# Fundamental Theorem of Projective Geometry over Semirings

Ayush Kumar Tewari

###### Abstract

We state the fundamental theorem of projective geometry for semimodules over
semirings, which is facilitated by recent work in the study of bases in
semimodules defined over semirings. In the process we explore in detail the
linear algebra setup over semirings. We also provide more explicit results to
understand the implications of our main theorem on maps between tropical lines
in the tropical plane. Along with this we also look at geometrical connections
to the rich theory of tropical geometry.

###### Contents

  1. 1 Introduction
  2. 2 Linear Algebra over Semirings
  3. 3 Main Theorem
  4. 4 Projectivity in the tropical plane
  5. 5 Conclusions

##  1 Introduction

Maps between projective spaces and their underlying algebraic structures have
been a classical area of study in algebraic geometry. This has been immensely
enriched with the seminal work of Tits on the theory of buildings and the
connections with the study of classical groups. One of the prominent results
in this case is the well known fundamental theorem of projective geometry
which is a structure theorem and elaborates on the structure of the
collinearity preserving maps between projective spaces associated with vector
spaces over fields. This in turn also helps in assessing the automorphism
group of such spaces and the group of isomorphisms of the underlying field.
The theorem has appeared in various forms throughout the literature in the
19th and 20th century, with various attributions attached to the different
versions. The first occurrence of some versions of the statement can be dated
back to the work of Darboux, Segre, Veblen and Von Staudt [VS47], [Dar80],
[Seg90].

Since then considerable work has been done to understand these maps and their
outcomes in terms of the associated geometry. Various versions of the
statement have been proved for linear systems over algebraic structures other
than fields, for e.g. free modules over commutative rings [OS69]. Our goal in
this article is to understand these results in the realm of semirings and
semimodules over them. The notion of linear independence and basis elements is
a bit nuanced in this setting. The one stark difference in this setting is
that two basis of a semimodule over a semiring may not have the same number of
elements. We use recent findings about the properties of basis over
semimodules [Tan14] and develop a background to define the notion of semiring
semi-linear maps and semiring collineations.

Modules (Vector Spaces)SemimodulesRings (Fields)Semirings Fund. Theorem of
Projective Geometry Fund. Theorem of (Tropical) Projective Geometry

Our goals in this work are twofold; firstly we describe an algebraic setup of
semimodules over semirings along with a geometric aspect to understand how
maps act on "lines" in "planes", with special emphasis in the case when the
underlying semiring is the _tropical semiring_. One of our first observations
is that _collinearity_ is geometrically well represented by _coaxiality_ in
the tropical setup. In Section 2, we discuss the basic background of linear
algebra over semirings, and describe the results regarding bases of
semimodules over _linear_ semirings. In Section 3 we establish our main
contribution in the form of Theorem 3, which we state as the fundamental
theorem of projective geometry over semirings, which can be also understood as
a fundamental theorem of tropical projective geometry. In Section 4, we see a
more explicit version where we prove a variant of the fundamental theorem,
restricted to the case of the tropical plane.

In Sections 5 we discuss possible generalizations of our results in terms of a
possible theory of buildings and complexes for tropical flag manifolds and
flag varieties, which is inspired by some recent work in this field
[BEZ21],[BH20]. We also recognize that the theory of finitely generated
semimodules ties up with _tropical convexity_ , as finitely generated
semimodules are also considered as _tropical cones_ in the literature
[MS09],[Jos22]. We introduce some of the connections of our work to the theory
of tropical convexity and are pretty hopeful to take it further in subsequent
work. We also highlight the fact that our results also complements the study
on matrix semigroups over semirings [IJK18], [GJN20] and can have fruitful
connections to problems concerning semigroups. We close our discussion with
possible definitions of _tropical cross ratios_ which might be suitable in
order to study tropical projective geometry further.

### Acknowledgements

I would like to thank Marianne Johnson and Hannah Markwig who provided their
valuable comments and suggestions on this draft. I would also like to thank
Michael Joswig for sharing his thoughts on the topic with me. I am also
grateful to the Mathematisches Forschungsinstitut Oberwolfach for providing
excellent facilities and for their warm hospitality during my stay. This
research was supported through the programme "Oberwolfach Leibniz Fellows" by
the Mathematisches Forschungsinstitut Oberwolfach in 2021.

##  2 Linear Algebra over Semirings

A semiring is defined as follows [Gol13],

######  Definition 1.

A semiring is a set $R$ equipped with two operations $+$ and . such that
$(R,+)$ is an abelian monoid with identity element $0$ and $(R,.)$ is also a
monoid with identity element 1, along with a distributive law which respects
the two operations.

An element $a$ in $R$ is called _invertible_ if there exists $b\in R$ such
that $ab=ba=1$.We denote by $U(R)$ the set of all invertible elements in $R$.
A semiring $R$ is called a semidomain if $ab=ac$ implies $b=c$ for all $b,c\in
R$ and all nonzero $a\in R$.

As is clear from the definition, no element of a semiring has a additive
inverse. For our discussions we would be considering the multiplicative
operation to be commutative, unless otherwise stated.

We define a semimodule [Gol13], which is based on the definition of a module
over a commutative ring.

######  Definition 2.

Let $R$ be a semiring. A $R$-semimodule is a commutative monoid $(M,+)$ with
additive identity $\theta$ for which we have a function $R\times
M\longrightarrow M$, denoted by $(\lambda,\alpha)\rightarrow\lambda\alpha$ and
called a scalar multiplication, which satisfies the following conditions for
all $\lambda,\mu$ in $R$ and $\alpha,\beta$ in $M$:

  1. 1.

$(\lambda\mu)\alpha=\lambda(\mu\alpha);$

  2. 2.

$\lambda(\alpha+\beta)=\lambda\alpha+\lambda\beta;$

  3. 3.

$(\lambda+\mu)\alpha=\lambda\alpha+\mu\alpha;$

  4. 4.

$1\alpha=\alpha$

  5. 5.

$\lambda\theta=\theta=0\alpha$

######  Example 2.0.1.

For a semiring $R$, $R$ is also a $R$\- semimodule over itself. Similarly,
$R^{d}$ is also a $R$\- semimdoule.

A nonempty subset $N$ of $M$ is said to be a subsemimodule of $M$ if it is
closed under addition and scalar multiplication.

Let $S$ be a nonempty subset of a $R$-semimodule $M$. Then the intersection of
all subsemimodules of $M$ containing $S$ is a subsemimodule of $M$, called the
subsemimodule generated by $S$ and denoted by $RS$ [Tan14].

$RS=\\{\sum_{i=1}^{k}\lambda_{i}\alpha_{i}|\lambda_{i}\in R,\alpha_{i}\in
S,i\in\underline{k},k\in\mathbb{N}\\}$

The expression $\sum_{i=1}^{k}\lambda_{i}\alpha_{i}$ is called a _linear
combination_ of the elements $\alpha_{1},\alpha_{2},\ldots\alpha_{k}$. If
$RS=M$, then $S$ is called a _generating set_ for $M$. We state some
definitions from [Gol13],

######  Definition 3.

Let $M$ be an $R$-semimodule. A non-empty subset $S$ of $M$ is called
_linearly independent_ if $\alpha\not\in R(S\setminus\\{\alpha\\})$ for any
$\alpha$ in $S$. If $S$ is not linearly independent then it is called
_linearly dependent_. The set $S$ is called _free_ if each element in $M$ can
be expressed as a linear combination of elements in $S$ in at most one way. It
is clear that any free set is linearly independent.

######  Definition 4.

Let $M$ be an $R$-semimodule. A linearly independent generating set for $M$ is
called a _basis_ for $M$ and a free generating set for $M$ is called a _free
basis_ for $M$. A $R$-semimodule having a free basis is called a _free
R-semimodule_.

For any commutative semiring $R$, let $\kappa(R)$ = max $\\{t\in\mathbb{N}$
$|$ the $R$-semimodule $R$ has a basis with $t$ elements $\\}$.

With linear combination of elements defined over semirings, the notion of a
basis is a quite natural extension. However, this as a notion is a bit subtle
in this case. Firstly, for a general semimodule, bases might not have the same
cardinality, i.e., two bases can have different number of elements. However,
in [Tan14] it is shown that if we restrict ourselves to certain semimodules
over a certain class of semirings, this can be restricted, and is stated in
the form of the following result,

######  Theorem 1 (Theorem 4.3 [Tan14]).

For any commutative semiring $R$, the following statements are equivalent,

  1. 1.

$\kappa(R)=1$

  2. 2.

For any $u,v\in R,1=u+v$ implies that either $u\in U(R)$ or $v\in U(R)$

  3. 3.

Any two bases for a finitely generated free $R$-semimodule $M$ have the same
cardinality.

We refer to a semiring with $\kappa(R)=1$ as a semi-linear semiring.

An important fact mentioned in [Tan14] is that the tropical semiring (both max
and min) is a semi-linear semiring.

For a free, finitely generated semimodule over a semi-linear semiring, we
refer to the cardinality of its basis as its _dimension_ or _rank_ of the
semimodule.

We recall the definition of an ideal over a semiring [Tan14].

######  Definition 5.

A nonempty subset $I$ of a semiring $R$ is said to be an ideal of $R$, if
$a+b\in I$ for all $a,b\in I$ and $ra\in I$ for all $r\in R$ and $a\in I$.

######  Definition 6.

An ideal $I$ of a semiring $R$ is called principal if $I=\\{ra:r\in R\\}$ for
some $a\in R$.

A semiring $R$ is called a principal ideal semidomain if $R$ is a semidomain
and all its ideals are principal.

######  Example 2.0.2.

The tropical semiring $\mathbb{T}=\\{\mathbb{R}\cup-\infty,max,+\\}$ is a
principal ideal semidomain. Firstly, it is easy to see that $\mathbb{T}$ is a
semidomain. Also, we observe that the only ideals of this semiring are,
$I_{0}=\emptyset$, the empty ideal, the single element ideal
$I_{1}=\\{-\infty\\}$, i.e the ideal containing the element $-\infty$ and the
full semiring $\mathbb{T}$. We see that $\mathbb{T}=\langle 0\rangle$, i.e the
ideal generated by $0$ is equal to the the whole semiring. Also,
$I_{1}=\\{-\infty\\}=\langle-\infty\rangle$. Hence, all ideals are principal,
and therefore $\mathbb{T}$ is a principal ideal semidomain.

We now look for a tropical counterpart of some classical results concerning
free and finitely generated modules over principal ideal domains; the proof
for Theorem 1 that we provide here follows in a mostly straight forward way
from the classical case [Mcn] and is stated here for the sake of completeness.
For this we consider free and finitely generated semimodules over principal
ideal semidomains.

######  Theorem 1.

Let $R$ be a principal ideal semidomain, let $F$ be a free and finitely
generated $R$-semimodule and let $E$ be a subsemimodule of $F$. Then $E$ is a
free and finitely generated $R$-semimodule and the rank of $E$ is at most the
rank of F.

###### Proof.

We first begin with some basic observations. We know that $R$ is a free
$R$-semimodule of rank 1. Consider $E$ to be a subsemimodule of the free
$R$-semimodule $R$. It is clear that $E$ is an ideal and that the ideals of
$R$ coincide with the subsemimodules of $R$. In case $E$ is trivial we see
that $E$ is the free $R$-semimodule of rank 0. So we consider the case that
$E$ is nontrivial. Since $R$ is a principal ideal semidomain we pick $w\neq 0$
so that $E$ is generated by $w$. That is $E=\\{rw|r\in R\\}$. Since we know
that $R$ has $\\{1\\}$ as a basis, we see that the map that sends $1$ to $w$
extends to a unique semimodule homomorphism from $R$ onto $E$. Indeed, notice
$h(r\cdot 1)=r\cdot h(1)=rw$ for all $r\in R$. But the homomorphism $h$ is
also one-to-one since

$\begin{split}h(r)&=h(s)\\\ rh(1)&=sh(1)\\\ rw&=sw\\\ &r=s\end{split}$

Therefore, we see that $E$ is isomorphic to the free $R$-module of rank $1$.
Also, we conclude that subsemimodules of the free $R$-module of rank 1 are
themselves free and have either rank 0 or rank 1.

