subscheme
$X=\operatorname{Proj}(B\\!\sslash\\!I)$ of ${\mathbb{P}}^{n}_{k}$ whose
underlying space is the subspace

$X\ =\ \big{\\{}{\mathfrak{p}}\in{\mathbb{P}}^{n}_{k}\,\big{|}\,\text{if
$a-b\in I$ and $a\in{\mathfrak{p}}$, then also $b\in{\mathfrak{p}}$}\big{\\}}$

of ${\mathbb{P}}^{n}_{k}$ and whose structure sheaf is determined by its
values

${\mathcal{O}}_{X}\big{(}U_{h}\cap X\big{)}\ =\
\big{(}(B\\!\sslash\\!I)[h^{-1}]\big{)}_{0}$

on the intersections with principal opens $U_{h}$. The inclusion
$\myiota:X\to{\mathbb{P}}^{n}_{k}$ together with the surjection
$\myiota^{\\#}:{\mathcal{O}}_{{\mathbb{P}}^{n}_{k}}\to{\mathcal{O}}_{X}$ of
sheaves is a closed immersion. Conversely, every closed subscheme of
${\mathbb{P}}^{n}_{k}$ is of this form.

######  Example  (Grassmannian).

An example of primary interest are Grassmannians $\operatorname{Gr}(r,E)$ over
${{\mathbb{F}}_{1}^{\pm}}$ for $E=\\{1,\dotsc,n\\}$ and $r$ an integer. The
ambient projective space

${\mathbb{P}}^{n^{r}-1}\ =\
\operatorname{Proj}\big{(}{{\mathbb{F}}_{1}^{\pm}}[T_{\textbf{x}}\mid\textbf{x}\in
E^{r}]\big{)}$

has homogeneous coordinates $T_{\textbf{x}}$, indexed by $r$-tuples
$\textbf{x}=(x_{1},\dotsc,x_{r})$ of elements of $E$. The homogeneous null