# Geometric interpretation of valuated term (pre)orders

Netanel Friedenberg  Department of Mathematics, Tulane University, New
Orleans, LA 70118, USA<EMAIL_ADDRESS>and  Kalina Mincheva
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
<EMAIL_ADDRESS>
###### Abstract.

Valuated term orders are studied for the purposes of Gröbner theory over
fields with valuation. The points of a usual tropical variety correspond to
certain valuated terms preorders. Generalizing both of these, the set of all
“well-behaved” valuated term preorders is canonically in bijection with the
points of a space introduced in our previous work on tropical adic geometry.
In this paper we interpret these points geometrically by explicitly
characterizing them in terms of classical polyhedral geometry. This
characterization gives a bijection with equivalence classes of flags of
polyhedra as well as a bijection with a class of prime filters on a lattice of
polyhedral sets. The first of these also classifies valuated term orders. The
second bijection is of the same flavor as the bijections from [14] in non-
archimedean analytic geometry and indicates that the results of that paper may
have analogues in tropical adic geometry.

##  1\. Introduction

Classical Gröbner theory uses a term order which compares two monomials
$m=a\underline{x}^{\underline{u}}$ and
$m^{\prime}=a^{\prime}\underline{x}^{\underline{u^{\prime}}}$ by only
considering $\underline{u}$ and $\underline{u^{\prime}}$. In [2] Chan and
Maclagan build Gröbner theory in $k[x_{1},\ldots,x_{n}]$, when $k$ is a field
with nontrivial valuation $v:k\to\mathbb{R}\cup\\{-\infty\\}=:\mathbb{T}$.
Their valuated Gröbner theory takes into account the coefficients by picking a
weight vector $\underline{w}\in\mathbb{R}^{n}$ and comparing
$v(a)+\underline{w}\cdot\underline{u}$ with
$v(a^{\prime})+\underline{w}\cdot\underline{u}^{\prime}$, and then using a
term order to break any ties.

Valuated Gröbner theory has a natural connection to tropical geometry. More
specifically, the tropicalization of a subvariety of the torus is the set of
$\underline{w}\in\mathbb{R}^{n}$ for which each polynomial in its defining
ideal has at least two monomials of maximal weight. In particular, tropical
geometry studies valuated monomial preorders, rather than orders. A _valuated
monomial preorder_ is a multiplicative total preorder $\leq$ on the set of
monomials $a\underline{x}^{\underline{u}}$ such that if $v(a)\leq v(b)$ then
$a\leq b$. Valuated monomial _orders_ are those for which
$a\underline{x}^{\underline{u}}\equiv b\underline{x}^{\underline{\mu}}$ if and
only if $v(a)=v(b)$ and $u=\mu$. In light of this, we treat
$a\underline{x}^{\underline{u}}$ and $b\underline{x}^{\underline{u}}$ as being
the same if $v(a)=v(b)$.

When working with valuated monomial orders, one is usually only interested in
those in which the valuation of the coefficients play a primary role. These
are exactly the orders that arise by using a weight vector and term order as
above. We call such orders Chan-Maclagan orders; they also occur in ongoing
work of Vaccon and Verron [18] on universal Gröbner bases in Tate algebras.

Recent work of Amini and Iriarte [1] aimed at the study of Newton-Okounkov
bodies investigates spaces of (quasi)monomial preorders. An understanding of
valuated monomial preorders is needed to extend their theory from varieties to
semistable models over a valuation ring.

We see from the above that a weight vector $\underline{w}\in\mathbb{R}^{n}$
provides a geometric interpretation for certain valuated monomial preorders.
In fact, if $v(k^{\times})\subseteq\mathbb{Q}$, then most points in
$\mathbb{R}^{n}$ define a valuated monomial _order_. For if
$\underline{w}\in\mathbb{R}^{n}$ has irrational entries which are linearly
independent over $\mathbb{Q}$, then distinct monomials will always have
different weights. This shows that certain valuated monomial orders have a
geometric interpretation; however, not all of them arise this way.

The goal of the paper is to derive a geometric meaning for each of the
following: (1) all valuated monomial preorders, (2) Chan-Maclagan orders, and
(3) those valuated monomial preorders determined by a weight vector and a term
_preorder_ as in the construction of the Chan-Maclagan orders. We do this as
an application of the theory proposed in [6], thus verifying that it carries
geometric information.

The theory from [6] applies more generally, and so we work with $k[M]$ where
$M$ is a toric monoid. We are then able to quickly reduce to the torus case,
where $k[M]=k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$.

Let $\Gamma=v(k^{\times})$ be the value group of $v$. We can now state our
main results.

######  Theorem A (Corollary 3.38).

There is an explicit bijection from the set of valuated monomial preorders on
$k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ to the set of $\Gamma$-local
equivalence classes of flags of polyhedral cones in $\mathbb{R}_{\geq
0}\times\mathbb{R}^{n}$.

See Definition 3.5 for the definition of $\Gamma$-local equivalence.

######  Theorem B (Corollary 3.37).

There is an explicit bijection from the set of Chan-Maclagan preorders on
$k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ to the set of $\Gamma$-local
equivalence classes of flags of polyhedra in $\mathbb{R}^{n}$.

######  Theorem C (Proposition 3.44).

The bijection of Theorem B gives a bijection from the set of Chan-Maclagan
orders on $k[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ to the set of $\Gamma$-local
equivalence classes of complete flags of polyhedra in $\mathbb{R}^{n}$.

The classification provided by Theorem C can be seen as complementary to an
earlier result in the non-valuated case. Usual monomial orders have been
classified in [16] in terms of their defining matrices. Since two matrices can
give the same monomial order, the classification proceeds by providing a
canonical representative for the equivalence class. In contrast, Corollary
3.42 gives an alternative, geometric criterion for when two matrices give the
same valuated or non-valuated monomial order.

We now interpret the Chan-Maclagan preorders in the context of (additively
idempotent) semirings, motivated by adic geometry. We first recall that for a
topological (usually f-adic) ring $R$ the set of equivalence classes of
continuous valuations $v:R\to G\cup\\{0\\}$, where $G$ can be any
multiplicatively-written totally ordered abelian group, is called the
continuous spectrum of $R$, denoted $\operatorname{Cont}R$. To consider
continuity of valuations, a certain canonical topology is placed on
$G\cup\\{0\\}$.

The semiring analogue of a valuation is a homomorphism to a totally ordered
semifield. Just as equivalence classes of homomorphisms from a ring $R$ to a
field are given by prime ideals, equivalence classes of valuations on a
semiring $A$ are given by certain equivalence relations on $A$ called _prime
congruences_ 111first introduced in [11].

For any topological semiring $A$ we consider the set of prime congruences $P$
on $A$ for which $A/P$ is not finite (a technical hypothesis that is redundant
in the cases we consider) and the map from $A$ to the residue semifield
$\kappa(P)$ is continuous. Here $\kappa(P)$, the semifield generated by $A/P$,
is given a canonical topology. We call the set of such $P$ the _continuous
spectrum of $A$_, and denote it by $\operatorname{Cont}A$. These spaces are
directly linked to the spaces that occur in adic geometry: a continuous
generalized valuation in the sense of [7, Definition 2.5.1] from a ring $R$ to
a semiring $A$ induces a map from $\operatorname{Cont}{A}$ to
$\operatorname{Cont}{R}$.

In this language, the set of Chan-Maclagan preorders on $k[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$ is canonically identified with the continuous
spectrum of $S[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$. Here $S$ is the set
$\Gamma\cup\\{-\infty\\}$ endowed with operations that make it a totally
ordered semifield, and $S[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$ is given a
topology coming from the inclusion map $S\hookrightarrow S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$. We denote this continuous spectrum by
$\operatorname{Cont}_{S}S[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$. Thus Theorem B
can be restated as follows.

######  Theorem B’.

There is an explicit bijection from $\operatorname{Cont}_{S}S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$ to the set of $\Gamma$-local equivalence classes of
flags of polyhedra in $\mathbb{R}^{n}$.

The motivation to interpret certain valuated monomial preorders as points on a
tropical adic space, and thus prime congruences, comes from the connection
between tropical geometry and Berkovich spaces. It is also inspired by the
broader program to endow tropical varieties with more structure; in [6] we
take an analytic approach to doing so, which led us to the results in this
work. There are alternative algebraic approaches using many different
frameworks. Among these are blueprints [12], tropical ideals [13], tropical
schemes [7], super-tropical algebra [10], and systems [17].

Our last major result gives an alternative classification of the points of
$\operatorname{Cont}_{S}S[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$. This
classification is reminiscent of a theorem for adic spaces.

######  Theorem D (Theorem 3.35).

There is an explicit bijection between $\operatorname{Cont}_{S}S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$ and the set of prime filters $\mathcal{F}$ on the
lattice of $\Gamma$-rational polyhedral sets in $\mathbb{R}^{n}$ such that
$\mathcal{F}$ contains a polytope.

This bijection between points of $\operatorname{Cont}_{S}S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$ and prime filters on the lattice of
$\Gamma$-rational polyhedral sets is of the same flavor as certain results in
[14]. If $X$ is a rigid affinoid space, then it is shown in [14] that the
points of the Huber adic space corresponding to $X$ are in bijection with the
prime filters on the lattice of special subsets of $X$.

We also show that, for any $P\in\operatorname{Cont}_{S}S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$, the minimum dimension of any element of the
corresponding filter is equal to the transcendence degree of the semifield
extension $\kappa(P)/S$; see Proposition 3.43.

## Acknowledgements

The authors thank Hernán Iriarte for exciting discussions and Bernd Sturmfels
for pointing us to the work of Vaccon and Verron in valuated Gröbner theory.
The second author is partially supported by Louisiana Board of Regents
Targeted Enhancement Grant number 090ENH-21.

##  2\. Preliminaries

######  Definition 2.1.

By a semiring $R$ we mean a commutative semiring with multiplicative unit.
That is, $R$ is a set which is a commutative monoid with respect to each of
two binary operations: an addition operation $+_{R}$, whose identity is
denoted $0_{R}$, and a multiplication operation $\cdot_{R}$, whose identity is
denoted $1_{R}$. Furthermore, we require that $0_{R}\neq 1_{R}$, $0_{R}$ is
multiplicatively absorbing, and multiplication distributes over addition. We
omit the subscripts from the operations whenever this will not cause
ambiguity. A semifield is a semiring in which all nonzero elements have a
multiplicative inverse.

######  Definition 2.2.

We call a semiring $R$ additively idempotent if $a+a=a$ for all $a\in R$. We
refer to additively idempotent semirings as just idempotent.

If $R$ is an idempotent semiring, then the addition defines a canonical
partial order on $R$ in the following way: $a\geq b\Leftrightarrow a+b=a$.
With respect to this order, $a+b$ is the least upper bound of $a$ and $b$.
When we consider a totally ordered idempotent semiring, we mean an idempotent
semiring for which this canonical order is a total order.

Some of the idempotent semifields that we use throughout the paper are:

  * •

The Boolean semifield $\mathbb{B}$ is the semifield with two elements
$\\{1,0\\}$, where $1$ is the multiplicative identity, $0$ is the additive
identity and $1+1=1$. $\mathbb{B}$ is the only finite additively idempotent
semifield.

  * •

The tropical semifield $\mathbb{T}$ is defined on the set
$\mathbb{R}\cup\\{-\infty\\}$, by setting the $+$ operation to be the usual
maximum and the multiplication operation to be the usual addition, with
$-\infty=0_{\mathbb{T}}$.