Since $F$ is a free and finitely generated semimodule, therefore Theorem 1
implies that all basis for $F$ have the same cardinality. Let $B$ be a basis
for $F$ and $C\subseteq B$. Because $F$ is not the trivial module, we see that
$B$ is not empty. Let $F_{C}$ be the subsemimodule of $F$ generated by $C$.
Let $E_{C}=E\cap F_{C}$. Evidently, $C$ is a basis for $F_{C}$. To see that
$E_{C}$ is free and finitely generated we will have to find a basis for it.

Suppose, for a moment, that $C$ has been chosen so that $E_{C}$ is known to be
free and finitely generated and consider an element $w\in B$ with $w\not\in
C$. Put $D:=C\cup\\{w\\}$. Consider the map defined on $D$ into $R$ that sends
all the elements of $C$ to $0$ and that sends $w$ to $1$. This map extends
uniquely to a homomorphism of semimodules $\phi$ from $F_{D}$ onto $R$ and it
is easy to check that the kernel of $\phi$ is just $F_{C}$. Contrary to the
case of modules, semimodules in the context of universal algebras have salient
versions of isomorphism theorems involving congruence relations and we notice
that Ker $\phi$ defines a congruence relation on $F_{D}$ in this case. We
invoke such isomomorphism theorem for semimodules [PR13, Theorem 2.6] and as
$\phi$ is one-one by definition we conclude that $F_{D}/F_{C}$ is isomorphic
to $R$ and that it is free of rank 1. Observe that $E_{C}=E\cap F_{C}=E\cap
F_{D}\cap F_{C}=E_{D}\cap F_{C}$. Again using the isomorphism theorem, we
obtain a second isomorphism theorem as follows,

$E_{D}/E_{C}=E_{D}/E_{D}\cap F_{C}\cong E_{D}+F_{C}/F_{C}$

But $E_{D}+F_{C}/F_{C}$ is a subsemimodule of $F_{D}/F_{C}$. This is a free
R-semimodule of rank 1. We noted earlier that every subsemimodule of a free
$R$-semimodule of rank 1 must be itself a free $R$-semimodule and have rank
either 0 or 1. In this way, we find that either $E_{D}=E_{C}$ (in the rank 0
case) or else $E_{D}/E_{C}$ is a free $R$-semimodule of rank 1. Let us take up
this latter case. Let $X$ be a basis for $E_{C}$, which we assumed, for the
moment, was free and finite. Pick $u\in E_{D}$ so that $\\{u/E_{C}\\}$ is a
basis for $E_{D}/E_{C}$.

We claim that $X\cup\\{u\\}$ is a basis for $E_{D}$. Suppose
$x_{0},\ldots,x_{n-1}$ are distinct elements of $X$, $r_{0},\ldots,r_{n}\in R$
and

$0=r_{0}x_{0}+\ldots+r_{n-1}x_{n-1}+r_{n}u$

Also,

$r_{n}(u/E_{C})=r_{n}u/E_{C}=(r_{0}x_{0}+\ldots+r_{n-1}x_{n-1}+r_{n}u)/E_{C}=0/E_{C}$

Since ${u/E_{C}}$ is a basis for $E_{D}/E_{C}$, we must have $r_{n}=0$. This
leads to

$0=r_{0}x_{0}+\ldots+r_{n-1}x_{n-1}$

But now since $X$ is a basis for $E_{C}$ we see that $0=r_{0}=\ldots=r_{n-1}$.
So we find that $X\cup\\{u\\}$ is linearly independent. To see that
$X\cup\\{u\\}$ generates $E_{D}$, pick $z\in E_{D}$. Since ${u/E_{C}}$ is a
basis for $E_{D}/E_{C}$, pick $r\in R$ so that $z/E_{C}=ru/E_{C}$. This means
that $z-ru\in E_{C}$. But $X$ is a basis of $E_{C}$. So pick
$x_{0},\ldots,x_{n-1}\in X$ and $r_{0},\ldots,r_{n-1}\in R$ so that
$z-ru=r_{0}x_{0}+\ldots+r_{n-1}x_{n-1}$. Surely this is enough to see that $z$
is in the subsemimodule generated by $X\cup\\{u\\}$. So this set generates
$E_{D}$ and we conclude that it must be a basis of $E$.

With this preliminary setup we try to generalize our argument for finding the
free and finite basis for $E$. Notice that $E=E\cap F=E\cap F_{B}$. So
$E=E_{B}$. We start with $\emptyset\subseteq B$. We observe that
$F\emptyset=E\emptyset$ is the module whose sole element is $0$. It is free of
rank $0$. Next we select an element $w\in B$ and form
$\emptyset\cup\\{w\\}=\\{w\\}$. We find that $E_{\\{w\\}}$ is free of rank $0$
or rank $1$. We select other elements until finally all the elements of $B$
have been selected. At this point we have $E_{B}$ which is free and finitely
generated and its rank can be no more than the total number of elements we
selected, namely $|B|$ which is the rank of $F$.

Let $\mathcal{F}=\\{f|f\text{is a function with dom}\>f\subseteq B\>\text{and
range f is a basis for}\>E_{domf}\\}$. We see that $\mathcal{F}$ is partially
ordered by set inclusion. We note that $\mathcal{F}$ is not empty since the
empty function is a member of $\mathcal{F}$. To invoke Zorn’s lemma, let $C$
be any chain included in $\mathcal{F}$. Let $h=\bigcup C$. Evidently
$f\subseteq h$ for all $f\in C$. So $h$ is an upper bound of $C$. We also
conclude that $h\in\mathcal{F}$. It is also evident that
$\text{dom}\>h=\bigcup\>\\{\text{dom}\>f|f\in C\\}$ and that
$\text{range}\>h=\bigcup\>\\{\text{range}\>f|f\in C\\}$. It remains to show
that range $h$ is a basis for $E_{\text{dom h}}$. To see that range $h$ is a
generating set, let $z$ be an arbitrary element of $E_{\text{dom h}}=E\cap
F_{\text{dom h}}$. Hence $z$ must be generated by some finitely many elements
belong in dom h. This means there are finitely many functions
$f_{0},\ldots,f_{n-1}\in C$ so that $z$ is generated by finitely many elements
of $\text{dom}f_{0}\cup\ldots\cup\text{dom}f_{n-1}$. But
$\text{dom}f_{0}\cup\ldots\cup\text{dom}f_{n-1}$, under rearrangement, forms a
chain under inclusion. So $z\in F_{\text{dom}f_{l}}$ for some $l<n$. Hence
$z\in E_{\text{dom}f_{l}}$. But $\text{range}f_{l}$ is a basis for
$E_{\text{dom}f_{l}}$. Because $\text{range}f_{l}\subseteq\text{range}h$ we
find that range $h$ has enough elements to generate $z$. Since $z$ was an
arbitrary element of $E_{\text{dom}h}$ we conclude that range $h$ generates
$E_{\text{dom}h}$. It remains to show that range $h$ is linearly independent.
But range $h$ is the union of the chain $\\{\text{range}f|f\in C\\}$. We
recall that the union of any chain of linearly independent sets must also be
linearly independent. This implies that $h$ belongs to $\mathcal{F}$. By
Zorn’s lemma, let $g$ be a maximal element of $\mathcal{F}$. We are done if
$\text{dom}g=B$, since then $E=E\cap F=E\cap F_{B}=E_{B}=E_{\text{dom}g}$. In
which case, range $g$ would be a basis for $E$ and rank $E$ =
$|\text{range}\>g|\leq|\text{dom}\>g|=|B|=\text{rank}\>F$.

Consider the possibility that dom $g$ is a proper subset of $B$. Put $C$ = dom
$g$ and put $X$ = range $g$. Let $w\in B$ with $w\not\in$ dom $g$. Put
$D=C\cup\\{w\\}$. As we have seen above, either $E_{D}=E_{C}$ or
$X\cup\\{u\\}$ is a basis for $E_{D}$, for some appropriately chosen $u$. We
can now extend $g$ to a function $g^{\prime}$ by letting $g^{\prime}(w)$ be
any element of range $g$ in the case when $E_{D}=E_{C}$ and by letting
$g^{\prime}(w)=u$ in the alternative case. In this way,
$g^{\prime}\in\mathcal{F}$, contradicting the maximality of $g$. So we negate
this possibility.

This completes the proof.

∎

We refer to a semi-linear semiring which is also a principal ideal semidomain
as a linear semiring.

######  Example 2.0.3.

The tropical semiring
$\mathbb{T}=\\{\mathbb{R}\cup\\{-\infty\\},\>\text{max},\>+\\}$ is a linear
semiring.

We now define the notion of a projective space over a semiring,

######  Definition 7.

Given a semimodule $M$ over a semiring $R$, the projective space over a
semimodule is the quotient space of the semimodule (omitting the additive
identity $0$) under scalar multiplication, omitting multiplication by the
scalar additive identity $0$.

$R(M)=(M\setminus 0)/(R\setminus 0)$

######  Example 2.0.4.

The _tropical projective space_ of dimension $d-1$ is [Jos22],

$\begin{split}{\mathbb{T}\mathbb{P}}^{d-1}=(\mathbb{T}^{d}\setminus\mathbf{\infty})\>/\>\mathbb{R}\cdot
1\\\ \>\text{where},\>\infty=(\infty,\ldots,\infty)\end{split}$

As elaborated in [Art11], a vector space $V$ can be attached with the notion
of a corresponding projective space $\overline{V}$, in which its elements are
the subspaces $U\subset V$, and each subspace $U$ has a projective dimension
dim${}_{p}U$ = dim $U$ \- 1. Similarly, we define $\overline{V}$ to be the
associated projective space corresponding to a free, finitely generated
semimodule $V$ over a linear semiring $R$, which has its elements as
subsemimodules $U$ of $V$ and the dimension of $U$ = #(basis($U$)) - 1. This
evidently coincides with the notion of a projective space in Definition 7, as
defined earlier, and this is illustrated in the following example.

######  Example 2.0.5.

Let us consider the semimodule
$V=\mathbb{T}^{3},\>\text{then}\>\mathbb{T}\mathbb{P}^{2}=(\mathbb{T}^{3}\setminus\\{-\infty\\})/\mathbb{R}\cdot
1$. In this case we also see that for $V=\mathbb{T}^{3}$ the associated
projective space consists of subsmsemimodules of $\mathbb{T}^{3}$, like
$\mathbb{T}^{2}$ of dimension two and $\mathbb{T}^{1}$ of dimension one. These
subsemimodules correspond to the points in $\mathbb{T}\mathbb{P}^{2}$, like
$\mathbb{T}^{2}$ corresponds to $(0,0,\infty)/\mathbb{R}\cdot 1$ and
$\mathbb{T}^{1}$ corresponds to $(0,\infty,\infty)/\mathbb{R}\cdot 1$. We
refer the reader to Figure 4 for a complete description of
$\mathbb{T}\mathbb{P}^{2}$. We realize that the scalar multiplication in the
subsemimodules in the definition of the associated projective space is
compensated by the action of $\mathbb{R}\cdot 1$ in Definition 7 and hence
these two definitions coincide.

Therefore, the "points" of $V$ are subspaces of projective dimension $0$ and
the "lines" are subspaces of projective dimension 1. Thus the lines of $V$
become the "points" of $\overline{V}$ and the planes of $V$ become the "lines"
of $\overline{V}$ [Art11].

We now take a close look at the behaviour of points and lines over a semiring
. A line over a semiring $R$ is defined as a one-dimensional subsemimodule of
a semimodule $V$ defined over $R$.

In the case of the tropical plane, a tropical line is defined as a
hypersurface by the linear polynomial

$L\equiv p(x,y)=a\otimes x\oplus b\otimes y\oplus c,\>\>a,b,c\in R$

where $L$ is a one dimensional subsemimodule of $V=\mathbb{R}^{2}$ over
$R=\mathbb{R}$, which is equal to the corner locus of three half rays
emanating from the point $(c-a,c-b)$, which we refer as the _vertex_ of the
tropical line, in the primitive directions of $(-1,0),(0,-1),(1,1)$ [MS09], as
described in Figure 1.

######  Remark 2.0.6.

At this juncture we would also like to point the reader to [Jos22, Section
5.2] and [MS09, Remark 5.2.2] which relate equivalence of tropical cones and
semimodules and equivalence of tropically convex sets and subsemimodules in
$\mathbb{T}^{d}\>/\mathbb{R}\cdot 1$, respectively. For readers familiar with
tropical convexity, our discussion regarding semimodules over semirings can
also be visualized in the context of tropical convexity, at least in the case
when the underlying semiring is the tropical semiring. We elaborate on the
connections between our results and tropical convexity in Section 3.

Figure 1: A tropical line $p_{1}\quad$$p_{2}\quad$ Figure 2: Two points
defining infinite number of tropical lines

A feature of such a geometric definition of a tropical line is that two
tropical lines can intersect at more than one point; they can intersect over a
a common half ray. Two tropical lines which have a unique intersection are
said to be in _general position_. Also, the tropical plane allows projective
duality [BJLR18], hence considering the projective dual, two points in the
tropical plane can define infinite number of lines passing through them (cf.
Figure 2). This leads us to the following definition [BJLR18],

######  Definition 8.

Two points are said to be _coaxial_ if they lie on the same axis of a tropical
line containing them.

Two tropical lines are said to be coaxial if there vertices are coaxial. As is
evident, this definition of coaxial lines is specific to the tropical
semiring.

We also recall the definition of a _stable tropical line_ from [Tew20],

######  Definition 9.

Consider $(L,p_{1},\ldots,p_{n}),(n\geq 2)$ where $L$ is a tropical line with
the points $(p_{1},\ldots,p_{n})$ on the line $L$, then
$(L,p_{1},\ldots,p_{n})$, is called stable if

  1. 1.

either $L$ is the unique line passing through the $p_{i}$’s, or

  2. 2.

one of the points $p_{1},\ldots,p_{n}$ is the vertex of $L$.

Figure 3: A stable tropical line $(L,p_{0},p_{1})$

We would now consider a line over an arbitrary semiring. We first recall the
axioms which govern incidence of points and lines classically,

######  Axiom 2.0.7 (Axiom 1 [Art11]).

Given two points $P$ and $Q$, there exists a unique line $l$ such that $P$
lies on $l$ and $Q$ lies on $l$.

The unique line containing $P$ and $Q$ is referred as $l=P+Q$.

Tropically, we define the following axiom which underlines point line
incidence over semirings,

######  Axiom 2.0.8.

Given two points $P$ and $Q$, there either exists a unique line $l$ such that
$P$ lies on $l$ and $Q$ lies on $l$ or there exist infinite number of lines
such that $P$ and $Q$ both lie on them.

We can now provide an axiomatic description of _coaxiality_ , similar to the
classical axiomatic description of _collinearity_.

######  Definition 10.

Given two points $P$ and $Q$, they are said to be _coaxial_ if there exists an
infinite number of lines containing both $P$ and $Q$.

We also refer to the lines containing two coaxial points $P$ and $Q$, as
coaxial lines. For two coaxial points, among the infinite number of lines
containing them, coaxiality defines an equivalence relation on the infinite
coaxial lines.

Given two points $P$ and $Q$, $l=P\oplus Q$ denotes either the unique line
containing them or the class of infinite coaxial lines containing $P$ and $Q$.

We now recall definitions of two classical maps regarding vector spaces over
fields and their associated projective space. Let $V$ and $V^{\prime}$ be two
vector spaces over the fields $k$ and $k^{\prime}$, and let $\mu$ be an
isomorphism between $k$ and $k^{\prime}$.

######  Definition 11 (Definition 2.10, [Art11]).

A map $\lambda:V\rightarrow V^{\prime}$ is called a semi-linear with respect
to the isomorphism $\mu$ if

  1. 1.

$\lambda(X+Y)=\lambda(X)+\lambda(Y),$

  2. 2.

$\lambda(\alpha X)=\alpha^{\mu}\lambda(X)$ for all $X,Y\in V$ and for all
$\alpha\in k.$

We refer to the group of semilinear transformations as $\pi(V,V^{\prime})$. In
the case of $V=V^{\prime}$, this is a group of automorphisms which contains a
normal subgroup isomorphic to $k^{*}$. We refer to the quotient of $\pi(V)$
with this normal subgroup as $P\pi(V)$.

######  Definition 12 (Definition 2.11, [Art11]).

A map $\sigma:\overline{V}\rightarrow\overline{V^{\prime}}$ of the elements of
a projective space $\overline{V}$ onto the elements of a projective space
$\overline{V^{\prime}}$ is called a collineation if

  1. 1.

dim $V$ = dim $V^{\prime}$,

  2. 2.

$\sigma$ is one to one and onto,

  3. 3.

$U_{1}\subset U_{2}\implies\sigma U_{1}\subset\sigma U_{2}$.

We now define tropical analogues of these two maps between two semimodules $V$
and $V^{\prime}$ defined over two linear semirings $R$ and $R^{\prime}$, which
are isomorphic with respect to an isomorphism $\mu$.

######  Definition 13.

A map $\lambda:V\rightarrow V^{\prime}$ is called semiring semi-linear with
respect to the isomorphism $\mu$ if

  1. 1.

$\lambda(X+Y)=\lambda(X)+\lambda(Y),$

  2. 2.

$\lambda(\alpha X)=\alpha^{\mu}\lambda(X)$ for all $X,Y\in V$ and for all
$\alpha\in R.$

######  Definition 14.

A map $\sigma:\overline{V}\rightarrow\overline{V^{\prime}}$ of the elements of
a associated projective space $\overline{V}$ onto the elements of a associated
tropical space $\overline{V^{\prime}}$ is called a semiring collineation if

  1. 1.

dim $V$ = dim $V^{\prime}$,

  2. 2.

$\sigma$ is one to one and onto,

  3. 3.

$U_{1}\subset U_{2}\implies\sigma U_{1}\subset\sigma U_{2}$.

##  3 Main Theorem

We first state the classical Fundamental Theorem of Projective Geometry as
stated in [Art11],

######  Theorem 2 (Theorem 2.26 [Art11]).

Let $V$ and $V^{\prime}$ be left vector spaces of equal dimensions $n\geq 3$
over fields $k$ and $k^{\prime}$ respectively, $\overline{V}$ and
$\overline{V^{\prime}}$ be the corresponding projective spaces. Let $\sigma$
be a one-to-one (onto) correspondence which has the following property:
Whenever three distinct "points" $L_{1},L_{2}$ and $L_{3}$ are collinear :
$L_{1}\subset L_{2}+L_{3}$, then there images are collinear: $\sigma
L_{1}\subset\sigma L_{2}+\sigma L_{3}$. There exists an isomorphism $\mu$ of
$k$ onto $k^{\prime}$ and a semi-linear map $\lambda$ of $V$ onto $V^{\prime}$
(with respect to $\mu$) such that the collineation which $\lambda$ induces on
$\overline{V}$ agrees with $\sigma$ on the points of $\overline{V}$. If
$\lambda_{1}$ is another semi-linear map with respect to an isomorphism
$\mu_{1}$ of $k$ onto $k^{\prime}$ which also induces this collineation, then
$\lambda_{1}(X)=\lambda(\alpha X)$ for some fixed $\alpha\neq 0$ of $k$ and
the isomorphism $\mu_{1}$ is given by $x^{\mu_{1}}={(\alpha
x\alpha^{-1})}^{\mu}$. For any $\alpha\neq 0$ the map $\lambda(\alpha X)$ will
be semi-linear and induce the same collineation as $\lambda$.

With the help of definitions discussed earlier, we now state a version of
Fundamental Theorem of Projective Geometry over semirings as follows,

######  Theorem 3 (Fundamental Theorem of Projective Geometry over semirings).

Let $V$ and $V^{\prime}$ be free, finitely generated semimodules of equal
dimensions $n\geq 3$ over linear semirings $R$ and $R^{\prime}$ respectively.
Let $\overline{V}$ and $\overline{V^{\prime}}$ be the corresponding associated
projective spaces. Let $\sigma$ be a one-to-one (onto) correspondence of the
"points" of $\overline{V}$ and "points" of $\overline{V^{\prime}}$ which has
the following property: whenever three distinct "points" $L_{1},L_{2}$ and
$L_{3}$ are coaxial : $L_{1}\subset L_{2}\oplus L_{3}$, then there images are
coaxial: $\sigma L_{1}\subset\sigma L_{2}\oplus\sigma L_{3}$. There exists an
isomorphism $\mu$ of $R$ onto $R^{\prime}$ and a semiring semi-linear map
$\lambda$ of $V$ onto $V^{\prime}$ (with respect to $\mu$) such that the
semiring collineation which $\lambda$ induces on $\overline{V}$ agrees with
$\sigma$ on the points of $\overline{V}$. For any $\alpha\neq 0$ the map
$\lambda(\alpha X)$ is semiring semi-linear and induces the same semiring
collineation as $\lambda$.

###### Proof.

We begin the proof similar to the setup used in the classical case in [Art11]
and [Put]. We divide the proof in various small results,

######  Lemma 3.0.1.

Let $v_{1},\ldots,v_{p}\in V$ and $\langle v_{i}\rangle=L_{i}$ (for all $1\leq
i\leq p)\in\overline{V}$. Also, let $v^{\prime}_{1},\ldots,v^{\prime}_{p}\in
V^{\prime}$ and $\langle v^{\prime}_{i}\rangle=L^{\prime}_{i}$ (for all $1\leq
i\leq p)\in\overline{V^{\prime}}$. Then

$\sigma(\langle v_{1},\ldots,v_{p}\rangle)=\langle
v^{\prime}_{1},\ldots,v^{\prime}_{p}\rangle$

###### Proof.

The subsemimodule $\langle v_{1},\ldots,v_{p}\rangle$, by definition is the
minimal subsemimodule of V containing all $\langle v_{i}\rangle=L_{i}$. Since
$\sigma$ is a map which preserves dimension and is one to one and onto, hence
$\sigma(\langle v_{1},\ldots,v_{p}\rangle)$ is the minimal subsemimodule of
$V^{\prime}$ containing each $\sigma(\langle v_{i}\rangle)=\langle
v^{\prime}_{i}\rangle=L^{\prime}_{i}$. Hence, the claim. ∎

Let $\\{v_{1},\ldots,v_{n}\\}$ be a basis for $V$ and we want to construct a
basis $\\{v^{\prime}_{1},\ldots,v^{\prime}_{n}\\}$ for $V^{\prime}$. Let
$v^{\prime}_{1}\in V^{\prime}$, such that $\sigma(\langle
v_{1}\rangle)=\langle v^{\prime}_{1}\rangle.$ This choice of $v^{\prime}_{1}$
will be our only arbitrary choice; everything else will be determined by it.
We now construct $\\{v^{\prime}_{2},\ldots,v^{\prime}_{n}\\}$.

######  Lemma 3.0.2.

For $2\leq i\leq n$, there exists a unique $v^{\prime}_{i}\in V^{\prime}$ such
that

$\sigma(\langle v_{i}\rangle)=\langle
v^{\prime}_{i}\rangle\quad\text{and}\quad\sigma(\langle
v_{1}+v_{i}\rangle)=\langle v^{\prime}_{1}+v^{\prime}_{i}\rangle$

Moreover, the set $\\{v^{\prime}_{1},\ldots,v^{\prime}_{n}\\}$ is a basis for
$V^{\prime}$.