  * •

The semifield $\mathbb{Q}_{\max}$ is a sub-semifield of $\mathbb{T}$. As a set
it is $\mathbb{Q}\cup\\{-\infty\\}$ and the operations are the restrictions of
the operations on $\mathbb{T}$. We also use the same notation when
$\mathbb{Q}$ is replaced by any other additive subgroup of $\mathbb{R}$. In
fact, every sub-semifield of $\mathbb{T}$ arises this way.

  * •

The semifield $\mathbb{T}^{(n)}$ is defined on the set
$\mathbb{R}^{n}\cup\\{-\infty\\}$, by setting the addition operation to be
lexicographical maximum and the multiplication operation to be the usual
pointwise addition.

Throughout this paper all monoids, rings and semirings are assumed to be
commutative. All rings and semirings have a multiplicative unit. Whenever we
use the word semiring without further qualification, we refer to an additively
idempotent semiring. Our focus will be on totally ordered semifields which are
different from $\mathbb{B}$. Henceforth, when we refer to totally ordered
semifields, we implicitly assume that they are not isomorphic to $\mathbb{B}$.
In particular, whenever we refer to a sub-semifield of $\mathbb{T}$, we
implicitly assume that it is not $\mathbb{B}$. Totally ordered semifields can
be seen as the image of a non-archimedean valuation. The semifield
$\mathbb{B}$ then is the image of the trivial valuation.

In order to distinguish when we are considering a real number as being in
$\mathbb{R}$ or being in $\mathbb{T}$ we introduce some notation. For a real
number $a$, we let $t^{a}$ denote the corresponding element of $\mathbb{T}$.
In the same vein, given $\mathfrak{a}\in\mathbb{T}$, we write
$\log(\mathfrak{a})$ for the corresponding element of
$\mathbb{R}\cup\\{-\infty\\}$. This notation is motivated as follows. Given a
non-archimedean valuation $\nu:K\rightarrow\mathbb{R}\cup\\{-\infty\\}$ on a
field and $\lambda\in\mathbb{R}$ with $\lambda>1$, we get a non-archimedean
absolute value $|\cdot|_{\nu}:K\rightarrow[0,\infty)$ by setting
$|x|_{\nu}=\lambda^{\nu(x)}$. Since $\mathbb{T}$ is isomorphic to the
semifield $\big{(}[0,\infty),\max,\cdot_{\mathbb{R}}\big{)}$, we use a
notation for the correspondence between elements of
$\mathbb{R}\cup\\{-\infty\\}$ and elements of $\mathbb{T}$ that is analogous
to the notation for the correspondence between $\nu(x)$ and $|x|_{\nu}$. This
notation is also convenient as we get many familiar identities such as
$\log(1_{\mathbb{T}})=0_{\mathbb{R}}$ and $t^{a}t^{b}=t^{a+_{\mathbb{R}}b}$.

######  Definition 2.3.

Let $R$ be a semiring and let $a\in R\backslash\\{0\\}$. We say that $a$ is a
_cancellative element_ if for all $b,c\in R$, whenever $ab=ac$ then $b=c$. If
all elements of $R\backslash\\{0\\}$ are cancellative, then we say that $R$ is
a cancellative semiring.

######  Definition 2.4 (Congruences).

  * •

A congruence $E$ on a semiring $R$ is an equivalence relation on $R$ that
respects the operations of $R$.

  * •

The _trivial congruence_ on $R$ is the diagonal $\Delta\subseteq R\times R$,
for which $R/\Delta\cong R$.

  * •

We call a proper congruence $P$ of a semiring $R$ prime if $R/P$ is totally
ordered and cancellative (cf. Definition 2.3 and Proposition 2.10 in [11]).

We will need some more notation around prime congruences.

######  Definition 2.5.

Let $P$ be a prime congruence on a semiring $A$. The residue semifield of $A$
at $P$, denoted $\kappa(P)$, is the total semiring of fractions222The total
semiring of fractions is a classical concept and is defined in [8]. An
alternative reference following the notation of this paper is in our previous
work [6, Definition 2.13]. of $A/P$. We denote the canonical homomorphism
$A\to A/P\to\kappa(P)$ by $a\mapsto|a|_{P}$.

######  Notation 2.6.

Let $R$ be semiring and let $P$ be a prime on $R$. For two elements
$r_{1},r_{2}\in R$ we say that $r_{1}\leq_{P}r_{2}$ (resp. $r_{1}<_{P}r_{2}$,
resp. $r_{1}\equiv_{P}r_{2}$) whenever $|r_{1}|_{P}\leq|r_{2}|_{P}$ (resp.
$|r_{1}|_{P}<|r_{2}|_{P}$, resp. $|r_{1}|_{P}=|r_{2}|_{P}$) in $R/P$.

We now focus our attention on the case when $R$ is a monoid algebra. Given a
totally ordered semifield $S$ and a monoid $M$, the corresponding monoid
$S$-algebra is denoted $S[M]$. Elements of $S[M]$ are finite sums of
expressions of the form $a\chi^{u}$ with $a\in S$ and $u\in M$, which are
called _monomials_ or _terms_. We call elements of $S[M]$ _generalized
semiring polynomials_ or _polynomials_ when no confusion will arise. We will
be most interested in $S[\mathbb{Z}^{n}]$, the _semiring of tropical Laurent
polynomials in $n$ variables_.

When $S\subseteq\mathbb{T}$ we describe the prime congruences on
$S[\mathbb{Z}^{n}]$ in terms of their _defining matrices_ and the monomial
preorder the give rise to. Every prime congruence on $S[\mathbb{Z}^{n}]$ has a
defining matrix; see [11], where it is shown that the matrix can be taken to
be particularly nice.

Let $R=S[\mathbb{Z}^{n}]$ where $S$ is a subsemifield of $\mathbb{T}$. A
$k\times(n+1)$ real valued matrix $C$ such that the first column of $C$ is
lexicographically greater than or equal to the zero vector gives a prime
congruence on $R$ as follows: For any monomial $m=t^{a}\chi^{u}\in S[M]$ we
let $\Phi(m)=C\begin{pmatrix}a\\\ u\end{pmatrix}\in\mathbb{R}^{k}$, where we
view $u\in\mathbb{Z}^{n}$ as a column vector. We call $\begin{pmatrix}a\\\
u\end{pmatrix}$ the _exponent vector_ of the monomial $m$. For any nonzero
$f\in S[M]$, write $f$ as a sum of monomials $m_{1},\ldots,m_{r}$ and set
$\Phi(f)=\max_{1\leq i\leq r}\Phi(m_{i})$, where the maximum is taken with
respect to the lexicographic order. Finally, set $\Phi(0)=-\infty$. We specify
the prime congruence $P$ by saying that $f$ and $g$ are equal modulo $P$ if
$\Phi(f)=\Phi(g)$. In this case we say that $C$ is a defining matrix for $P$.
The first column of $C$ is called the column corresponding to the coefficient
or the column corresponding to $S$. For $1\leq i\leq n$, we say that the
$(i+1)^{\text{st}}$ column of $C$ is the column corresponding to $x_{i}$.

When can also explicitly describe the order that $P$ gives rise to. Given any
defining matrix for $P$ and any $f,g\in R$, $f\leq_{P}g$ exactly if
$\Phi(f)\leq_{lex}\Phi(g)$.

It will be convenient to know that we can choose a defining matrix with a
particular form. To accomplish this, we will use the following lemma.

######  Lemma 2.7 ([6] Lemma 2.12, Downward gaussian elimination).

Let $S\subseteq\mathbb{T}$ and $M=\mathbb{Z}^{n}$ and let $P$ be a prime
congruence of $S[M]$ with defining matrix $C$. Then the following elementary
row operations do not change the prime.

  * •

multiplying a row by a positive constant.

  * •

adding a multiple of a row to any row below it.

Note that, if a row of the matrix is all zero, then removing the row does not
change the prime congruence that the matrix defines. Thus, using downward
gaussian elimination, we can always choose a defining matrix in which the rows
are linearly independent (over $\mathbb{R}$).

In [6] we define a set $\operatorname{Cont}_{S}A$ of certain prime congruences
on $A$. When $A=S[M]$ with $M=\mathbb{Z}^{n}$, the following proposition
completely characterizes $\operatorname{Cont}_{S}A$.

######  Proposition 2.8 ([6] Corollary 4.16).

Let $S$ be a subsemifield of $\mathbb{T}$ and let $M\cong\mathbb{Z}^{n}$. Let
$P$ be a prime on $S[M]$ and let $C$ be any defining matrix for $P$. Then
$P\in\operatorname{Cont}_{S}S[M]$ if and only if the $(1,1)$ entry of $C$ is
positive. In particular, if $P\in\operatorname{Cont}_{S}S[M]$, then we can
choose a defining matrix for $P$ whose first column is the first standard
basis vector $e_{1}$.

######  Proposition 2.9 ([6] Proposition 9.10).

Let $M\cong\mathbb{Z}^{n}$. A relation $\preceq$ on the terms of $S[M]$ is the
same as $\leq_{P}$ for a prime $P$ on $S[M]$ if and only if the following
conditions hold:

  1. (1)

$\preceq$ is a total preorder,

  2. (2)

$\preceq$ is multiplicative, and

  3. (3)

$\preceq$ respects the order on $S$.

Moreover, $P\in\operatorname{Cont}_{S}{S[M]}$ if and only if

  1. (4)

for any term $a\chi^{u}$ of $S[M]$, $\exists b\in S^{\times}$, such that
$b\prec a\chi^{u}$.

######  Definition 2.10.

The _height_ of a prime congruence $P$ on a semiring $A$ is the maximum length
$\operatorname{ht}(P)$ of a chain $P_{0}\subsetneq
P_{1}\subsetneq\cdots\subsetneq P_{k}=P$ of primes under $P$.

In [11] the _dimension_ of a semiring $A$, denoted $\dim A$, is defined to be
the number of strict inclusions in a chain of prime congruences on $A$ of
maximal length. We refine this notion in the next definition.

######  Definition 2.11.

When $S\subseteq\mathbb{T}$ is a sub-semifield and $A$ is a $S$-algebra, the
_relative dimension of $A$ over_ $S$ is the number $\dim_{S}A$ of strict
inclusions in a longest chain of prime congruences in
$\operatorname{Cont}_{S}{A}$.

We illustrate the distinction between dimension and relative dimension in the
following example.

######  Example 2.12.

Let $S$ be a sub-semifield of $\mathbb{T}$. Then $\dim S$ is equal to the
number of proper convex subgroups of $S^{\times}$ which is 1. On the other
hand, $\dim_{S}S=0$. $\diamondsuit$

We now briefly introduce some notation from toric varieties. In this paper we
follow conventions from [15]. We will use the notation to make a statement
about the dimension of the semiring $S[M]$ and its geometric interpretation.