###### Proof.

Pick $w_{i}$s $\in V^{\prime}$, such that $\sigma(\langle
v_{i}\rangle)=\langle w_{i}\rangle$. Using Lemma 3.0.1, we have

$\sigma(\langle v_{1}+v_{i}\rangle)\subset\sigma(\langle
v_{1},v_{i}\rangle)=\langle v^{\prime}_{1},w_{i}\rangle$

Since $\sigma(\langle v_{1}+v_{i}\rangle)\neq\langle w_{i}\rangle$, therefore
it follows that there exists a unique $\gamma_{i}\in R^{\prime}$ such that

$\sigma(\langle v_{1}+v_{i}\rangle)=\langle
v^{\prime}_{1}+\gamma_{i}w_{i}\rangle$

therefore the desired vector $v^{\prime}_{i}=\gamma_{i}w_{i}$. To deduce that
$\\{v^{\prime}_{1},\ldots,v^{\prime}_{n}\\}$ forms a basis, we use Lemma 3.0.1
and the fact that $\sigma$ is one to one and onto. ∎

We now try to construct the isomorphism map $\mu:R\longrightarrow R^{\prime}$,

######  Lemma 3.0.3.

For $2\leq i\leq n$, there exists a unique set map $\mu:R\longrightarrow
R^{\prime}$ such that

$\sigma(\langle v_{1}+cv_{i}\rangle)=\langle
v^{\prime}_{1}+\mu_{i}(c)\>v^{\prime}_{i}\rangle$

($c\in R$)

###### Proof.

We define $\mu_{i}$ as follows, consider $c\in R$. We apply Lemma 3.0.1 to get

$\sigma(\langle v_{1}+cv_{i}\rangle)\subset\sigma(\langle
v_{1},v_{i}\rangle)=\langle v^{\prime}_{1},v^{\prime}_{i}\rangle$

Since $\sigma(\langle v_{1}+cv_{i}\rangle)\neq\langle v^{\prime}_{i}\rangle$,
therefore there exists a unique $\mu_{i}(c)\in R^{\prime}$, such that

$\sigma(\langle v_{1}+cv_{i}\rangle)=\langle
v^{\prime}_{1}+\mu_{i}(c)\>v^{\prime}_{i}\rangle$

The uniqueness of $\mu_{i}$ follows from the fact that
$\mu_{i}(0_{R})=0_{R^{\prime}}$ and $\mu_{i}(1_{R})=1_{R^{\prime}}$, where
$0_{R}$ and $0_{R^{\prime}}$ represents the additive identities of the
semirings $R$ and $R^{\prime}$ respectively. Similarly, $1_{R}$ and
$1_{R^{\prime}}$ represents the multiplicative identities of the semirings $R$
and $R^{\prime}$ respectively.

∎

######  Lemma 3.0.4.

For distinct $2\leq i,j\leq n$, we have $\mu_{i}=\mu_{j}$.

###### Proof.

Consider a non-zero $c\in R$. We have

$\langle v_{i}-v_{j}\rangle\subset\langle
v_{i},v_{j}\rangle\quad\text{and}\quad\langle v_{i}-v_{j}\rangle\subset\langle
v_{1}+cv_{i},v_{1}+cv_{j}\rangle$

Applying Lemma 3.0.1, we get

$\sigma(\langle v_{i}-v_{j}\rangle)\subset\langle
v^{\prime}_{i},v^{\prime}_{j}\rangle\quad\text{and}\quad\sigma(\langle
v_{i}-v_{j}\rangle)\subset\langle
v^{\prime}_{1}+\mu_{i}(c)\>v^{\prime}_{i},v^{\prime}_{1}+\mu_{j}(c)\>v^{\prime}_{j}\rangle$

We have

$\langle v^{\prime}_{i},v^{\prime}_{j}\rangle\cap\langle
v^{\prime}_{1}+\mu_{i}(c)\>v^{\prime}_{i},v^{\prime}_{1}+\mu_{j}(c)\>v^{\prime}_{j}\rangle=\langle\mu_{i}(c)\>v^{\prime}_{i}-\mu_{j}(c)\>v^{\prime}_{j}\rangle$

Therefore,

$\sigma(\langle
v_{i}-v_{j}\rangle)=\langle\mu_{i}(c)v^{\prime}_{i}-\mu_{j}(c)v^{\prime}_{j}\rangle$

since the left hand side is independent of $c$ in this equation, therefore the
right hand side should also be independent of $c$. Hence,

$\langle
v^{\prime}_{i}-v^{\prime}_{j}\rangle=\langle\mu_{i}(1_{R})v^{\prime}_{i}-\mu_{j}(1_{R})v^{\prime}_{j}\rangle=\langle\mu_{i}(c)v^{\prime}_{i}-\mu_{j}(c)v^{\prime}_{j}\rangle$

Therefore,

$\mu_{i}(c)=\mu_{j}(c),\quad\forall c\in R$

∎

######  Lemma 3.0.5.

For $c_{1},\ldots,c_{n}\in V$, we have

$\sigma(\langle v_{1}+c_{2}v_{2}+\ldots+c_{n}v_{n}\rangle)=\langle
v^{\prime}_{1}+\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{n})\>v^{\prime}_{n}\rangle$

###### Proof.

We assume,

$\sigma(\langle v_{1}+\ldots+c_{p}v_{p}\rangle)=\langle
v^{\prime}_{1}+\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{p})\>v^{\prime}_{p}\rangle$

for all $2\leq p\leq n$, by using induction on $p$. The base case is $p=2$ and
it holds true as that is the defining property of $\mu$, so we consider
$2<p\leq n$. Applying Lemma 3.0.1 and the induction hypothesis we get,

$\sigma(\langle v_{1}+\ldots+c_{p}v_{p}\rangle)\subset\sigma(\langle
v_{1}+\ldots+c_{p-1}v_{p-1},v_{p}\rangle)=\langle
v^{\prime}_{1}+\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{p-1})\>v^{\prime}_{p-1},v^{\prime}_{p}\rangle$

Moreover, $\sigma(\langle v_{1}+c_{2}v_{2}+\ldots c_{p}v_{p}\rangle)$ is not
$\langle v^{\prime}_{p}\rangle$, so we deduce that there exists some $d\in
R^{\prime}$, such that

$\sigma(\langle v_{1}+c_{2}v_{2}+\ldots+c_{p}v_{p}\rangle)=\langle
v^{\prime}_{1}+\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{p-1})\>v^{\prime}_{p-1}+dv^{\prime}_{p}\rangle$

Applying Lemma 3.0.1 and defining property of $\mu$, we see

$\sigma(\langle
v_{1}+c_{2}v_{2}+\ldots+c_{p}v_{p}\rangle)\subset\sigma(\langle
v_{1}+c_{p}v_{p},v_{2},\ldots,v_{p-1}\rangle)=\langle
v^{\prime}_{1}+\mu(c_{p})\>v^{\prime}_{p},v^{\prime}_{2},\ldots,v^{\prime}_{p-1}\rangle$

Therefore, $d=\mu(c_{p})$. ∎

######  Lemma 3.0.6.

For $c_{2},\ldots,c_{n}\in R$, we have

$\sigma(\langle
c_{2}v_{2}+\ldots+c_{n}v_{n}\rangle)=\langle\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{n})\>v^{\prime}_{n}\rangle$

###### Proof.

By Lemma 3.0.1, we have

$\sigma(\langle c_{2}v_{2}+\ldots+c_{n}v_{n}\rangle)\subset\sigma(\langle
v_{2},\ldots,v_{n}\rangle)=\langle
v^{\prime}_{2},\ldots,v^{\prime}_{n}\rangle$

also with Lemma 3.0.1 and Lemma 3.0.5, we have

$\sigma(\langle c_{2}v_{2}+\ldots+c_{n}v_{n}\rangle)\subset\sigma(\langle
v_{1},v_{1}+c_{2}v_{2},+\ldots+c_{n}v_{n}\rangle)=\langle
v^{\prime}_{1},v^{\prime}_{1}+\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{n})\>v^{\prime}_{n}\rangle$

and we get,

$\sigma(\langle
c_{2}v_{2}+\ldots+c_{n}v_{n}\rangle)=\langle\mu(c_{2})\>v^{\prime}_{2}+\ldots+\mu(c_{n})\>v^{\prime}_{n}\rangle$

∎

We now show that the map $\mu$ is an isomorphism of semirings.

######  Lemma 3.0.7.

For $c,d\in R$ we have $\mu(c+d)=\mu(c)+\mu(d)$.

###### Proof.

By Lemma 3.0.5, we have

$\sigma(\langle v_{1}+(c+d)v_{2}+v_{3}\rangle)=\langle
v^{\prime}_{1}+\mu(c+d)\>v^{\prime}_{2}+v^{\prime}_{3}\rangle$

By combining Lemma 3.0.1 with Lemma 3.0.5 and Lemma 3.0.6, we have

$\sigma(\langle v_{1}+(c+d)v_{2}+v_{3}\rangle)\subset\sigma(\langle
v_{1}+cv_{2},dv_{2}+v_{3}\rangle)=\langle
v^{\prime}_{1}+\mu(c)\>v^{\prime}_{2},\mu(d)\>v^{\prime}_{2}+v^{\prime}_{3}\rangle$

combining these equations we get

$\langle
v^{\prime}_{1}+\mu(c+d)v^{\prime}_{2}+v^{\prime}_{3}\rangle\subset\langle
v^{\prime}_{1}+\mu(c)\>v^{\prime}_{2},\mu(d)\>v^{\prime}_{2}+v^{\prime}_{3}\rangle$

this can only hold true if

$\mu(c+d)=\mu(c)+\mu(d)$

∎

######  Lemma 3.0.8.

For $c,d\in R$, we have

$\mu(cd)=\mu(c)\cdot\mu(d)$

###### Proof.

By Lemma 3.0.5, we have

$\sigma(\langle v_{1}+cdv_{2}+cv_{3}\rangle)=\langle
v^{\prime}_{1}+\mu(cd)\>v^{\prime}_{2}+\mu(c)\>v^{\prime}_{3}\rangle$

combining Lemma 3.0.1 and Lemma 3.0.6, we get

$\sigma(\langle v_{1}+cdv_{2}+cv_{3}\rangle)\subset\sigma(\langle
v_{1},dv_{2}+v_{3}\rangle)=\langle
v^{\prime}_{1},\mu(d)\>v^{\prime}_{2}+v^{\prime}_{3}\rangle$

combining these two equations we get

$\langle
v^{\prime}_{1}+\mu(cd)\>v^{\prime}_{2}+\mu(c)\>v^{\prime}_{3}\rangle\subset\langle
v^{\prime}_{1},\mu(d)\>v^{\prime}_{2}+v^{\prime}_{3}\rangle$

the only way the this holds is iff

$\mu(cd)=\mu(c)\cdot\mu(d)$

∎

######  Lemma 3.0.9.

The map $\mu:R\longrightarrow R^{\prime}$ is an isomorphism of semirings.

###### Proof.

We know that $\mu(0_{R})=\mu(0_{R^{\prime}})$ and
$\mu(1_{R})=\mu(1_{R^{\prime}})$, therefore $\mu$ is a one-one onto map. Also,
by Lemma 3.0.7 and Lemma 3.0.8 we conclude that $\mu$ is a semiring
homomorphism. Hence, $\mu$ is an isomorphism of semirings. ∎

We now have constructed our basis $\\{v^{\prime}_{1},\ldots,v^{\prime}_{n}\\}$
for $V^{\prime}$ and the isomorphism $\mu:R\longrightarrow R^{\prime}$ , so we
can define semiring semi-linear map $\lambda:V\longrightarrow V^{\prime}$ via
the formula

$\lambda(c_{1}v_{1}+\ldots+c_{n}v_{n})=\mu(c_{1})\>v^{\prime}_{1}+\ldots+\mu(c_{n})\>v^{\prime}_{n}\quad(c_{1},\ldots,c_{n}\in
R)$

######  Lemma 3.0.10.

The semiring semi-linear map $\lambda:V\longrightarrow V^{\prime}$ induces the
semiring collineation $\sigma$.