Let $\Lambda$ be a finitely generated free abelian group and let
$\Lambda_{\mathbb{R}}$ be the vector space
$\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$. We write
$N:=\Lambda^{*}=\operatorname{Hom}(\Lambda,\mathbb{Z})$ and
$N_{\mathbb{R}}=\operatorname{Hom}(\Lambda,\mathbb{R})\cong
N\otimes_{\mathbb{Z}}\mathbb{R}$. We will use the pairing
$N_{\mathbb{R}}\times\Lambda_{\mathbb{R}}\to\mathbb{R}$ given by
$(v,u)\mapsto\left\langle v,u\right\rangle:=v(u)$. A cone $\sigma$ is a
strongly convex rational polyhedral cone in $N_{\mathbb{R}}$ if
$\sigma=\sum\limits_{i=1}^{r}\mathbb{R}_{\geq 0}v_{i}$ for $v_{i}\in N$ and
$\sigma$ contains no line. We denote by $\sigma^{\vee}$ the dual cone of
$\sigma$,

$\sigma^{\vee}=\\{u\in\Lambda_{\mathbb{R}}:\left<v,u\right>\leq 0,\forall
v\in\sigma\\}=\bigcap\limits_{i=1}^{r}\\{u\in\Lambda_{\mathbb{R}}:\left<v_{i},u\right>\leq
0\\}.$

The monoids that we consider in this paper are of the form
$M=\sigma^{\vee}\cap\Lambda$. Because $\sigma$ is strongly convex, we can
identify $\Lambda$ with the groupification of $M$.

######  Lemma 2.13 ([6] Lemma 9.13).

Let $S$ be sub-semifield of $\mathbb{T}$ and let $M\subseteq\Lambda$ be a
toric monoid corresponding to a cone $\sigma$ in $N_{\mathbb{R}}$. For any
$P\in\operatorname{Cont}^{\circ}_{S}S[M]$,
$\operatorname{ht}(P)=\operatorname{rk}(\Lambda)-\operatorname{rk}(\kappa(P)^{\times}/S^{\times})$.

######  Proposition 2.14 ([6] Proposition 9.18).

Let $S$ be a sub-semifield of $\mathbb{T}$, let $\Lambda$ be a finitely
generated free abelian group, let $\sigma$ be a cone in $N_{\mathbb{R}}$, and
let $M=\sigma^{\vee}\cap\Lambda$. Then
$\dim_{S}S[M]=\operatorname{rk}\Lambda.$

##  3\. Results

We provide a geometric interpretation of the points of
$\operatorname{Cont}_{S}{S[M]}$ with $S$ a sub-semifield of $\mathbb{T}$ and
$M$ a toric monoid. We will denote by $\Gamma:=\log(S^{\times})$ the subgroup
of $\mathbb{R}$ corresponding to $S$.

By [6, Proposition 9.14], fixing the ideal-kernel of
$P\in\operatorname{Cont}_{S}{S[M]}$ decomposes $\operatorname{Cont}_{S}{S[M]}$
into toric strata, which are tori. Thus, we only need to deal with the torus
case, i.e., the case where $M$ is a finitely generated free abelian group.

The main results of this paper show that, in this case, there are explicit
bijections between the points of $\operatorname{Cont}_{S}{S[M]}$, the set of
prime filters on the lattice of $\Gamma$-rational polyhedral sets in
$N_{\mathbb{R}}$ where the filter contains a polytope, and certain equivalence
classes of flags of polyhedra.

###  3.1. Flags and prime congruences.

By a _cone_ we will mean a strongly convex polyhedral cone which is not
necessarily rational. We note that there is a bijective correspondence between
the set of non-empty strongly convex polyhedra $\mathcal{P}\subseteq
N_{\mathbb{R}}$ and the set of strongly convex cones
$\sigma\subseteq\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$, not contained in
$\\{0\\}\times N_{\mathbb{R}}$. Under this correspondence we map a polyhedron
$\mathcal{P}$ to the closed cone over it, denoted $c(\mathcal{P})$, and a cone
$\mathcal{C}$ to the restriction of $\mathcal{C}$ to $\\{1\\}\times
N_{\mathbb{R}}$, by which we mean the inverse image of $\mathcal{C}$ under the
map $N_{\mathbb{R}}\to\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ given by
$x\mapsto(1,x)$.

Additionally, we can think of a point $w=(r,\xi)\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}$ as a homomorphism $w:\Gamma\times M\to\mathbb{R}$, mapping
$(\gamma,u)\mapsto r\gamma+\left\langle\xi,u\right\rangle$. We also use
$\left\langle\bullet,\bullet\right\rangle$ to denote the resulting pairing
$(\mathbb{R}_{\geq 0}\times N_{\mathbb{R}})\times(\Gamma\times
M)\to\mathbb{R}$. Thus, for any $\mathfrak{S}\subseteq\Gamma\times M$ we can
consider $\mathfrak{S}^{\vee}=\\{w\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}\;:\;\left\langle w,\mathfrak{u}\right\rangle\leq 0\text{ for
all }\mathfrak{u}\in\mathfrak{S}\\}.$

######  Definition 3.1.

We denote by
$\mathcal{P}_{\bullet}=(\mathcal{P}_{0}\leq\mathcal{P}_{1}\leq\cdots\leq\mathcal{P}_{k})$
a flag of polyhedra, where $\dim\mathcal{P}_{i}=i$. We assume that all
polyhedra are strongly convex, non-empty and live in $N_{\mathbb{R}}$.

Similarly, we let
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\mathcal{C}_{1}\leq\cdots\leq\mathcal{C}_{k})$
denote a _flag of cones_ in $\mathbb{R}_{\geq_{0}}\times N_{\mathbb{R}}$ with
$\dim\mathcal{C}_{i}=i+1$. We say that $\mathcal{C}_{\bullet}$ is _simplicial_
if each $\mathcal{C}_{i}$ is a simplicial cone.

Given a flag $\mathcal{P}_{\bullet}$ of polyhedra, the corresponding flag of
cones is $c(\mathcal{P}_{\bullet})=(c(\mathcal{P}_{0})\leq
c(\mathcal{P}_{1})\leq\cdots\leq c(\mathcal{P}_{k}))$. The flags of cones
$\mathcal{C}_{\bullet}$ which arise this way are exactly those for which
$\mathcal{C}_{0}$ is not contained in $\\{0\\}\times N_{\mathbb{R}}$. For such
flags $\mathcal{C}_{\bullet}$, we write $\mathcal{C}_{\bullet}|_{\\{1\\}\times
N_{\mathbb{R}}}$ for the corresponding flag of polyhedra.

Consider the map taking a matrix $C$ whose first column is lexicographically
at least 0 to the prime it defines on $S[M]$. Here, because we have not fixed
an identification $M\cong\mathbb{Z}^{n}$, $C$ is of the form
$C=\begin{pmatrix}r_{0}&\xi_{0}\\\ r_{1}&\xi_{1}\\\ \vdots&\vdots\\\
r_{k}&\xi_{k}\end{pmatrix}$ where $r_{0},\ldots,r_{k}\in\mathbb{R}$ and
$\xi_{0},\ldots,\xi_{k}\in N_{\mathbb{R}}$. We now show how this map factors
through the set of simplicial flags of cones. Moreover, we show that if we
restrict to matrices giving primes in $\operatorname{Cont}_{S}S[M]$, the map
factors through the corresponding set of flags of polyhedra.

Given a matrix $C$ as above, we can use downward gaussian elimination as in
Lemma 2.7 and removal of zero rows to get all of the entries in the first
column non-negative and to get the rows linearly independent without changing
the prime that $C$ defines. Then we let $\mathcal{C}_{\bullet}(C)$ be the
simplicial flag of cones defined by letting $\mathcal{C}_{i}(C)$ be the cone
generated by $(r_{0},\xi_{0}),\ldots,(r_{i},\xi_{i})$ for each $0\leq i\leq
k$.

For any simplicial flag of cones $\mathcal{C}_{\bullet}$ in $\mathbb{R}_{\geq
0}\times N_{\mathbb{R}}$, we get a prime congruence on $S[M]$ as follows.

Let $\mathcal{C}_{-1}=\\{0\\}$ and pick points
$w_{i}\in\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$ for $0\leq i\leq k$.
Since $w_{i}\in\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$, we can think of each
$w_{i}$ as a homomorphism $w_{i}:\Gamma\times M\to\mathbb{R}$. Thus, we can
consider the group homomorphism

$\boldsymbol{w}=(w_{0},\ldots,w_{k}):\Gamma\times
M\rightarrow\mathbb{R}^{k+1},$

which preserves the order on $\Gamma$ when we give $\mathbb{R}^{k+1}$ the
lexicographic order. Therefore, $\boldsymbol{w}$ gives rise to a semiring
homomorphism

$\varphi_{\boldsymbol{w}}:S[M]\rightarrow\mathbb{T}^{(k+1)},\quad
a\chi^{u}\mapsto(w_{0},\ldots,w_{k})(\log a,u).$

We let $P_{\boldsymbol{w}}=\ker\varphi_{\boldsymbol{w}}$. That is,
$P_{\boldsymbol{w}}$ is the prime on $S[M]$ defined by the matrix
$\begin{pmatrix}w_{0}\\\ \vdots\\\ w_{k}\end{pmatrix}$. By Proposition 2.8,
$P_{\boldsymbol{w}}$ is in $\operatorname{Cont}_{S}S[M]$ if and only if the
(1,1) entry of this matrix is positive. Since
$w_{0}\in\mathcal{C}_{0}\backslash\\{0\\}\subseteq\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}$, this happens exactly when $\mathcal{C}_{0}$ is not contained
in $\\{0\\}\times N_{\mathbb{R}}$, i.e., when
$\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$ for some flag
$\mathcal{P}_{\bullet}$ of polyhedra.

In order for this construction to give a prime defined by
$\mathcal{C}_{\bullet}$, we must show that $P_{\boldsymbol{w}}$ is independent
of the choice of $\boldsymbol{w}$. Set $v_{0}=w_{0}$. For $1\leq i\leq k$,
$\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$ contains a unique ray of
$\mathcal{C}_{i}$. Fix a generator of this ray and call it $v_{i}$. Consider
the vector $\boldsymbol{v}=(v_{0},\ldots,v_{k})$. Analogously to
$\varphi_{\boldsymbol{w}}$ and $P_{\boldsymbol{w}}$, we can define
$\varphi_{\boldsymbol{v}}$ and $P_{\boldsymbol{v}}$. The following proposition
gives us that $P_{\boldsymbol{v}}$ and $P_{\boldsymbol{w}}$ are the same.

######  Proposition 3.2.

There is an automorphism $\psi$ of $\mathbb{T}^{(k+1)}$ given by a linear
transformation on $\mathbb{R}^{k+1}$ such that
$\varphi_{\boldsymbol{w}}=\psi\circ\varphi_{\boldsymbol{v}}$.

###### Proof.

We proceed by induction on $k$. The case $k=0$ is trivial. Assume the
statement is true for some $k=r$; we want to show that it is true for $k=r+1$.
Let $\tilde{\varphi}_{\boldsymbol{w}}$, $\tilde{\varphi}_{\boldsymbol{v}}$ and
$\tilde{\psi}$ be the maps we obtain for the flag
$(\mathcal{P}_{0}\leq\mathcal{P}_{1}\leq\cdots\leq\mathcal{P}_{r})$. Write

$w_{r+1}=\sum_{i=0}^{r+1}\alpha_{i}v_{i},\text{ where
}\alpha_{i}\in\mathbb{R}_{\geq 0}\text{ and }\alpha_{r+1}>0.$

Thus, we have

$\begin{pmatrix}w_{0}\\\ w_{1}\\\ \vdots\\\ w_{r}\\\
w_{r+1}\end{pmatrix}=\left(\begin{array}[]{@{}c|c@{}}\tilde{\psi}&\begin{matrix}0\\\
0\\\ \vdots\\\ 0\end{matrix}\\\
\hline\cr\begin{matrix}\alpha_{0}&\alpha_{1}&\cdots&\alpha_{r}\end{matrix}&\alpha_{r+1}\end{array}\right)\begin{pmatrix}v_{0}\\\
v_{1}\\\ \vdots\\\ v_{r}\\\ v_{r+1}\end{pmatrix},$

where we call the matrix in this product $\psi$. Note that it is invertible
since $\tilde{\psi}$ is and $\alpha_{r+1}>0$. Moreover, $\psi$ is order
preserving, i.e., $\boldsymbol{h}>\boldsymbol{h}^{\prime}$ implies that
$\psi\boldsymbol{h}>\psi\boldsymbol{h^{\prime}}$. We can show this by
induction; the base case is trivial. If $\tilde{\psi}$ is order preserving and
then $\psi$ is, since $v_{r+1}>v_{r+1}^{\prime}$ if and only if
$\alpha_{r+1}v_{r+1}>\alpha_{r+1}v_{r+1}^{\prime}$ and $\alpha_{r+1}>0$. ∎

Now consider $\boldsymbol{w}^{\prime}=(w^{\prime}_{0},\ldots,w^{\prime}_{k})$
with $w^{\prime}_{i}\in\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$. In the
same way as we got $\boldsymbol{v}$ from $\boldsymbol{w}$, we now get
$\boldsymbol{v}^{\prime}$. Since $v_{i}$ and $v^{\prime}_{i}$ are generators
of the same ray, they are positive multiples of each other. Thus,
$P_{\boldsymbol{w}^{\prime}}=P_{\boldsymbol{v}^{\prime}}=P_{\boldsymbol{v}}=P_{\boldsymbol{w}}$.
In light of this we can make the following definition.

######  Definition 3.3.

Let $\mathcal{C}_{\bullet}$ be a simplicial flag of cones. The _prime
congruence $P_{\mathcal{C}_{\bullet}}$ defined by $\mathcal{C}_{\bullet}$_ is
$P_{\boldsymbol{w}}$ for any choice of $\boldsymbol{w}=(w_{0},\ldots,w_{k})$
with $w_{i}\in\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$. If
$\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$, then
$P_{\mathcal{P}_{\bullet}}=P_{c(\mathcal{P}_{\bullet})}$ is the _prime
congruence defined by $\mathcal{P}_{\bullet}$_.

If $\mathcal{C}_{\bullet}=\mathcal{C}_{\bullet}(C)$ for some matrix $C$, then
we can chose $w_{i}$ to be the $(i+1)^{\text{st}}$ row of $C$, so $C$ is a
defining matrix for $P_{\mathcal{C}_{\bullet}}$. In particular, for every
prime congruence $P$ on $S[M]$, there is a simplicial flag
$\mathcal{C}_{\bullet}$ of cones such that $P=P_{\mathcal{C}_{\bullet}}$. If
$P\in\operatorname{Cont}_{S}S[M]$, then we know that
$\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$ for some flag
$\mathcal{P}_{\bullet}$ of polyhedra, so $P=P_{\mathcal{P}_{\bullet}}$.

###  3.2. Equivalence of flags.

Recall that a $\Gamma$-rational polyhedron is a polyhedron in $N_{\mathbb{R}}$
which can be written as $\\{x\in N_{\mathbb{R}}\ :\ \left\langle
x,u_{i}\right\rangle\leq\gamma_{i}\text{ for }i=1,\ldots,q\\}$ for some
$u_{1},\ldots,u_{q}\in M$ and $\gamma_{1},\ldots,\gamma_{q}\in\Gamma$. A
$\Gamma$-rational polyhedral set is a finite union of $\Gamma$-rational
polyhedra.

A cone in $\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ is _$\Gamma$ -admissible_
if it can be written as

$\\{(r,x)\in\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}\ :\
r\gamma_{i}+\left\langle x,u_{i}\right\rangle\leq 0\text{ for
}i=1,\ldots,q\\}$

for some $u_{1},\ldots,u_{q}\in M$ and
$\gamma_{1},\ldots,\gamma_{q}\in\Gamma$. A polyhedron $\mathcal{Q}$ is
$\Gamma$-rational if and only if $c(\mathcal{Q})$ is $\Gamma$-admissible. A
cone contained in $\\{0\\}\times N_{\mathbb{R}}$ is $\Gamma$-admissible if and
only if it is rational. We call a finite union of $\Gamma$-admissible cones a
_$\Gamma$ -admissible fan support set_. Note that the set of $\Gamma$-rational
polyhedral sets in $N_{\mathbb{R}}$ and the set of $\Gamma$-admissible fan
support sets each form a lattice when ordered by inclusion. In both of these
lattices, meet and join are given by intersection and union, respectively.

######  Definition 3.4.

Let $\mathcal{P}_{\bullet}$ be a flag of polyhedra and let
$\mathcal{C}_{\bullet}$ be a flag of cones. A $\Gamma$-rational neighborhood
of $\mathcal{P}_{\bullet}$ is a $\Gamma$-rational polyhedron $\mathcal{Q}$
which meets the relative interior of each $\mathcal{P}_{i}$. A _$\Gamma$
-admissible neighborhood_ of $\mathcal{C}_{\bullet}$ is a $\Gamma$-admissible
cone $\mathcal{D}$ which meets the relative interior of each
$\mathcal{C}_{i}$.

######  Definition 3.5.

We say that two flags $\mathcal{P}_{\bullet}$ and
$\mathcal{P}^{\prime}_{\bullet}$ of polyhedra are ($\Gamma$-)locally
equivalent if the $\Gamma$-rational neighborhoods of $\mathcal{P}_{\bullet}$
are exactly the $\Gamma$-rational neighborhoods of
$\mathcal{P}^{\prime}_{\bullet}$. Similarly, we say that a two flags
$\mathcal{C}_{\bullet}$ and $\mathcal{C}^{\prime}_{\bullet}$ are
($\Gamma$-)_locally equivalent_ if they have the same $\Gamma$-admissible
neighborhoods.

Note that $\mathcal{Q}$ is a $\Gamma$-rational neighborhood of
$\mathcal{P}_{\bullet}$ if and only if $c(\mathcal{Q})$ is a
$\Gamma$-admissible neighborhood of $c(\mathcal{P}_{\bullet})$. Thus
$\mathcal{P}_{\bullet}$ and $\mathcal{P}^{\prime}_{\bullet}$ are locally
equivalent if and only if $c(\mathcal{P}_{\bullet})$ and
$c(\mathcal{P}^{\prime}_{\bullet})$ are.

The following lemma provides an alternate characterization of neighborhoods of
flags. To prove it we will use the fact that a point $x$ is in the relative
interior of a cone $\mathcal{C}$ of dimension $k$ if and only if $x$ can be
written as a positive linear combination of $k$ linearly independent points in
$\mathcal{C}$. See [19, Lemmas 2.8 and 2.9] for the proof of the corresponding
fact about polytopes. The proof for cones is directly analogous.

######  Lemma 3.6.

A $\Gamma$-admissible cone $\mathcal{D}$ is a neighborhood of
$\mathcal{C}_{\bullet}$ if and only if $\mathcal{D}$ meets
$\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$ for $0\leq i\leq k$. In
particular, a $\Gamma$-rational polyhedron $\mathcal{Q}$ is a neighborhood of
$\mathcal{P}_{\bullet}=(\mathcal{P}_{0}\leq\mathcal{P}_{1}\leq\cdots\leq\mathcal{P}_{k})$
if and only if $\mathcal{P}_{0}\in\mathcal{Q}$ and $\mathcal{Q}$ meets
$\mathcal{P}_{i}\setminus\mathcal{P}_{i-1}$, for $1\leq i\leq k$.

###### Proof.

One direction follows from the fact that
$\operatorname{rel.int.}\mathcal{C}_{i}\subseteq\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$.
For the other direction, we will proceed by induction on $k$. The case $k=0$
is trivial because
$\operatorname{rel.int.}\mathcal{C}_{0}=\mathcal{C}_{0}\backslash\mathcal{C}_{-1}$.
Assume the statement holds for $k-1$. By the inductive hypothesis,
$\mathcal{D}$ meets the relative interior of $\mathcal{C}_{i}$ for $1\leq
i<k$, so we only need to show that $\mathcal{D}$ meets the relative interior
of $\mathcal{C}_{k}$. Pick a point
$x\in\mathcal{D}\cap\operatorname{rel.int.}\mathcal{C}_{k-1}$. So $x$ is a
positive linear combination of $k$ linearly independent points in
$\mathcal{C}_{k-1}$. Because $\mathcal{D}$ meets
$\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$, we can pick a point
$y\in\mathcal{D}\cap\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$. Then $x+y$ is
a positive linear combination of $k+1$ linearly independent points in
$\mathcal{C}_{k}$, so $x+y\in\operatorname{rel.int.}\mathcal{C}_{k}$. Since
$x,y\in\mathcal{D}$ and $\mathcal{D}$ is a cone, we get
$x+y\in\mathcal{D}\cap\operatorname{rel.int.}\mathcal{C}_{k}$.

The claim about polyhedra follows by considering
$\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$ and
$\mathcal{D}=c(\mathcal{Q})$ and noting that, because $\mathcal{C}_{0}$ is a
ray,
$\mathcal{C}_{0}\backslash\mathcal{C}_{-1}=\mathcal{C}_{0}\backslash\\{0\\}$
meets the cone $\mathcal{D}$ if and only if $\mathcal{C}_{0}\backslash\\{0\\}$
is contained in $\mathcal{D}$. ∎

We now work towards showing that every flag of cones is locally equivalent to
one that is simplicial. To do this, we will need a technical lemma.

######  Lemma 3.7.

Let $\mathcal{C}$ be a cone and let $\mathcal{B}$ be a facet of $\mathcal{C}$.
Let $\mathcal{B}^{\prime}$ be a subcone of $\mathcal{B}$ with such that
$\dim\mathcal{B}^{\prime}=\dim\mathcal{B}$ and let $\mathcal{C}^{\prime}$ be a
subcone of $\mathcal{C}$ with $\dim\mathcal{C}^{\prime}=\dim\mathcal{C}$ and
$\mathcal{B}^{\prime}$ a face of $\mathcal{C}^{\prime}$. Suppose that
$v\in\operatorname{rel.int.}\mathcal{B}^{\prime}$ and
$w\in\operatorname{rel.int.}\mathcal{C}$. Then there is a real number
$\epsilon>0$ such that $v+\epsilon w\in\mathcal{C}^{\prime}$.

###### Proof.

Since $\mathcal{B}$ is a face of $\mathcal{C}$, there is some $u\in
M_{\mathbb{R}}$ such that such that $\mathcal{C}\subseteq u^{\vee}$ and
$\mathcal{B}=\mathcal{C}\cap u^{\perp}$. In particular, $v\in u^{\perp}$ and
$w\in u^{\vee}$. Since $\mathcal{B}^{\prime}=\mathcal{C}^{\prime}\cap
u^{\perp}$ is a facet of $\mathcal{C}^{\prime}$ and
$v\in\operatorname{rel.int.}\mathcal{B}^{\prime}$, the star of
$\mathcal{C}^{\prime}$ at $v$ is $u^{\vee}$. That is,
$u^{\vee}=\mathbb{R}_{\geq 0}(\mathcal{C}^{\prime}-v)$. Since $w\in u^{\vee}$
is nonzero, this means that we can write $w=r(z-v)$ for some $r>0$ and
$z\in\mathcal{C}^{\prime}$. So, letting $\epsilon=\frac{1}{r}$, we have
$v+\epsilon w=v+(z-v)=z\in\mathcal{C}^{\prime}$. ∎

This lemma will help us with an inductive argument. The following definition
will also facilitate this argument.