###### Proof.

Consider a subspace $U$ of $V$. We can write $U=\langle
u_{1},\ldots,u_{p}\rangle$, where each $u_{i}$ is either of the form
$v_{1}+c_{2}v_{2}+\ldots+c_{n}v_{n}$ or of the from
$c_{2}v_{2}+\ldots+c_{n}v_{n}$ for some $c_{2},\ldots,c_{n}\in R$. Combining
Lemma 3.0.1, Lemma 3.0.5 and Lemma 3.0.6, we see that
$\sigma(U)=\langle\lambda(u_{1}),\ldots,\lambda(u_{p})\rangle$, as desired. ∎

With all the above ten lemmas we complete the proof of the main theorem. ∎

The statement of Theorem 2 is very general and mostly in literature it is
stated in the case when $k=k^{\prime}$ and $V=V^{\prime}=k^{n}$ and the result
is stated in the following form,

######  Theorem 4.

If $k$ is a field and $n\geq 3$, Aut$(\overline{V})=P\pi(V)$

Similarly, when we consider in the case of semirings that $R=R^{\prime}$ and
$V=V^{\prime}=R^{n}$, then with Theorem 3, we have

######  Theorem 5.

If $R$ is a semiring and $n\geq 3$, Aut$(\overline{V})=P\pi^{\prime}(V)$

where $\pi^{\prime}(V)$ refers to the group of semiring semi-linear
automorphisms on $V$ and $P\pi^{\prime}(V)$ is the quotient of
$\pi^{\prime}(V)$ with the group of automorphisms of $R$.

We now highlight some connections between our results on semimodules and
_tropical convexity_ , wherein the underlying semiring is the tropical
semiring $\mathbb{T}=\\{\mathbb{R}\cup\\{-\infty\\},\text{max},+\\}$. We
recall some basic definitions concerning tropical convexity [MS09],

######  Definition 15.

A subset $S$ of $\mathbb{T}^{n}$ is _tropically convex_ if $x,y\in S$ and
$a,b\in T$ implies $a\odot x\oplus b\odot y\in S$. The _tropical convex hull_
of a given subset $V\subset\mathbb{T}^{n}$ is the smallest tropically convex
subset of $\mathbb{T}^{n}$ that contains $V$.

A _tropical polytope_ is the tropical convex hull of a finite subset $V$ in
the tropical projective space.

In [IJK18], the focus is studying on tropical matrix groups, especially over
the finitary tropical semiring $\mathbb{F}\mathbb{T}=(\mathbb{R},max,+)$ and
they consider tropical polytopes over this semiring and prove a similar result
as Theorem 3 in the case of this semiring,

######  Theorem 6 (Theorem 5.4, [IJK18]).

Every automorphism of a projective n-polytope in $\mathbb{F}\mathbb{T}^{n}$

  * •

extends to an automorphism of $\mathbb{F}\mathbb{T}^{n}$; and

  * •

is a (classical) affine linear map.

Since, all projective tropical polytopes are free and finitely generated,
hence the statement of Theorem 6 can be seen as a special case of the general
statement in Theorem 3 when the underlying linear semiring is
$\mathbb{F}\mathbb{T}$.

We now look at case when the semirings $R=R^{\prime}$ and $V=V^{\prime}$ in
Theorem 3, and one such example is the following,

######  Example 3.0.11.

We consider the case
$R=R^{\prime}=\mathbb{T}=\\{\mathbb{R}\cup\\{\infty\\},\text{max},+\\}$ and
$V=V^{\prime}=\mathbb{T}^{2}$. In this case, we obtain
$\overline{V}=\overline{V^{\prime}}=\mathbb{T}\mathbb{P}^{2}=(\mathbb{T}^{3}\setminus\\{-\infty\\})/\mathbb{R}\cdot
1$, which is referred as the _tropical projective plane_ and is an example of
a _tropical toric variety_ [MR18][Jos22].

We note that the map -log is a homeomorphism from ${\mathbb{R}}^{d}_{\geq 0}$
to $\mathbb{T}^{d}$ when taken componentwise,

$\begin{split}\text{-log}:\mathbb{R}^{d}&\longrightarrow\mathbb{T}^{d}\\\
(v_{1},\ldots,v_{n})&\longrightarrow(\text{-log}(v_{1}),\ldots,\text{-log}(v_{n}))\end{split}$

under this map the standard basis vector $e_{k}=(0,\ldots,0,1,0,\ldots,0)$ is
mapped to

$e^{\text{trop}}_{k}=(\infty,\ldots,\infty,0,\infty,\ldots,\infty)$

which provides us the description in Figure 4 of $\mathbb{T}\mathbb{P}^{2}$
[Jos22].

$e_{3}\quad$$0\quad$$e_{1}\quad$$e_{2}\quad$$(0,0,\infty)\quad$$e^{\text{trop}}_{2}=(\infty,0,\infty)\quad$$(0,\infty,\infty)=e^{\text{trop}}_{1}\quad$$e^{\text{trop}}_{3}=(\infty,\infty,0)\quad$
Figure 4: A pictorial description of $\mathbb{T}\mathbb{P}^{2}$

We see from the description of $\mathbb{T}\mathbb{P}^{2}$ in Figure 4 that all
the automorphisms of this space are just permutations of the coordinate
directions. This complements our result in Theorem 3, describing the structure
of each automorphism of the automorphism group of $\mathbb{T}\mathbb{P}^{2}$
as a conjugation of a semiring semi linear map of $\mathbb{T}^{2}$ and an
automorphism of the semiring $\mathbb{T}$.

We highlight the fact that the space $\overline{V}$ in Theorem 4 is an example
of a _Tits Building_. We elaborate on this connection in Section 5 and how we
could explore possible tropical equivalents of classical buildings that might
arise based on the results that we prove in this article.

##  4 Projectivity in the tropical plane

We recall some definitions from classical projective geometry concerning
pencils of points and lines in the plane,

######  Definition 16.

Given two lines $l_{1}$ and $l_{2}$ and a point $P$ not lying on both of them,
a perspectivity is a bijective mapping between the pencils of points on
$l_{1}$ and $l_{2}$, such that the lines incident with the corresponding
points of the two pencils is concurrent at $P$. The point $P$ is referred as
the center of the perspectivity.

If $A,B$ and $C$ is the pencil of points on $l_{1}$ and
$A^{\prime},B^{\prime}$ and $C^{\prime}$ is the pencil of points on $l_{2}$
such that the point $P$ is the center of the perspectivity between them, then
we represent this as

$ABC\>\>\stackrel{{\scriptstyle
P}}{{\doublebarwedge}}\>\>A^{\prime}B^{\prime}C^{\prime}$

######  Definition 17.

A finite composition of two or more perspectivities is called a projectivity.

$l_{1}$$l_{2}$$P$$A$$B$$C$$A^{\prime}$$B^{\prime}$$C^{\prime}$ Figure 5: A
example of a classical perspectivity

We now consider the following classical result, termed as the fundamental
theorem involving projectivity of lines in the plane [Ced04, Theorem 4.8]

######  Theorem 7.

A projectivity between two pencils is uniquely determined by three pairs of
corresponding points.

This result rests on the following axiom [Ced04, Axiom 6]

######  Axiom 1.

If a projectivity on a pencil of points leaves three distinct points of the
pencil invariant, it leaves every point of the pencil invariant.

We now define the set of corresponding definitions in the tropical plane,

######  Definition 18.

We say that $l(p,q,r)$ is a tropical pencil of points with a line $l$ and
$n=p+q+r$ points, if $l$ is a tropical line with $p$ points on the $(1,1)$
half ray, $q$ points on the $(-1,0)$ half ray and $r$ points on the $(0,-1)$
half ray.

######  Definition 19.

We define $l(\overline{a},\overline{b},\overline{c})$ to be the reduced
tropical pencil derived from a tropical pencil $l(p,q,r)$, where $l$ is the
tropical line, along with one point out of all coaxial points on each of its
half rays as a representative from the three families of coaxial points, as
illustrated in the Figure 6.

$a_{1}$$a_{2}$$a_{q}$$b_{1}$$b_{2}$$b_{r}$$c_{1}$$c_{2}$$c_{p}$$\overline{a}$$\overline{b}$$\overline{c}$
Figure 6: A reduced tropical pencil

In Figure 6, $\overline{a}=a_{i}$ for some $i$ such that $1\leq i\leq q$,
$\overline{b}=b_{j}$ for some $j$ such that $1\leq j\leq r$ and
$\overline{c}=c_{k}$ for some $k$ such that $1\leq k\leq p$. We also observe
that any reduced tropical pencil has at most three points. Essentially, a
reduced tropical pencil is obtained by using coaxiality as an equivalence
relation among points, and considering one representative from each
equivalence class of coaxial points.

######  Definition 20.

Two tropical pencils $l_{1}(p_{1},q_{1},r_{1})$ and $l_{2}(p_{2},q_{2},r_{2})$
are said to be compatible if the lines $l_{1}$ and $l_{2}$ are non-coaxial and
$p_{1}=p_{2},\>q_{2}=q_{2},\>r_{2}=r_{2}$.

$\overline{A}$$\overline{B}$$\overline{C}$$\overline{B^{\prime}}$$\overline{A^{\prime}}$${P}$$\overline{C^{\prime}}$
Figure 7: A tropical perspectivity

We now define tropical counterparts to perspectivity and projectivity,

######  Definition 21.

Given two reduced tropical pencils $l(\overline{A},\overline{B},\overline{C})$
and
$l^{\prime}(\overline{A^{\prime}},\overline{B^{\prime}},\overline{C^{\prime}})$
which are non-coaxial and a point $P$ not on both of them, a tropical
perspectivity is a bijective mapping between the points on $l$ and
$l^{\prime}$, such that the tropical lines incident with the corresponding
points of the two pencils is concurrent at $P$.

In Figure 7, we see a tropical perspectivity which maps $\overline{A}$ to
$\overline{A^{\prime}}$, $\overline{B}$ to $\overline{B^{\prime}}$ and
$\overline{C}$ to $\overline{C^{\prime}}$, i.e,

$\overline{A}\>\overline{B}\>\overline{C}\>\>\stackrel{{\scriptstyle
P}}{{\doublebarwedge}}\>\>\overline{A^{\prime}}\>\overline{B^{\prime}}\>\overline{C^{\prime}}$

######  Remark 4.0.1.

We observe that a tropical perspectivity always maps at most three points from
one line to the other.

######  Definition 22.

A tropical projectivity is a bijective mapping between compatible tropical
pencils obtained as a finite composition of tropical perspectivities.

We now establish a tropical counterpart to Theorem 7,

######  Theorem 8.

A tropical projectivity between two compatible tropical pencils is uniquely
determined (up to equivalence) by three pairs of corresponding points.

###### Proof.

We consider two compatible tropical pencils $l_{1}(p,q,r)$ and
$l_{2}(p^{\prime},q^{\prime},r^{\prime})$, and consider three points $A,B,C$
and $A^{\prime},B^{\prime},C^{\prime}$ on $l_{1}$ and $l_{2}$ respectively
such that the corresponding reduced tropical pencils are
$l_{1}(\overline{A},\overline{B},\overline{C})$ and
$l_{2}(\overline{A^{\prime}},\overline{B^{\prime}},\overline{C^{\prime}})$. We
consider in our construction that the chosen six points lie on different rays
of the given tropical line, however as we will see in the latter part of the
proof, the argument for the other cases is also similar to the one we consider
here.