######  Definition 3.8.

Let
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\mathcal{C}_{1}\leq\cdots\leq\mathcal{C}_{k})$
be a flag of cones. For any $j\leq k$, the truncation of
$\mathcal{C}_{\bullet}$ at $j$ is
$\mathcal{C}_{\bullet}^{(j)}=(\mathcal{C}_{0}\leq\mathcal{C}_{1}\leq\cdots\leq\mathcal{C}_{j})$.

######  Proposition 3.9.

Any flag $\mathcal{C}_{\bullet}$ of cones is locally equivalent to a
simplicial flag $\mathcal{C}_{\bullet}^{\prime}$ of cones. Any flag
$\mathcal{P}_{\bullet}$ of polyhedra is locally equivalent to a flag
$\mathcal{P}_{\bullet}^{\prime}$ of cones such that
$c(\mathcal{P}_{\bullet}^{\prime})$ is simplicial.

###### Proof.

Consider any flag of cones
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\cdots\leq\mathcal{C}_{k})$ and,
for each $0\leq i\leq k$ pick a ray $\rho_{i}$ of $\mathcal{C}_{i}$ not
contained in $\mathcal{C}_{i-1}$. Define a simplicial flag
$\mathcal{C}_{\bullet}^{\prime}$ of cones by setting
$\mathcal{C}_{i}^{\prime}=\displaystyle\sum_{j=0}^{i}\rho_{j}$. Note that
$\mathcal{C}_{i}^{\prime}$ is a subcone of $\mathcal{C}_{i}$ with
$\dim\mathcal{C}_{i}^{\prime}=i+1=\dim\mathcal{C}_{i}$ and
$\mathcal{C}_{i}^{\prime}\backslash\mathcal{C}_{i-1}^{\prime}\subseteq\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$.
In particular, any $\Gamma$-admissible neighborhood of
$\mathcal{C}_{\bullet}^{\prime}$ is a $\Gamma$-admissible neighborhood of
$\mathcal{C}_{\bullet}$.

We now prove that any $\Gamma$-admissible neighborhood of
$\mathcal{C}_{\bullet}$ is a $\Gamma$-admissible neighborhood of
$\mathcal{C}_{\bullet}^{\prime}$ by induction on $k$. For the base of $k=0$,
note that $\mathcal{C}_{0}^{\prime}=\mathcal{C}_{0}$, so we are done. Now
assume that $k\geq 1$ and the result is true for $k-1$. Suppose that
$\mathcal{D}$ is a $\Gamma$-admissible neighborhood of
$\mathcal{C}_{\bullet}$. Then $\mathcal{D}$ is a neighborhood of
$\mathcal{C}_{\bullet}^{(k-1)}$ so, by the inductive hypothesis, $\mathcal{D}$
is a neighborhood of $\mathcal{C}_{\bullet}^{\prime(k-1)}$. Thus $\mathcal{D}$
meets the relative interior of $\mathcal{C}_{i}^{\prime}$ for $i<k$, and it
suffices to show that $\mathcal{D}$ meets
$\mathcal{C}_{k}^{\prime}\backslash\mathcal{C}_{k-1}^{\prime}$. We can pick
$v\in(\operatorname{rel.int.}\mathcal{C}_{k-1}^{\prime})\cap\mathcal{D}$ and
$w\in(\operatorname{rel.int.}\mathcal{C}_{k})\cap\mathcal{D}$. By Lemma 3.7,
there is an $\epsilon>0$ such that $v+\epsilon w\in\mathcal{C}_{k}^{\prime}$.
Since $\epsilon w\in\operatorname{rel.int.}\mathcal{C}_{k}$ and
$\mathcal{C}_{k-1}$ is a proper face of $\mathcal{C}_{k}$,
$v\in\mathcal{C}_{k-1}$ gives us that $v+\epsilon
w\notin\mathcal{C}_{k-1}\supseteq\mathcal{C}_{k-1}^{\prime}$. Thus,
$v+\epsilon w\in\mathcal{C}_{k}^{\prime}\backslash\mathcal{C}_{k-1}^{\prime}$.
Since $v,w\in\mathcal{D}$ and $\mathcal{D}$ is a cone, $v+\epsilon
w\in\mathcal{D}$.

Given any flag $\mathcal{P}_{\bullet}$ of polyhedra, consider the
corresponding flag $\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$ of cones
and let $\mathcal{C}_{\bullet}^{\prime}$ be as above. Since
$\mathcal{C}_{0}^{\prime}=\mathcal{C}_{0}$ is not contained in $\\{0\\}\times
N_{\mathbb{R}}$, there is a flag $\mathcal{P}_{\bullet}^{\prime}$ of polyhedra
such that $\mathcal{C}_{\bullet}^{\prime}=c(\mathcal{P}_{\bullet}^{\prime})$.
Since $c(\mathcal{P}_{\bullet})$ and $c(\mathcal{P}_{\bullet}^{\prime})$ are
locally equivalent, so are $\mathcal{P}_{\bullet}$ and
$\mathcal{P}_{\bullet}^{\prime}$. ∎

###  3.3. The filter of a prime congruence.

Now we introduce the next key player in this paper. For finitely many Laurent
polynomials $f_{0},\ldots,f_{n}\in S[M]$ we define the rational set
$R(f_{0},\ldots,f_{n})$ to be

$R(f_{0},\ldots,f_{n})=\\{x\in N_{\mathbb{R}}\,:\,f_{0}(x)\geq f_{i}(x),\text{
for }1\leq i\leq n\\}.$

Here, when $f=\sum_{u\in M}f_{u}\chi^{u}\in S[M]$ and $x\in N_{\mathbb{R}}$,
we let $f(x)=\max_{u\in M}\left(\log(f_{u})+_{\mathbb{R}}\left\langle
x,u\right\rangle\right)$.

For any $f\in S[M]$, we can form the homogenized function $\widetilde{f}$
given by

$\widetilde{f}(r,x)=\max_{u\in M}\left(r\log(f_{u})+_{\mathbb{R}}\left\langle
x,u\right\rangle\right)$

for any $(r,x)\in\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$. We let

$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})=\left\\{w\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}\,:\,\widetilde{f}_{0}(w)\geq\widetilde{f}_{i}(w)\text{ for
}1\leq i\leq n\right\\}.$

Note that $R(f_{0},f_{1},\ldots,f_{n})$ is the restriction of
$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})$ to $\\{1\\}\times N_{\mathbb{R}}$.

######  Lemma 3.10.

Say $f_{0},\ldots,f_{n}\in S[M]$ and write $f_{i}=\sum_{u\in
M}f_{i,u}\chi^{u}$. Then

  1. (1)

$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})=\bigcap_{i=1}^{n}\widetilde{R}(f_{0},f_{i})$,

  2. (2)

$\widetilde{R}(f_{0},f_{i})=\bigcap_{u\in
M}\widetilde{R}(f_{0},f_{i,u}\chi^{u})=\bigcup_{u\in
M}\widetilde{R}(f_{0,u}\chi^{u},f_{i})$, and

  3. (3)

for any $\mu\in M$,
$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})\supseteq\widetilde{R}(f_{0,\mu}\chi^{\mu},f_{1},\ldots,f_{n})$.

The same applies if we replace $\widetilde{R}$ with $R$.

###### Proof.

(1) follows from the definition. (2) and (3) follow from the fact that, in the
evaluation of $\widetilde{f}(w)$, we take the maximum over the terms of $f$.
The final claim follows by restriction to $\\{1\\}\times N_{\mathbb{R}}$. ∎

######  Definition 3.11.

For any prime congruence $P$ on $S[M]$, the _prime filter
$\widetilde{\mathcal{F}}_{P}$ that $P$ defines on the lattice of
$\Gamma$-admissible fan support sets_ is the collection of $\Gamma$-admissible
fan support sets $\widetilde{U}$ for which there are
$f_{0},f_{1},\ldots,f_{n}\in S[M]$ such that
$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})\subseteq\widetilde{U}$ and
$|f_{0}|_{P}\geq|f_{i}|_{P}$ for $1\leq i\leq n$. If
$P\in\operatorname{Cont}_{S}{S[M]}$, then the _prime filter $\mathcal{F}_{P}$
that $P$ defines on the lattice of $\Gamma$-rational polyhedral sets_ is the
collection of $\Gamma$-rational polyhedral sets $U$ in $N_{\mathbb{R}}$ for
which there are $f_{0},f_{1},\ldots,f_{n}\in S[M]$ such that
$R(f_{0},f_{1},\ldots,f_{n})\subseteq U$ and $|f_{0}|_{P}\geq|f_{i}|_{P}$ for
$1\leq i\leq n$. In this case, a $\Gamma$-rational polyhedral set $U$ is in
$\mathcal{F}_{P}$ if and only if $c(U)$ is in $\widetilde{\mathcal{F}}_{P}$.

We will justify these names later in this section. We begin our study of
$\widetilde{\mathcal{F}}_{P}$ and $\mathcal{F}_{P}$ by observing that, in the
above definitions, it is enough to consider monomials.

######  Lemma 3.12.

Let $P$ be a prime congruence on $S[M]$ and $\widetilde{U}$ a
$\Gamma$-admissible fan support set. Then
$\widetilde{U}\in\widetilde{\mathcal{F}}_{P}$ if and only if there are terms
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})\subseteq\widetilde{U}$
and $1_{\kappa(P)}\geq|a_{i}\chi^{u_{i}}|_{P}$ for $1\leq i\leq n$.

If $P\in\operatorname{Cont}_{S}S[M]$ and $U$ is a $\Gamma$-rational polyhedral
set, then $U\in\mathcal{F}_{P}$ if and only if there are terms
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})\subseteq U$ and
$1_{\kappa(P)}\geq|a_{i}\chi^{u_{i}}|_{P}$ for $1\leq i\leq n$.

###### Proof.

The “if” direction is clear.

For the other direction, suppose
$\widetilde{U}\in\widetilde{\mathcal{F}}_{P}$, i.e., there are
$f_{0},f_{1},\ldots,f_{n}\in S[M]$ such that
$\widetilde{R}(f_{0},f_{1},\ldots,f_{n})\subseteq\widetilde{U}$ and
$|f_{0}|_{P}\geq|f_{i}|_{P}$ for $1\leq i\leq n$. Let $a_{0}\chi^{u_{0}}$ be a
term of $f_{0}$ such that the maximum in $|f_{0}|_{P}=\displaystyle\sum_{u\in
M}|f_{0,u}\chi^{u}|_{P}$ is attained at $|a_{0}\chi^{u_{0}}|_{P}$. Then we
have that $|a_{0}\chi^{u_{0}}|_{P}\geq|f_{i}|_{P}$ for $1\leq i\leq n$ and, by
Lemma 3.10 (3),
$\widetilde{R}(a_{0}\chi^{u_{0}},f_{1},\ldots,f_{n})\subseteq\widetilde{R}(f_{0},f_{1},\ldots,f_{n})\subseteq\widetilde{U}$.
Thus, we may assume without loss of generality that $f_{0}=a_{0}\chi^{u_{0}}$.
Also, $a_{0}\chi^{u_{0}}$ is a unit in $S[M]$ and
$|a_{0}\chi^{u_{0}}|_{P}\geq|f_{i}|_{P}$ gives us that
$1_{\kappa(P)}\geq|(a_{0}\chi^{u_{0}})^{-1}f_{i}|_{P}$. Since also
$\widetilde{R}(a_{0}\chi^{u_{0}},f_{1},\ldots,f_{n})=\widetilde{R}(1_{S},(a_{0}\chi^{u_{0}})^{-1}f_{1},\ldots,(a_{0}\chi^{u_{0}})^{-1}f_{n})$,
we may assume without loss of generality that $f_{0}=1_{S}$.

By Lemma 3.10 (1) and (2),
$\widetilde{R}(1_{S},f_{1},\ldots,f_{n})=\displaystyle\bigcap_{i=1}^{n}\widetilde{R}(1_{S},f_{i})=\displaystyle\bigcap_{i=1}^{n}\displaystyle\bigcap_{u\in
M}\widetilde{R}(1_{S},f_{i,u}\chi^{u})$. So, if we relabel those
$f_{i,u}\chi^{u}$ with $f_{i,u}\neq 0_{S}$ as
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}$, then we have
$\widetilde{R}(1_{S},f_{1},\ldots,f_{n})=\displaystyle\bigcap_{i=1}^{n}\widetilde{R}(1_{S},a_{i}\chi^{u_{i}})=\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$.

The proof of the statement for $\mathcal{F}_{P}$ is analogous. ∎

Having reduced to considering the sets
$R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ and
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$, we now
consider the geometric nature of these sets.

######  Lemma 3.13.

For any terms $a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$,
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ is a
$\Gamma$-admissible cone. If $P\in\operatorname{Cont}_{S}S[M]$ and
$1_{\kappa(P)}\geq|a_{i}\chi^{u_{i}}|_{P}$ for $1\leq i\leq n$ then
$R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ is a $\Gamma$-rational
polyhedron.

###### Proof.

Note that, for $1\leq i\leq n$,
$\tilde{R}(1_{S},a_{i}\chi^{u_{i}})=\\{(r,x)\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}\;:\;0_{\mathbb{R}}\geq r\log(a_{i})+_{\mathbb{R}}\left\langle
x,u\right\rangle\\}$ is a $\Gamma$-admissible half-space. So
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})=\displaystyle\bigcap_{i=1}^{n}\widetilde{R}(1_{S},a_{i}\chi^{u_{i}})$
is a $\Gamma$-admissible cone.

Now say $P\in\operatorname{Cont}_{S}S[M]$. Since
$R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ can be written as the
restriction of
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ to
$\\{1\\}\times N_{\mathbb{R}}$, it remains only to show that
$R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})$ is nonempty. By
Proposition 2.8, we can pick a defining matrix for $P$ of the form
$\begin{pmatrix}1&\xi_{0}\\\ 0&\xi_{1}\\\ \vdots&\vdots\\\
0&\xi_{k}\end{pmatrix}$ with $\xi_{0},\ldots,\xi_{k}$ in $N_{\mathbb{R}}$.
Since $1_{\kappa(P)}\geq|a_{i}\chi^{u_{i}}|_{P}$, we have that
$0\geq\log(a_{i})+_{\mathbb{R}}\left\langle\xi_{0},u_{i}\right\rangle$, i.e.,
$\xi_{0}\in R(1_{S},a_{i}\chi^{u_{i}})$. Therefore,
$\xi_{0}\in\displaystyle\bigcap_{i=1}^{n}R(1_{S},a_{i}\chi^{u_{i}})=R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}).$

∎

At the end of the proof of the previous lemma, we took advantage of the fact
that we could evaluate whether $1_{\kappa(P)}\geq|a\chi^{u}|_{P}$ by picking a
defining matrix for $P$ and using the lexicographic order. We now spell out
the definition of the lexicographic order as it applies here and work towards
translating it into geometric conditions. Let

$C=\begin{pmatrix}v_{0}\\\ v_{1}\\\ \vdots\\\ v_{k}\end{pmatrix}$

be a defining matrix for a prime congruence $P$ on $S[M]$. Let
$\mathfrak{u}=\begin{pmatrix}\log(a)\\\ u\end{pmatrix}$ be the exponent vector
of $a\chi^{u}$. Then $1_{S}\geq|a\chi^{u}|_{P}$ is equivalent to

$\begin{pmatrix}\left\langle v_{0},\mathfrak{u}\right\rangle\\\ \left\langle
v_{1},\mathfrak{u}\right\rangle\\\ \vdots\\\ \left\langle
v_{k},\mathfrak{u}\right\rangle\end{pmatrix}\leq_{lex}\begin{pmatrix}0\\\ 0\\\
\vdots\\\ 0\end{pmatrix},$

which happens whenever

$\displaystyle\left\langle v_{0},\mathfrak{u}\right\rangle$
$\displaystyle<0\text{, or }$ $\displaystyle\left\langle
v_{0},\mathfrak{u}\right\rangle$ $\displaystyle=0\text{ and }\left\langle
v_{1},\mathfrak{u}\right\rangle<0\text{, or }$ $\displaystyle\ \ \ \vdots$ (1)
$\displaystyle\left\langle v_{0},\mathfrak{u}\right\rangle$
$\displaystyle=0,\ldots,\left\langle
v_{k-2},\mathfrak{u}\right\rangle=0\text{, and }\left\langle
v_{k-1},\mathfrak{u}\right\rangle<0\text{, or }$ $\displaystyle\left\langle
v_{0},\mathfrak{u}\right\rangle$ $\displaystyle=0,\ldots,\left\langle
v_{k-1},\mathfrak{u}\right\rangle=0\text{, and }\left\langle
v_{k},\mathfrak{u}\right\rangle\leq 0.$

Equivalently,

$\displaystyle\left\langle v_{0},\mathfrak{u}\right\rangle$
$\displaystyle<0\text{, or }$ $\displaystyle\left\langle
v_{0},\mathfrak{u}\right\rangle$ $\displaystyle\leq 0\text{ and }\left\langle
v_{1},\mathfrak{u}\right\rangle<0\text{, or }$ $\displaystyle\ \ \ \vdots$ (2)
$\displaystyle\left\langle v_{0},\mathfrak{u}\right\rangle$ $\displaystyle\leq
0,\ldots,\left\langle v_{k-2},\mathfrak{u}\right\rangle\leq 0\text{, and
}\left\langle v_{k-1},\mathfrak{u}\right\rangle<0\text{, or }$
$\displaystyle\left\langle v_{0},\mathfrak{u}\right\rangle$ $\displaystyle\leq
0,\ldots,\left\langle v_{k-1},\mathfrak{u}\right\rangle\leq 0\text{, and
}\left\langle v_{k},\mathfrak{u}\right\rangle\leq 0.$

For convenience, we label the conditions in (3.3) as (3.3.0), (3.3.1),
$\ldots$, (3.3.$k-1$), and (3.3.$k$). Similarly, we label the conditions in
(3.3) as (3.3.0), (3.3.1), $\ldots$, (3.3.$k-1$), and (3.3.$k$).

######  Remark 3.14.

Suppose $C^{\prime}$ is obtained from $C$ by downwards gaussian elimination.
The proof of Lemma 2.7 not only shows that
$C\mathfrak{u}\leq_{lex}\boldsymbol{0}$ if and only if
$C^{\prime}\mathfrak{u}\leq_{lex}\boldsymbol{0}$, but that, for $0\leq i\leq
k$, (3.3.$i$) is satisfied for $C\mathfrak{u}\leq_{lex}\boldsymbol{0}$ if and
only if it is satisfied for $C^{\prime}\mathfrak{u}\leq_{lex}\boldsymbol{0}$.
Since (3.3.$i$) is satisfied exactly if (3.3.$j$) is satisfied for some $j\leq
i$, we also get that (3.3.$i$) is satisfied for
$C\mathfrak{u}\leq_{lex}\boldsymbol{0}$ if and only if it is satisfied for
$C^{\prime}\mathfrak{u}\leq_{lex}\boldsymbol{0}$.

The following proposition gives a geometric interpretation to the above
conditions. In order to state it, we define
$\widetilde{R}^{\circ}(f,g)=\\{w\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}:\widetilde{f}(w)>\widetilde{g}(w)\\}$. When we are considering
a half-space $\widetilde{R}(1_{S},a\chi^{u})$,
$\widetilde{R}^{\circ}(1_{S},a\chi^{u})$ is the corresponding strict half-
space.

######  Proposition 3.15.

Let $\mathcal{C}_{\bullet}$ be a simplicial flag of cones, and let
$P=P_{\mathcal{C}_{\bullet}}$. For any term $a\chi^{u}\in S[M]$,
$1_{\kappa(P)}\geq|a\chi^{u}|_{P}$ if and only if

$\displaystyle\operatorname{rel.int.}\mathcal{C}_{0}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or }$
$\displaystyle\operatorname{rel.int.}\mathcal{C}_{1}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or}$ $\displaystyle\qquad\qquad\quad\vdots$ (3)
$\displaystyle\operatorname{rel.int.}\mathcal{C}_{k-1}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or }$ $\displaystyle\mathcal{C}_{k}\subseteq\widetilde{R}(1_{S},a\chi^{u}).$

######  Remark 3.16.

As before, we label the conditions in (3.15) as (3.15.0), (3.15.1), $\ldots$,
(3.15.$k-1$), and (3.15.$k$). Note that, if (3.15.$i$) is satisfied, then, by
taking closures in $\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$, we get that
$\mathcal{C}_{i}\subseteq\widetilde{R}(1_{S},a\chi^{u})$ and so also
$\mathcal{C}_{j}\subseteq\widetilde{R}(1_{S},a\chi^{u})$ for all $j\leq i$. In
particular, if $1_{\kappa(P)}\geq|a\chi^{u}|_{P}$ then
$\mathcal{C}_{0}\subseteq\widetilde{R}(1_{S},a\chi^{u})$.

###### Proof of Proposition 3.15.

Fix a matrix $C=\begin{pmatrix}v_{0}\\\ v_{1}\\\ \vdots\\\ v_{k}\end{pmatrix}$
such that $\mathcal{C}_{\bullet}=\mathcal{C}_{\bullet}(C)$. Let $a\chi^{u}$ be
any term of $S[M]$ and let $\mathfrak{u}=\begin{pmatrix}\log(a)\\\
u\end{pmatrix}$ be the exponent vector of $a\chi^{u}$. Suppose that
$1_{\kappa(P)}\geq|a\chi^{u}|_{P}$, and, specifically, (3.3.$i$) is satisfied.
If $i=k$ then $v_{0},\ldots,v_{k}$ are contained in the half-space
$\widetilde{R}(1_{S},a\chi^{u})$, so the cone $\mathcal{C}_{k}$ that they
generate is also contained in $\widetilde{R}(1_{S},a\chi^{u})$, i.e.,
(3.15.$k$) is satisfied. So now suppose $i<k$, and consider any
$w\in\operatorname{rel.int.}\mathcal{C}_{i}$. Since $\mathcal{C}_{i}$ is the
simplicial cone generated by $v_{0},v_{1},\ldots,v_{i}$, we can write
$w=\displaystyle\sum_{j=0}^{i}\epsilon_{j}v_{j}$ with $\epsilon_{j}>0$. Since
we have $\left\langle v_{j},\mathfrak{u}\right\rangle\leq 0$ for $j<i$ and
$\left\langle v_{i},\mathfrak{u}\right\rangle<0$, we have

$\left\langle
w,\mathfrak{u}\right\rangle=\displaystyle\sum_{j=0}^{i}\epsilon_{j}\left\langle
v_{j},\mathfrak{u}\right\rangle\leq\epsilon_{i}\left\langle
v_{i},\mathfrak{u}\right\rangle<0,$

so $w\in\widetilde{R}^{\circ}(1_{S},a\chi^{u})$. Thus (3.15.$i$) is satisfied.