We begin by a constructive approach to show the desired projectivity exists
and the argument for the uniqueness would follow. Consider the points
$\overline{A},\overline{B}$ and $\overline{C}$ on $l_{1}$ and
$\overline{A^{\prime}},\overline{B^{\prime}}$ and $\overline{C^{\prime}}$ on
$l_{2}$, as shown in Figure 8. We construct the tropical line between the
points $A$ and $A^{\prime}$ and we choose a point $p^{\prime}$ on this line.
Let $l^{\prime}$ be an arbitrary line passing through $A^{\prime}$. Then
$B_{1}=Bp^{\prime}\cdot l^{\prime}$, i.e., $B_{1}$ is the point of stable
intersection between the lines $l^{\prime}$ and the tropical line passing
through $B$ and $p^{\prime}$. We observe that there exists a tropical
perspectivity between $l_{1}$ and $l^{\prime}$, defined as follows

$ABC\>\>\stackrel{{\scriptstyle
p^{\prime}}}{{\doublebarwedge}}\>\>A^{\prime}B_{1}C$ (1)

where $A$ is mapped to $A^{\prime}$, $B$ is mapped to $B_{1}$ and $C$ is
mapped to itself via $p^{\prime}$.

Subsequently, we consider the point $p^{\prime}=CC^{\prime}\cdot
B_{1}B^{\prime}$, a point lying on the intersection between the tropical line
passing through $C$ and $C^{\prime}$ and the tropical line passing through
$B_{1}$ and $B^{\prime}$. We see that we obtain another tropical perspectivity
via $p^{\prime}$ where

$A^{\prime}B_{1}C\>\>\stackrel{{\scriptstyle
p"}}{{\doublebarwedge}}\>\>A^{\prime}B^{\prime}C^{\prime}$ (2)

$A$ is mapped to itself, $B_{1}$ is mapped to $B^{\prime}$ and $C$ is mapped
to itself via $C^{\prime}$.

Using the equations 1 and 2 we obtain a tropical projectivity,

$ABC\>\>\barwedge\>\>A^{\prime}B^{\prime}C^{\prime}$ (3)

where the $p$ points on the $(1,1)$ ray of $l_{1}$ are mapped to the
$p^{\prime}=p$ points on the $(1,1)$ ray of $l_{2}$ via the reduced pencil and
similarly the $q$ points on the $(-1,0)$ ray of $l_{1}$ are mapped to the
$q^{\prime}=q$ points on the $(-1,0)$ ray of $l_{2}$ and the $r$ points on the
$(0,-1)$ ray of $l_{1}$ are mapped to the $r^{\prime}=r$ points on the
$(0,-1)$ ray of $l_{2}$. Since, the choice of the six points
$A,B,C,A^{\prime}B^{\prime}$ and $C^{\prime}$ fixes the projectivity, we have
the desired result.

$l_{1}$$A$$B$$C$$A^{\prime}$$B_{1}$$p^{\prime}$$B^{\prime}$$C^{\prime}$$p"$$l_{2}$$l^{\prime}$$l^{\prime}$$l^{\prime}$
Figure 8: Tropical projectivity between two reduced tropical pencil of points

∎

##  5 Conclusions

We would like to elaborate on the various connections and possible extensions
of our results presented here. As alluded to earlier, restricted to the case
of the tropical semiring, study regarding affine buildings has been carried
out in [JSY07], [Wer11]. Since, the fundamental theorem of projective geometry
is a structure theorem for classical groups, hence it plays an important role
in the study of classical groups and associated buildings. Therefore, a
natural successor of our results could be a possible definition of classical
groups over semimodules and associated buildings.

We recall some basic definitions from [Jos22] and [JSY07]. In this case, the
study focuses on the case when the underlying field is
$\mathbb{K}=\mathbb{C}((t))$, field of Laurent series with complex
coefficients. It has a valuation ring which is local, with the residue field
isomorphic to $\mathbb{C}$. The vector space $\mathbb{K}^{d}$ is a module over
the valuation ring and lattices over this ring are considered as submodules
generated by $d$ linearly independent vectors in $\mathbb{K}^{d}$. Two
lattices $\lambda_{1},\lambda_{2}\subset\mathbb{K}^{d}$ are called
_equivalent_ if $c\cdot\lambda_{1}=\lambda_{2}$, for some $c\in K^{x}$. Two
equivalence class of lattices $\langle\lambda_{1}\rangle$ and
$\langle\lambda_{2}\rangle$ are called _adjacent_ if there are representatives
$\lambda_{1}$ and $\lambda_{2}$ such that
$t\cdot\lambda_{2}\subset\lambda_{1}\subset\lambda_{2}$.

The adjacency relation on equivalence classes of lattices in $\mathbb{K}_{d}$
defines a graph, and its _flag simplicial complex_ is the _Bruhat–Tits
building_ $\mathbb{B}_{d}(\mathbb{K})$. A point in the tropical projective
space $\mathbb{TP}^{d-1}$ is a _lattice point_ if it is represented by a
vector $x$ in $\mathbb{Z}^{d}$. A _tropical lattice polytope_ is the tropical
convex hull of finitely many lattice points in $\mathbb{TP}^{d-1}$. Similar to
the classical case, we can define adjacency between tropical lattice points
with respect to the _natural metric_ defined in [JSY07]. This defines a graph
whose flag simplicial complex provides a _triangulation_ of the affine space
$\mathbb{TP}^{d-1}$.

We believe that similar to the case of modules over valuation rings, one can
study lattices over semimodules over an appropriately chosen semiring, much in
essence to the study which we carried out in this article. This would also
help in defining buildings over semimodules, and study of matrix groups over
semirings. Further details will be explored elsewhere.

We also would like to point the reader to the recent developments in the
theory of _ordered blueprints_ [BL18] and possible connections to our work. We
recall some basic definitions from [BL18]. An _ordered semiring_ is a semiring
$R$ with a partial order $\leq$ such that $x\leq y$ and $z\leq t$ implies
$x+z\leq y+t$ and $xz\leq yt$. An _ordered blueprint_ is a triple
$B=(B^{.},B^{+},\leq)$ where $(B^{+},\leq)$ is an ordered semiring and $B^{.}$
is a multiplicative subset of $B^{+}$ that generates $B^{+}$ as a semiring and
contains $0$ and $1$. An _ordered bluefield_ is an ordered blueprint $B$ with
$B^{.}=B^{\times}\cup\\{0\\}$. We suggest the reader to refer [BL18] for
further elaborate descriptions of _ordered blue schemes_ and the construction
of the functor _Proj_ , but we emphasize the main takeaway; the notion of a
projective space over ordered blueprints, and examples of projective line and
projective planes [BL18, Example 4.13]. This leads us to the following
question,

######  Question 5.0.1.

Can a version of fundamental theorem of projective geometry be established
over matroids in the background of ordered blueprints and the associated
projective space ?

We wish to explore this question further and connect with the existing notion
of projective geometry over ordered blueprints.

We now discuss one of the most prominent aspects of classical projective
geometry related to our results - _cross ratios_. Most of what we recall here
is referenced from the book by Jürgen Richter-Gebert [RG11]. We recall the
notation that a point $a\in\mathbb{R}^{2}$ is denoted as
$a=\begin{pmatrix}a_{1}\\\ a_{2}\\\ \end{pmatrix}$ and

$[a,b]=\text{det}\begin{pmatrix}a_{1}&b_{1}\\\ a_{2}&b_{2}\end{pmatrix}$

######  Definition 23.

Let $a,b,c,d$ be four nonzero vectors in $\mathbb{R}^{2}$. The _cross-ratio_
$(a,b;c,d)$ is the following quantity,

$(a,b;c,d):=\frac{[a,c][b,d]}{[a,d][b,c]}$

This definition extends naturally to complex numbers and also to arbitrary
fields. An important feature of cross ratios is that it is invariant under
projective transformations.

######  Lemma 5.0.2.

For any real nonzero parameters
$\lambda_{a},\lambda_{b},\lambda_{c},\lambda_{d}\in\mathbb{R}$ we have

$(a,b;c,d)=(\lambda_{a}a,\lambda_{b}b;\lambda_{c}c,\lambda_{d}d)$

######  Lemma 5.0.3.

Let $M$ be a $2\times 2$ matrix with nonvanishing determinant and let
$a,b,c,d$ be four vectors in $\mathbb{R}^{2}$. Then we have

$(a,b;c,d)=(M\cdot a,M\cdot b;M\cdot c,M\cdot d)$

This invariance can be seen in the following result regarding classical
perspectivities in the plane,

######  Corollary 1 (Corollary 4.1 [RG11]).

Let $o$ be a point and let $l$ and $l^{\prime}$ be two lines not passing
through $o$. If four points $a,b,c,d$ are on $l$ and are projected by the
viewpoint $o$ to four points $a^{\prime},b^{\prime},c^{\prime},d^{\prime}$ on
$l^{\prime}$, then the cross ratios satisfy
$(a,b;c,d)=(a^{\prime},b^{\prime};c^{\prime},d^{\prime})$.

We now explore the possibility of a tropical counterpart to the notion of a
cross ratio. Firstly, we recall the definition of the _tropical determinant_
[MS09]. For a $n\times n$ matrix $X=x_{ij}$, with entries from the tropical
semiring, the tropical determinant is defined as follows

$\begin{split}\text{tdet}(X)&=\bigoplus_{\sigma\in S_{n}}x_{1\sigma(1)}\odot
x_{2\sigma(2)}\odot\ldots\odot x_{n\sigma(n)}\\\ &=\max_{\sigma\in S_{n}}\
x_{1\sigma(1)}\odot x_{2\sigma(2)}\odot\ldots\odot x_{n\sigma(n)}\end{split}$

where $\sigma$ represents a cycle in $S_{n}$, the symmetric group on $n$
elements. A square matrix $X$ is said to be _tropically singular_ if
$\text{tdet(X)}=\infty$ or the tropical polynomial tdet vanishes on $X$
[Jos22].

We fix the following notation,

$[a,b]^{tr}=\text{tdet}\begin{pmatrix}a_{1}&b_{1}\\\ a_{2}&b_{2}\end{pmatrix}$

Let $a,b,c,d$ be four nonzero vectors in $\mathbb{R}^{2}$. We consider the the
following quantity,

$(a,b;c,d)^{tr}:=([a,c]^{tr}\odot[b,d]^{tr})-([a,d]^{tr}\odot[b,c]^{tr})$

We now try to see if $(a,b;c,d)^{tr}$ can be a candidate for a _tropical
cross-ratio_. We show invariance of $(a,b;c,d)^{tr}$ with respect to tropical
scalar multiplication,

######  Lemma 5.0.4.

For any real nonzero parameters
$\lambda_{a},\lambda_{b},\lambda_{c},\lambda_{d}\in\mathbb{R}$ we have

$(a,b;c,d)^{tr}=(\lambda_{a}\odot a,\lambda_{b}\odot b;\lambda_{c}\odot
c,\lambda_{d}\odot d)^{tr}$

###### Proof.

$\begin{split}(\lambda_{a}\odot a,\lambda_{b}\odot b;\lambda_{c}\odot
c,\lambda_{d}\odot d)^{tr}&=\left(\text{tdet}\begin{pmatrix}\lambda_{a}\odot
a_{1}&\lambda_{c}\odot c_{1}\\\ \lambda_{a}\odot a_{2}&\lambda_{c}\odot
c_{2}\end{pmatrix}\odot\text{tdet}\begin{pmatrix}\lambda_{b}\odot
b_{1}&\lambda_{d}\odot d_{1}\\\ \lambda_{b}\odot b_{2}&\lambda_{d}\odot
d_{2}\end{pmatrix}\right)\\\
&-\left(\text{tdet}\begin{pmatrix}\lambda_{a}\odot a_{1}&\lambda_{d}\odot
d_{1}\\\ \lambda_{a}\odot a_{2}&\lambda_{d}\odot
d_{2}\end{pmatrix}\odot\text{tdet}\begin{pmatrix}\lambda_{b}\odot
b_{1}&\lambda_{c}\odot c_{1}\\\ \lambda_{b}\odot b_{2}&\lambda_{c}\odot
c_{2}\end{pmatrix}\right)\\\ &=\text{max}(\lambda_{a}\odot
a_{1}+\lambda_{c}\odot c_{2},\lambda_{a}\odot a_{2}+\lambda_{c}\odot
c_{1})+\text{max}(\lambda_{b}\odot b_{1}+\lambda_{d}\odot d_{2},\\\
&\lambda_{b}\odot b_{2}+\lambda_{d}\odot d_{1})-(\text{max}(\lambda_{a}\odot
a_{1}+\lambda_{d}\odot d_{2},\\\ &\lambda_{a}\odot a_{2}+\lambda_{d}\odot
d_{1})+\text{max}(\lambda_{b}\odot b_{1}+\lambda_{c}\odot
c_{2},\lambda_{b}\odot b_{2}+\lambda_{c}\odot c_{1}))\\\
&=\lambda_{a}+\lambda_{c}+\text{max}(a_{1}+c_{2},a_{2}+c_{1})+\lambda_{b}+\lambda_{d}+\text{max}(b_{1}+d_{2},b_{2}+d_{1})-\\\
&(\lambda_{a}+\lambda_{d}+\text{max}(a_{1}+d_{2},a_{2}+d_{1})+\lambda_{b}+\lambda_{c}+\text{max}(b_{1}+c_{2},b_{2}+c_{1}))\\\
&=(\lambda_{a}+\lambda_{b}+\lambda_{c}+\lambda_{d})+\text{max}(a_{1}+c_{2},a_{2}+c_{1})+\text{max}(b_{1}+d_{2},b_{2}+d_{1})\\\
&-(\lambda_{a}+\lambda_{b}+\lambda_{c}+\lambda_{d})-\text{max}(a_{1}+d_{2},a_{2}+d_{1})-\text{max}(b_{1}+c_{2},b_{2}+c_{1}))\\\
&=(a,b;c,d)^{tr}\end{split}$

∎

However, we are not able to recover all classical results. For example, let us
consider the case of Lemma 5.0.3. We consider a tropically non-singular
$2\times 2$ matrix $M=\begin{pmatrix}m_{1}&m_{2}\\\ m_{3}&m_{4}\end{pmatrix}$.
Then for $a=\begin{pmatrix}a_{1}\\\ a_{2}\\\ \end{pmatrix}$ and
$b=\begin{pmatrix}b_{1}\\\ b_{2}\\\ \end{pmatrix}$ we get,

$M\odot a=\begin{pmatrix}m_{1}\odot a_{1}\oplus m_{2}\odot a_{2}\\\ m_{3}\odot
a_{1}\oplus m_{4}\odot a_{2}\end{pmatrix}\quad\text{and}\quad M\odot
b=\begin{pmatrix}m_{1}\odot b_{1}\oplus m_{2}\odot b_{2}\\\ m_{3}\odot
b_{1}\oplus m_{4}\odot b_{2}\end{pmatrix}$

$\begin{split}[M\odot a,M\odot b]^{tr}&=\text{tdet}\begin{pmatrix}m_{1}\odot
a_{1}\oplus m_{2}\odot a_{2}&m_{1}\odot b_{1}\oplus m_{2}\odot b_{2}\\\
m_{3}\odot a_{1}\oplus m_{4}\odot a_{2}&m_{3}\odot b_{1}\oplus m_{4}\odot
b_{2}\end{pmatrix}\\\ &=\underline{m_{1}\odot m_{3}\odot a_{1}\odot
b_{1}}\oplus m_{1}\odot m_{4}\odot a_{1}\odot b_{2}m_{2}\odot m_{3}a_{2}\odot
b_{1}\oplus\underline{m_{2}\odot m_{4}\odot a_{2}\odot b_{2}}\\\
&\oplus\underline{m_{1}\odot m_{3}\odot a_{1}\odot b_{1}}\oplus m_{3}\odot
m_{2}\odot a_{1}\odot b_{2}\oplus m_{1}\odot m_{4}\odot a_{2}\odot
b_{1}\oplus\underline{m_{2}\odot m_{4}\odot a_{2}\odot b_{2}}\\\ &=a_{1}\odot
b_{2}\odot(m_{1}\odot m_{4}\oplus m_{2}\odot m_{3})\oplus a_{2}\odot
b_{1}\odot(m_{1}\odot m_{4}\oplus m_{2}\odot m_{3})\\\ &\oplus m_{1}\odot
m_{3}\odot a_{1}\odot b_{1}\oplus m_{2}\odot m_{4}\odot a_{2}\odot b_{2}\\\
&=(m_{1}\odot m_{4}\oplus m_{2}\odot m_{3})\odot(a_{2}\odot b_{1}\oplus
a_{1}\odot b_{2})\oplus m_{1}\odot m_{3}\odot a_{1}\odot b_{1}\oplus
m_{2}\odot m_{4}\odot a_{2}\odot b_{2}\\\
&=\text{tdet}(M)\cdot[a,b]^{tr}\oplus m_{1}\odot m_{3}\odot a_{1}\odot
b_{1}\oplus m_{2}\odot m_{4}\odot a_{2}\odot b_{2}\end{split}$

We see that in the tropical case, we have an additional term along with
$\text{tdet}(M)\cdot[a,b]^{tr}$, which is due to the fact that there is no
additive inverse tropically and the tropical determinant is a tropical sum
whereas the classical determinant has a change of sign for terms which results
in cancellations. Therefore, for a tropically non-singular $2\times 2$ matrix
$M$ we cannot conclude that

$(a,b;c,d)^{tr}=(M\cdot a,M\cdot b;M\cdot c,M\cdot d)^{tr}$

We would like to explore other suitable variants of the definition of a
tropical cross ratio, in the context of tropical projective geometry, which
complement the definition of tropical perpsectivity and remain invariant under
projective transformations, which is true in the classical case.

######  Remark 5.0.5.

We do acknowledge that there is a related description of tropical double
ratios and tropical cross ratio in the context of curve counting discussed in
[Mik07] and [Gol21] respectively. Also, in [BL18] definition of a cross ratio
is defined over the basis-exchange graph of a matroid over a pasture. However,
for our purposes of looking at tropical projective geometry, we consider other
variants which are more suited for our discussion.

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MFO Oberwolfach
---
E-mail address<EMAIL_ADDRESS>


††thanks: M. J. Rendell and S. D. Liles contributed equally to this work

# Probing Fermi surface shifts with spin resolved transverse magnetic
focussing

M. J. Rendell  S. D. Liles  S. Bladwell  A. Srinivasan  School of Physics,
University of New South Wales, Sydney, NSW 2052, Australia  O. Klochan
University of New South Wales Canberra, Canberra, ACT 2600, Australia  I.
Farrer  Department of Electronic and Electrical Engineering, University of
Sheffield, Sheffield, S1 3JD, UK  D. A. Ritchie  Cavendish Laboratory,
University of Cambridge, Cambridge, CB3 0HE, UK  O. P. Sushkov  A. R. Hamilton
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Transverse magnetic focussing is the solid state equivalent of a mass
spectrometer. It is unique among 2D measurement techniques as it is able to
measure a well defined section of the Fermi surface, making it possible to
detect changes that would be averaged out over the whole Fermi surface. Here,
we utilise this unique property to probe non-adiabatic spin dynamics and spin
dependent scattering of holes. We combine spin-resolved magnetic focussing
with an additional independent in-plane magnetic field and observe a change in
focussing peak amplitude that is not symmetric with respect to the field
direction (i.e. $+B_{\parallel}\neq-B_{\parallel}$), and is extremely
sensitive to the magnitude of the in-plane magnetic field. We show that the
magnetic focussing signal is extremely sensitive to small changes in the Fermi
velocity, which can be used to detect small shifts in the Fermi surface caused
by an in-plane magnetic field. We also find that focussing can be used to
detect the proximity between spin-split Fermi surfaces, which cause non-
adiabatic spin dynamics.

††preprint: APS/123-QED

Techniques for measuring the Fermi surface have existed since the early 1930s
de Haas and van Alphen (1930); Schubnikow and W.J. de Haas (1930), and are a
powerful way to probe electronic and magnetic properties of metals,
semiconductors, superconductors and heavy fermion compounds Lonzarich (1988);
Hussey  _et al._ (2003); Doiron-Leyraud  _et al._ (2007); Danzenbächer  _et
al._ (2011). Magnetic focussing was originally proposed as a technique for
probing the Fermi surface of metals Sharvin (1965); Tsoi (1974, 1996), and is
unique among methods for measuring 2D Fermi surfaces in that it uses point
contacts as injection and detection points as a solid-state equivalent of a
charge-mass spectrometer. This means focussing is able to measure a well
defined section of the Fermi surface as the charge carriers do not complete a
full orbit van Houten  _et al._ (1989); Heremans  _et al._ (1992, 1994);
Taychatanapat  _et al._ (2013); Lee  _et al._ (2016); Bachmann  _et al._
(2019); Berdyugin  _et al._ (2020). In addition, the use of point contacts
makes focussing extremely sensitive, allowing it to probe spin and charge
dynamics including branched electron flow, small-angle scattering, spin
separation and spin precession Aidala  _et al._ (2007); Bladwell and Sushkov
(2018); Gupta  _et al._ (2021); Rendell  _et al._ (2023); Rokhinson  _et al._
(2004); Lo  _et al._ (2017).

Here we show that magnetic focussing of holes is an extremely sensitive probe
of the spin-dependent Fermi surface and spin dynamics in 2D systems. We use
focussing to observe non-adiabatic spin dynamics and spin-dependent scattering
created by in-plane magnetic fields (B∥). We observe a change in focussing
peak amplitude that is non-symmetric with respect to the polarity of the in-
plane magnetic field (+B${}_{\parallel}\neq$ -B∥). The change in peak
amplitude can be explained by B∥ causing a shift in the spin-split 2D Fermi
surfaces. For smaller in-plane fields (B${}_{\parallel}<$ 4T) the change in
peak amplitude is monotonic, and is consistent with B∥ changing the Fermi
velocity and hence the scattering rate of the spin states. The amplitude
change is visible for magnetic fields as small as B∥ = 0.1T, demonstrating the
sensitivity of focussing. At larger in-plane fields (B${}_{\parallel}>$ 4T) a
significant shift in the focussing peaks is observed for one polarity of B∥.
This is a result of the spin-split Fermi surfaces touching, creating non-
adiabatic spin dynamics. Both of these effects are only visible in focussing
measurements, as they would usually be averaged out over the full Fermi
surface.

Figure 1: Spin-resolved focussing with in-plane magnetic fields a) SEM of the
focussing sample. The overlay shows the orientation of the in-plane
(B${}_{\parallel})$ and out-of-plane (BFocus) magnetic fields, as well as the
electrical setup for measurements. Red and blue semicircles indicate the spin-
split focussing trajectories. b) The first 2D subband for holes. A Rashba
spin-orbit interaction causes a spin-splitting of the subband resulting in two
different momenta (k+ and k-) at the Fermi energy (horizontal dashed line).
This difference in momentum results in different focussing trajectories for
each spin. c) The spin split first focussing peak resulting from the two spin-
dependent focussing trajectories (labelled with red circle and blue square).
By fitting a double Gaussian to the first focussing peak the amplitude of both
peaks can be extracted. d) Spin split focussing with an in-plane magnetic
field. The change in peak amplitude is not symmetric when the direction of the
in-plane field is reversed.