Now suppose that (3.15.$i$) is satisfied for some $0\leq i\leq k$. If $i=k$
then
$v_{0},\ldots,v_{k}\in\mathcal{C}_{k}\subseteq\widetilde{R}(1_{S},a\chi^{u})$,
so $\left\langle v_{j},\mathfrak{u}\right\rangle\leq 0$ for all $0\leq j\leq
k$, i.e., (3.3.$k$) is satisfied. So now suppose $i<k$, giving us
$\operatorname{rel.int.}\mathcal{C}_{i}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})$
and thus $\mathcal{C}_{i}\subseteq\widetilde{R}(1_{S},a\chi^{u})$. In
particular, for all $j\leq i$,
$v_{j}\in\mathcal{C}_{i}\subseteq\widetilde{R}(1_{S},a\chi^{u})$ gives us that
$\left\langle v_{j},\mathfrak{u}\right\rangle\leq 0$. Pick a point
$w\in\operatorname{rel.int.}\mathcal{C}_{i}$, so $\left\langle
w,\mathfrak{u}\right\rangle<0$. Since $w$ is in the relative interior of the
simplicial cone generated by $v_{0},\ldots,v_{i}$, we can write
$w=\displaystyle\sum_{j=0}^{i}\epsilon_{j}v_{j}$ with $\epsilon_{j}>0$. Since

$0>\left\langle
w,\mathfrak{u}\right\rangle=\displaystyle\sum_{j=0}^{i}\epsilon_{j}\left\langle
v_{j},\mathfrak{u}\right\rangle,$

we must have that $\left\langle v_{j},\mathfrak{u}\right\rangle<0$ for some
$j<i$. Then (3.3.$j$) is satisfied. ∎

With this background, we are now prepared to show that
$\widetilde{\mathcal{F}}_{P}$ and $\mathcal{F}_{P}$ are prime filters.

######  Theorem 3.17.

Let $P$ be a prime congruence on $S[M]$. Then $\widetilde{\mathcal{F}}_{P}$ is
a prime filter on the lattice of fan support sets. If
$P\in\operatorname{Cont}_{S}{S[M]}$, then $\mathcal{F}=\mathcal{F}_{P}$ is a
prime filter on the lattice of $\Gamma$-rational polyhedral sets.

###### Proof.

We need to check that $\widetilde{\mathcal{F}}=\widetilde{\mathcal{F}}_{P}$
satisfies the conditions in the definition of prime filter on a lattice,
namely, (1) $\widetilde{\mathcal{F}}$ is not the whole lattice, (2)
$\widetilde{\mathcal{F}}$ is not empty, (3) if
$\widetilde{U}_{1},\widetilde{U}_{2}\in\widetilde{\mathcal{F}}$ then
$\widetilde{U}_{1}\cup\widetilde{U}_{2}\in\widetilde{\mathcal{F}}$, (4) if
$\widetilde{U}\in\widetilde{\mathcal{F}}$ and
$\widetilde{V}\supseteq\widetilde{U}$ is a $\Gamma$-admissible fan support
set, then $\widetilde{V}\in\mathcal{F}_{P}$, and (5) if
$\widetilde{U}=\displaystyle\bigcup_{i=1}^{m}\widetilde{U}_{i}$ is in
$\widetilde{\mathcal{F}}$ and each $\widetilde{U}_{i}$ is a
$\Gamma$-admissible fan support set, then some $\widetilde{U}_{i_{0}}$ is in
$\widetilde{\mathcal{F}}$.

(1) Write $P=P_{\mathcal{C}_{\bullet}}$ for some simplicial flag
$\mathcal{C}_{\bullet}$ of cones. If $\widetilde{U}\in\widetilde{\mathcal{F}}$
then $\mathcal{C}_{0}\subseteq\widetilde{U}$. So, for any $\Gamma$-admissible
ray $\rho\neq\mathcal{C}_{0}$, $\rho\notin\widetilde{\mathcal{F}}$.

(2) We have $\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}=\widetilde{R}(1_{S},0_{S})$, so $\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}\in\widetilde{\mathcal{F}}$.

(3) Say $\widetilde{U}_{1},\widetilde{U}_{2}\in\widetilde{\mathcal{F}}$, so
$\widetilde{U}_{1}\supseteq\displaystyle\bigcap_{j\in
J_{1}}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})$ and
$\widetilde{U}_{2}\supseteq\displaystyle\bigcap_{j\in
J_{2}}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})$ for some finite indexing sets
$J_{1}$ and $J_{2}$. Then
$\widetilde{U}_{1}\cap\widetilde{U}_{2}\supseteq\displaystyle\bigcap_{j\in
J_{1}\cup J_{2}}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})$, so
$\widetilde{U}_{1}\cap\widetilde{U}_{2}\in\widetilde{\mathcal{F}}$.

(4) This is immediate from the definition of $\widetilde{\mathcal{F}}$.

(5) Suppose $\widetilde{U}=\cup_{i=1}^{m}\widetilde{U}_{i}$ is in
$\widetilde{\mathcal{F}}$ with each $\widetilde{U}_{i}$ a $\Gamma$-admissible
fan support set; we want to show that some $\widetilde{U}_{i}$ is in
$\widetilde{\mathcal{F}}$. By writing each $\widetilde{U}_{i}$ as a finite
union of $\Gamma$-admissible cones, we may assume without loss of generality
that each $\widetilde{U}_{i}$ is a $\Gamma$-admissible cone. Let $n_{0}=0$.
Note that we can write

$U_{i}=\bigcap_{l=n_{i-1}+1}^{n_{i}}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}}),$

for some increasing sequence of integers $n_{0}<n_{1}<\cdots<n_{m}$ and terms
$a_{l}\chi^{u_{l}}\in S[M]$. Since $U\in\widetilde{\mathcal{F}}$, there are
terms $c_{1}\chi^{u_{1}},\ldots,c_{q}\chi^{u_{q}}\in S[M]$ such that
$U\supseteq\cap_{j=1}^{q}\widetilde{R}(1_{S},c_{j}\chi^{\mu_{j}}),$ and
$1_{S}\geq|c_{j}\chi^{\mu_{j}}|_{P}$. For $1\leq l\leq n_{m}$, set

$b_{l}\chi^{\theta_{l}}=\begin{cases}a_{l}\chi^{u_{l}}&\text{ if
}1_{S}\geq|a_{l}\chi^{u_{l}}|_{P}\\\ a_{l}^{-1}\chi^{-u_{l}}&\text{
otherwise}\end{cases}.$

We have $1_{S}\geq|b_{l}\chi^{\theta_{l}}|_{P}$ for $1\leq l\leq n_{m}$, so
the cone

$\mathfrak{P}=\bigcap_{l=1}^{n_{m}}\widetilde{R}(1_{S},b_{l}\chi^{\theta_{l}})\cap\bigcap_{j=1}^{q}\widetilde{R}(1_{S},c_{j}\chi^{\mu_{j}})$

is in $\widetilde{\mathcal{F}}$. Also,
$\mathfrak{P}\subseteq\bigcap_{j=1}^{q}\widetilde{R}(1_{S},c_{j}\chi^{\mu_{j}})\subseteq\widetilde{U}.$
To finish the proof it is enough to see the following claim: for any $i$,
either $\mathfrak{P}\subseteq U_{i}$ or
$U_{i}\cap\operatorname{rel.int.}(\mathfrak{P})=\emptyset$. For, if this claim
holds, then the fact that
$\mathfrak{P}\subseteq\widetilde{U}=\displaystyle\bigcup_{i=1}^{m}\widetilde{U}_{i}$
implies that $\mathfrak{P}$ is contained in some $\widetilde{U}_{i_{0}}$.

To prove the claim, consider the linear hyperplane arrangement in
$\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ with hyperplanes

$\displaystyle\\{(r,\xi)\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}:r\log(a_{l})+\left<\xi,u_{l}\right>=0\\}\text{ for all }l\text{
and }$ $\displaystyle\\{(r,\xi)\in\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}:r\log(c_{j})+\left<\xi,\mu_{j}\right>=0\\}\text{ for all }j.$

This hyperplane arrangement defines a fan $\Sigma$ such that
$\mathfrak{P}\in\Sigma$ and the support of $\Sigma$ is $\mathbb{R}_{\geq
0}\times N_{\mathbb{R}}$. Moreover, for every $i$, $\widetilde{U}_{i}$ is the
support of a subfan $\Sigma_{i}$ of $\Sigma$. Since $\widetilde{U}_{i}$ is the
disjoint union of $\operatorname{rel.int.}(\mathfrak{D})$ for
$\mathfrak{D}\in\Sigma_{i}$ and $\mathfrak{P}\in\Sigma$, if
$\operatorname{rel.int.}(\mathfrak{P})$ meets $\widetilde{U}_{i}$, then
$\mathfrak{P}\in\Sigma_{i}$, implying that
$\mathfrak{P}\subseteq\widetilde{U}_{i}$.

The proof for $\mathcal{F}=\mathcal{F}_{P}$ is analogous. ∎

Theorem 3.17 immediately gives us the following corollary.

######  Corollary 3.18.

For prime congruences $P$ on $S[M]$, $\widetilde{\mathcal{F}}_{P}$ is
determined by which $\Gamma$-admissible cones it contains. For
$P\in\operatorname{Cont}_{S}S[M]$, $\mathcal{F}_{P}$ is determined by which
$\Gamma$-rational polyhedra it contains.

###  3.4. Flags and filters.

We now directly relate the filters $\mathcal{F}$ and $\widetilde{\mathcal{F}}$
back to flags of polyhedra and cones. We start with a lemma that will
facilitate our inductive argument.

######  Lemma 3.19.

Let $\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\cdots\leq\mathcal{C}_{k})$ be
a simplicial flag of cones, let $P=P_{\mathcal{C}_{\bullet}}$, and let
$\widetilde{\mathcal{F}}=\widetilde{\mathcal{F}}_{P}$. For any $0\leq j<k$,
consider the truncation $\mathcal{C}_{\bullet}^{(j)}$, let
$P^{\prime}=P_{\mathcal{C}_{\bullet}^{(j)}}$, and let
$\widetilde{\mathcal{F}}^{\prime}=\widetilde{\mathcal{F}}_{P^{\prime}}$. Then
$\widetilde{\mathcal{F}}\subseteq\widetilde{\mathcal{F}}^{\prime}$.

###### Proof.

It suffices to show the result for $j=k-1$; the general result follows by
induction.

Recall that Proposition 3.15 gives explicit geometric conditions for when
$1_{\kappa(P)}\geq|a\chi^{u}|_{P}$. For convenience, we now state the
corresponding conditions in the case where $P$ is replaced by $P^{\prime}$.
That is, $1_{\kappa(P^{\prime})}\geq|a\chi^{u}|_{P^{\prime}}$ if and only if

$\displaystyle\operatorname{rel.int.}\mathcal{C}_{0}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or }$
$\displaystyle\operatorname{rel.int.}\mathcal{C}_{1}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or}$ $\displaystyle\qquad\qquad\quad\vdots$ (4)
$\displaystyle\operatorname{rel.int.}\mathcal{C}_{k-2}\subseteq\widetilde{R}^{\circ}(1_{S},a\chi^{u})\text{,
or }$ $\displaystyle\mathcal{C}_{k-1}\subseteq\widetilde{R}(1_{S},a\chi^{u}).$

As before, we label the conditions in (3.4) as (3.4.0), (3.4.1), $\ldots$,
(3.4.$k-2$), and (3.4.$k-1$).