The magnetic focussing sample is fabricated on a GaAs/AlGaAs heterostructure
with a 15nm GaAs quantum well confining the 2D hole gas (2DHG). The 2DHG is
induced in accumulation mode (no doping) by applying a negative voltage to an
overall top gate. Figure 1 shows an SEM of the sample with lithographic split
gates used to define the focussing geometry. The overlay indicates the
electrical measurement setup and the orientation of the in-plane magnetic
field. To perform focussing, a constant current of holes (I${}_{\text{SD}}$ =
5nA) is injected through a quantum point contact (QPC), and the resulting
focussing signal is measured as a voltage across a second QPC. This is
performed as a four terminal measurement with a pair of lock-in amplifiers at
low frequency (17 Hz). When the perpendicular magnetic field
(B${}_{\text{Focus}}$) is such that the focussing diameter
(d${}_{\text{Focus}}$) is equal to the spacing between QPCs, a peak is
observed in the focussing voltage. These peaks occur when the magnetic field
is an integer multiple of van Houten  _et al._ (1989)

$B_{\text{Focus}}=\frac{2\hbar k_{F}}{ed_{\text{Focus}}}$

Where kF is the Fermi momentum. All measurements in this work use a focussing
diameter d${}_{\text{Focus}}$ = 800nm at a 2D density of n${}_{\text{2D}}$ =
1.89x1011 cm-2, (V${}_{\text{TG}}$ = -1.35V). The QPCs have lithographic
dimensions of 300x300nm and are biased to G=$2e^{2}/h$ to inject and detect
both spin polarisations. The sample is measured in a He dilution system with a
9/5/1 T vector magnet at base temperature (30mK). In-plane and out-of-plane
fields are also measured using Hall sensors on the sample probe to correct for
any magnet hysteresis.

In GaAs hole systems, spin-orbit interactions have a significant effect on
magnetic focussing measurements. The Rashba spin-orbit interaction causes a
spin splitting of the first 2D subband which results in a different momentum
for each spin. Figure 1 b) shows the spin-split first 2D subband for holes in
GaAs, with the red and blue bands representing the different spins. The
splitting of the 2D subband results in a difference of momentum between the
spins (k+ and k-) at the Fermi energy (horizontal dashed line). The different
momentum creates a different focussing trajectory for each spin, which splits
the first focussing peak Rokhinson  _et al._ (2004). The splitting of the 2D
subband can also create a different scattering rate for each spin, as it
changes the slope of each subband and hence the velocity Rendell  _et al._
(2023). The relatively symmetric quantum well heterostructure used in this
work allows for visible spin splitting while also giving a similar scattering
rate for both spin states.

Figure 1 c) shows focussing with no in-plane magnetic field. A double peak is
observed consistent with spin-resolved focussing. A double Gaussian fit to the
split peak (red and blue peaks in Fig. 1 c) shows that both peaks have a
similar amplitude in the absence of an in-plane magnetic field. Figure 1 d)
shows the first focussing peak with a magnetic field applied in-plane parallel
to the QPC current direction (B∥). With B∥ = +2T (top trace - blue), a clear
change in the peak amplitude is observed compared to the peaks with no in-
plane field (black centre trace). This change in peak amplitude would
typically be interpreted as a change in the spin polarisation Rokhinson  _et
al._ (2004). However, when the direction of B∥ is reversed to -2T (bottom
trace in Fig. 1 d) the change in peak amplitude is not symmetric. This is not
consistent with a change in spin polarisation as the Zeeman splitting should
be the same for $\pm$B∥.

Figure 2: Focussing with small B in-plane parallel to the QPC orientation. a)
Shows the focussing signal for different in-plane fields of $\pm$ 0.5 T in 0.1
T steps. Central black trace is for zero field, top solid traces are for field
parallel to the QPC current and bottom dashed traces are for field
antiparallel. Data has been vertically offset for clarity. b) The amplitude of
each spin peak from a double Gaussian fit to the data in a). c) The calculated
energy dispersion of one of the spin-split 2D hole subbands for B∥ = +0.1T, 0T
and -0.1T using Eq 1. The dashed horizontal line is the Fermi energy. d) The
change in velocity of the spin subbands as a function of B∥. The velocity (vB)
is normalised by the average velocity of the two spins at B${}_{\parallel}=0$
(v${}_{\text{B}=0}$).

To rule out any Zeeman or spin polarisation effects, we next measure the
change in peak amplitude with small in-plane magnetic fields. Figure 2 a)
shows the results of focussing with small B∥ applied in 0.1T increments up to
$\pm$0.5T. A double Gaussian is fitted to each peak and the amplitude as a
function of B∥ is plotted in Fig. 2 b). A change in amplitude of the spin
peaks is observed for fields as small as $\pm$0.1T, far too small to be caused
by a change in spin polarisation. Even at 0.5T the change in peak amplitude
($\sim$ 10%) is significantly larger than the Zeeman energy ($\sim$1% of
E${}_{\text{F}}$) and therefore is too small to be a Zeeman effect. Instead,
we consider a shift in the Rashba spin-splitting of the 2D subbands caused by
B∥.

The Hamiltonian for the 2D subbands is of the form

$\mathcal{H}=\frac{\mathbf{p^{2}}}{2m^{*}}+\frac{i\alpha}{2}(\sigma_{+}p^{3}_{-}-\sigma_{-}p_{+}^{3})+\frac{g_{1}\mu_{B}}{2}(B_{+}p^{2}_{+}\sigma_{-}+B_{-}p^{2}_{-}\sigma_{+})\\\
+\frac{g_{2}\mu_{B}}{2}(B_{-}p^{4}_{+}\sigma_{-}+B_{+}p^{4}_{-}\sigma_{+})$
(1)

where $\frac{i\alpha}{2}(\sigma_{+}p^{3}_{-}-\sigma_{-}p_{+}^{3})$ is the
Rashba spin-orbit term,
$\frac{g_{1}\mu_{B}}{2}(B_{+}p^{2}_{+}\sigma_{-}+B_{-}p^{2}_{-}\sigma_{+})+\frac{g_{2}\mu_{B}}{2}(B_{-}p^{4}_{+}\sigma_{-}+B_{+}p^{4}_{-}\sigma_{+})$
are the Zeeman terms due to the in-plane magnetic field and $B_{\pm}=B_{x}\pm
iB_{y}$. $B_{\parallel}$ causes a small shift in the 2D spin subbands, which
is in opposite directions for the two subbands. Figure 2 c) shows the
calculated energy dispersion of one of the 2D hole subbands using Eq. 1. There
is a small shift in the subband dispersion, which is not symmetric for
$\pm\text{B}_{\parallel}$. This shift is too small to cause a measurable
change in the location of the magnetic focussing peaks, however it will still
cause a change in the subband curvature Bladwell and Sushkov (2015, 2018);
Marcellina  _et al._ (2018). The change in subband curvature causes the Fermi
velocity (v${}_{\text{F}}$) to change along the hole trajectory.
v${}_{\text{F}}$ is linked to the scattering rate of each spin, since holes
that travel slower will have more time to scatter (and vice versa). The
focussing peak amplitude is exponentially sensitive to v${}_{\text{F}}$:

$R_{Focus}\propto e^{-\pi d/(2v_{\text{F}}\tau_{\text{Focus}})}$ (2)

where $d$ is the focussing diameter and v${}_{\text{F}}\tau_{\text{Focus}}$ is
the characteristic focussing scattering length Heremans  _et al._ (1992);
Rendell  _et al._ (2015, 2023). Figure 2 d) shows the calculated change in
v${}_{\text{F}}$ for small B∥. This matches the trend in focussing peak
amplitude shown in Fig. 2 b). This change in focussing peak amplitude for
small B∥ demonstrates the high sensitivity of focussing to small changes in
the Fermi surface.

Figure 3: Focussing with a large B in-plane parallel to the QPC current. a)
The focussing signal with an in-plane field applied along the QPC up to +7 T
in 1 T steps. The black trace is for zero in-plane field. Each trace has been
vertically offset by 1 k$\Omega$ for clarity. b) The same as in a) with the
opposite polarity of in-plane magnetic field. c) The amplitude of each spin
peak as from a fit to each focussing trace.

Next, we consider shifts in the Fermi surface caused by large B∥. Figure 3 a)
shows the evolution of the first focussing peak with a large in-plane magnetic
field applied parallel to the QPC current direction (B∥). As B∥ increases in
magnitude the peak amplitude changes, which would typically interpreted as a
change in spin polarisation. Again, when the polarity of B∥ is reversed (Fig.
3 b), the change in peak amplitude is not symmetric. This is not consistent
with a change in spin polarisation as Zeeman splitting should be the same for
$\pm\text{B}_{\parallel}$. To make this clearer, the amplitude of the spin
peaks is plotted as a function of B∥ in Fig. 3 c). The peak amplitude is
clearly not symmetric around B${}_{\parallel}=0$, and similar results are
obtained when considering the peak area. If the change in peak amplitude was
due to Zeeman spin polarisation there should be a monotonic response in the
amplitude. This is not visible in Fig. 3 c), providing further evidence that
the change in peak amplitude is not a Zeeman effect or a change in spin
polarisation. In addition, the asymmetry in the amplitude change is not an
artefact of the setup, as it meets the Onsager reciprocity conditions ($\pm$B∥
symmetry is restored if the field direction is reversed and the current and
voltage probes are swapped - see supplementary info).

Figure 4: Calculated spin split Fermi surface shift with B${}_{\parallel}=\pm$
4T (double check field) a) With no B∥. Red and blue circles correspond to the
2D spin sub-bands. Solid lines indicates the section of the Fermi surfaces
over which focussing measurements are performed. b) With B∥ = +4T, separation
of the two spin-split Fermi surfaces is increased over focussing trajectories.
c) With B∥ = -4T, Fermi surfaces almost touch leading to non-adiabatic spin
dynamics.

The non-monotonic change in the focussing peak amplitude can be explained by a
shift in the spin-split Fermi surfaces caused by B∥. Figure 4 shows the
calculated shift in the spin-split Fermi surfaces using Eq. 1. Figure 4 a)
shows the section of the Fermi surface measured by focussing in the absence of
B∥. The red and blue lines indicate the calculated spin-split Fermi surface.
The solid section corresponds to the focussing trajectory, while the dashed
side does not contribute to focussing. At B${}_{\parallel}=0$T the spacing
between the blue and red subbands is the same for all points on the Fermi
surface. Figure 4 b) shows the change in Fermi surface for B∥ = +4T. The in-
plane field causes the blue Fermi surface to shift towards k$=0$, while the
red surface is shifted away. Since only the solid parts of the Fermi surfaces
contribute to focussing, these paths move further apart, leading to more
adiabatic transport. When the direction of B∥ is reversed (Fig. 4 c), the blue
Fermi surface shifts away from k$=0$, while the red surface shifts towards
k$=0$. The solid sections of the Fermi surfaces now almost touch, allowing
mixing between the spin states. This results in non-adiabatic spin evolution
Zener (1932) and causes a significant shift in the focussing peaks at
B${}_{\parallel}=-4$T as shown in Fig. 3 c). This asymmetry in the Fermi
surface shift is not observed in other 2D measurements (e.g. Shubnikov-de Haas
oscillations) since they sample the full Fermi surface and hence would see the
same result for +B∥ and -B∥

In summary, we have used magnetic focussing to measure a shift in spin-split
Fermi surfaces caused by B∥. For small B∥, the Fermi surface shift results in
a change in velocity and hence scattering along each focussing trajectory. At
large B∥, Fermi surface shifts lead to non-adiabatic transport in one
direction of B∥. This non adiabatic transport causes a significant shift in
the focussing peaks. Neither of these effects can be explained by a change in
spin polarisation or a Zeeman effect. These results show the sensitivity of
magnetic focussing as a technique for probing changes in the 2D Fermi surface.
In addition, these effects can only be measured via magnetic focussing as it
is able to probe a section of the Fermi surface, unlike other 2D measurements
such as Shubnikov-de Haas oscillations.

The authors would like to thank U. Zülicke and Z. Krix for many valuable
discussions. Devices were fabricated at the UNSW node of the Australian
National Fabrication Facility (ANFF). This research was funded by the
Australian Government through the Australian Research Council Discovery
Project Scheme; Australian Research Council Centre of Excellence FLEET
(project number CE170100039); and by the the UK Engineering and Physical
Sciences Research Council (Grant No. EP/R029075/1). All experimental data and
calculation code is available at http://dx.doi.org/10.5281/zenodo.8368876

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