Suppose $\widetilde{U}\in\widetilde{\mathcal{F}}$, so there are terms
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$\widetilde{U}\supseteq\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$
and $1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$ for $1\leq l\leq n$. Note that,
to get that $\widetilde{U}\in\widetilde{\mathcal{F}}^{\prime}$, it suffices to
show that $1_{\kappa(P^{\prime})}\geq|a_{l}\chi^{u_{l}}|_{P^{\prime}}$ for
each $1\leq l\leq n$. Fix such an $l$. Since
$1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$, there is some $0\leq i_{l}\leq k$
such that $a_{l}\chi^{u_{l}}$ satisfies (3.15.$i_{l}$).

If $i_{l}=k$ then
$\mathcal{C}_{k-1}\subseteq\mathcal{C}_{k}\subseteq\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$,
so $a_{l}\chi^{u_{l}}$ satisfies (3.4.$k-1$), giving us
$1_{\kappa(P^{\prime})}\geq|a_{l}\chi^{u_{l}}|_{P^{\prime}}$. If $i_{l}=k-1$
then, by Remark 3.16,
$\mathcal{C}_{k-1}\subseteq\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$, i.e,
$a_{l}\chi^{u_{l}}$ satisfies (3.4.$k-1$), and so
$1_{\kappa(P^{\prime})}\geq|a_{l}\chi^{u_{l}}|_{P^{\prime}}$ again.

So now suppose $i_{l}<k-1$. Then (3.4.$i_{l}$) and (3.15.$i_{l}$) are the same
condition, so we get that
$1_{\kappa(P^{\prime})}\geq|a_{l}\chi^{u_{l}}|_{P^{\prime}}$. ∎

######  Theorem 3.20.

For any prime $P$ on $S[M]$, let
$\widetilde{\mathcal{F}}=\widetilde{\mathcal{F}}_{P}$ and pick a simplicial
flag $\mathcal{C}_{\bullet}$ of cones such that $P=P_{\mathcal{C}_{\bullet}}$.
Then, for any $\Gamma$-admissible cone $\widetilde{U}$ in $\mathbb{R}_{\geq
0}\times N_{\mathbb{R}}$, $\widetilde{U}\in\widetilde{\mathcal{F}}$ if and
only if $\widetilde{U}$ is a neighborhood of $\mathcal{C}_{\bullet}$.

If $P\in\operatorname{Cont}_{S}{S[M]}$ then we can write
$P=P_{\mathcal{P}_{\bullet}}$ for some flag $\mathcal{P}_{\bullet}$ of
polyhedra and consider $\mathcal{F}=\mathcal{F}_{P}$. For any
$\Gamma$-rational polyhedron $U$ in $N_{\mathbb{R}}$, $U\in\mathcal{F}$ if and
only if $U$ is a neighborhood of the flag $\mathcal{P}_{\bullet}$.

###### Proof.

We first consider $\widetilde{\mathcal{F}}$ and $\mathcal{C}_{\bullet}$.
Suppose that $\widetilde{U}$ is a neighborhood of $\mathcal{C}_{\bullet}$, so
for $0\leq i\leq k$ we can pick a point
$w_{i}\in\widetilde{U}\cap\operatorname{rel.int.}(\mathcal{C}_{i})$. The
matrix $C=\begin{pmatrix}w_{0}\\\ \vdots\\\ w_{k}\end{pmatrix}$ defines a
homomorphism $\varphi:S[M]\to\mathbb{T}^{(k+1)}$ and, by Definition 3.3,
$P=\ker\varphi$. Thus, for any $f,g\in S[M]$, $|f|_{P}\leq|g|_{P}$ if and only
if $\varphi(f)\leq\varphi(g)$. To get that
$\widetilde{U}\in\widetilde{\mathcal{F}}$, we need to show that there are
terms $a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$\widetilde{U}\supseteq\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})=\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$
and $1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$.

Since $\widetilde{U}$ is $\Gamma$-admissible, there are
$\mathfrak{u}_{1}=(\alpha_{1},u_{1}),\ldots,\mathfrak{u}_{n}=(\alpha_{n},u_{n})\in\Gamma\times
M$ such that
$\widetilde{U}=\\{\mathfrak{u}_{1},\ldots,\mathfrak{u}_{n}\\}^{\vee}=\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$,
where $a_{l}=t^{\alpha_{l}}\in S^{\times}$. Since $\mathfrak{u}_{l}$ is the
exponent vector of $a_{l}\chi^{u_{l}}$, we have that
$1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$ if and only if

$\begin{pmatrix}0\\\ \vdots\\\
0\end{pmatrix}\geq_{lex}C\mathfrak{u}_{l}=\begin{pmatrix}\left\langle
w_{0},\mathfrak{u}_{l}\right\rangle\\\ \vdots\\\ \left\langle
w_{k},\mathfrak{u}_{l}\right\rangle\end{pmatrix}.$

Since each
$w_{i}\in\widetilde{U}=\\{\mathfrak{u}_{1},\ldots,\mathfrak{u}_{n}\\}^{\vee}$,
we have that each $\left\langle w_{i},\mathfrak{u}_{l}\right\rangle$ is non-
positive, so we get that $1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$. Thus,
$\widetilde{U}\in\widetilde{\mathcal{F}}$.

For the other direction, suppose $\widetilde{U}\in\widetilde{\mathcal{F}}$. We
proceed to prove that $\widetilde{U}$ is a neighborhood of
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\cdots\leq\mathcal{C}_{k})$ by
induction on $k$. For the base case of $k=0$, applying Proposition 3.15 shows
that $\widetilde{U}\in\widetilde{\mathcal{F}}$ gives us that there are
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$\widetilde{U}\supseteq\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})\supseteq\mathcal{C}_{0}$,
so $\widetilde{U}$ is a neighborhood of
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0})$. For the inductive step, suppose
that $k\geq 1$ and the statement is true for $k-1$. Fix
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}\in S[M]$ such that
$\widetilde{U}\supseteq\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$
and $1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$. In particular,
$\mathfrak{P}:=\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})\in\widetilde{\mathcal{F}}$
and it suffices to show that $\mathfrak{P}$ is a neighborhood of
$\mathcal{C}_{\bullet}$. Consider the truncation
$\mathcal{C}_{\bullet}^{(k-1)}$ of $\mathcal{C}_{\bullet}$, let
$P^{\prime}=P_{\mathcal{C}_{\bullet}^{(k-1)}}$, and let
$\widetilde{\mathcal{F}}^{\prime}=\widetilde{\mathcal{F}}_{P^{\prime}}$. Lemma
3.19 tells us that $\mathfrak{P}\in\widetilde{\mathcal{F}}^{\prime}$ and so,
by the inductive hypothesis, $\mathfrak{P}$ is a neighborhood of
$\mathcal{C}_{\bullet}^{(k-1)}$. Since $\mathfrak{P}$ is a neighborhood of
$\mathcal{C}_{\bullet}$ if and only if it is a neighborhood of
$\mathcal{C}_{\bullet}^{k-1}$ and meets
$\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$, it now suffices to show that
$\mathfrak{P}$ meets $\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$.

For each $1\leq l\leq n$, $1_{\kappa(P)}\geq|a_{l}\chi^{u_{l}}|_{P}$, so
$a_{l}\chi^{u_{l}}$ satisfies (3.15.$i_{l}$) for some $0\leq i_{l}\leq k$. If
$i_{l}=k$ then
$\mathcal{C}_{k}\subseteq\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$, so

$\mathfrak{P}\cap(\mathcal{C}_{k}\backslash\mathcal{C}_{k-1})=\left(\displaystyle\bigcap_{\begin{subarray}{c}1\leq
j\leq n\\\ j\neq
l\end{subarray}}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})\right)\cap(\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}).$

Hence, it suffices to show that
$\displaystyle\bigcap_{\begin{subarray}{c}1\leq j\leq n\\\ j\neq
l\end{subarray}}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})$ meets
$\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$. Thus, we may assume without loss
of generality that $0\leq i_{l}\leq k-1$ for all $l$, so
$\operatorname{rel.int.}\mathcal{C}_{i_{l}}\subseteq\widetilde{R}^{\circ}(1_{S},a_{l}\chi^{u_{l}})$.

Since $\mathfrak{P}$ is a neighborhood of $\mathcal{C}_{\bullet}^{(k-1)}$ and
$0\leq i_{l}\leq k-1$, we can pick a point
$w_{l}\in\operatorname{rel.int.}(\mathcal{C}_{i_{l}})\cap\mathfrak{P}$, so, in
particular,
$w_{l}\in\operatorname{rel.int.}(\mathcal{C}_{i_{l}})\subseteq\widetilde{R}^{\circ}(1_{S},a_{l}\chi^{u_{l}})$.
Also, $w_{l}\in\mathcal{C}_{k}\cap\mathfrak{P}$. Fix
$v\in\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$. Since $w_{l}$ is in the
interior of $\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$, there is some
$\epsilon_{l}>0$ such that
$w_{l}+[0,\varepsilon_{l}]v\subseteq\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$.
Letting $\epsilon=\min_{l}\epsilon_{l}$, we get that $w_{l}+\epsilon
v\in\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$ for all $l$. Also, recall that
each
$w_{l}\in\mathfrak{P}=\displaystyle\bigcap_{j=1}^{n}\widetilde{R}(1_{S},a_{j}\chi^{u_{j}})$.

Consider the point $\omega=\epsilon v+\displaystyle\sum_{l=1}^{n}w_{l}$. If we
fix $l$ then we can write $\omega=(w_{l}+\epsilon v)+\displaystyle\sum_{j\neq
l}w_{j}$ with each of $w_{l}+\varepsilon v$ and $w_{j}$ in the linear half-
space $\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$, so
$\omega\in\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$. Since this is true for all
$l$, we conclude that
$\omega\in\displaystyle\bigcap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})=\mathfrak{P}$.
Since $\displaystyle\sum_{l=1}^{n}w_{l}\in\mathcal{C}_{k}$ and $\varepsilon
v\in\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$, we also have
$\omega\in\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$. So $\mathfrak{P}$ meets
$\mathcal{C}_{k}\backslash\mathcal{C}_{k-1}$, concluding the proof that
$\widetilde{U}\in\widetilde{\mathcal{F}}$ if and only if $\widetilde{U}$ is a
neighborhood of $\mathcal{C}_{\bullet}$.

Now suppose $P\in\operatorname{Cont}_{S}S[M]$ and consider
$\mathcal{P}_{\bullet}$ and $\mathcal{F}$ as in the statement of the theorem.
Letting $\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$, we have
$P=P_{\mathcal{C}_{\bullet}}$. So, using the earlier part of the theorem, we
have

$\displaystyle U\in\mathcal{F}$ $\displaystyle\iff
c(U)\in\widetilde{\mathcal{F}}$ $\displaystyle\iff c(U)\text{ is a
neighborhood of }\mathcal{C}_{\bullet}=c(\mathcal{P}_{\bullet})$
$\displaystyle\iff U\text{ is a neighborhood of }\mathcal{P}_{\bullet}.$

∎

######  Corollary 3.21.

Consider simplicial flags $\mathcal{C}_{\bullet}$ and
$\mathcal{C}_{\bullet}^{\prime}$ of cones and let
$P=P_{\mathcal{C}_{\bullet}}$ and
$P^{\prime}=P_{\mathcal{C}_{\bullet}^{\prime}}$. Then
$\widetilde{\mathcal{F}}_{P}=\widetilde{\mathcal{F}}_{P^{\prime}}$ if and only
if $\mathcal{C}_{\bullet}$ and $\mathcal{C}_{\bullet}^{\prime}$ are locally
equivalent.

If $P=P_{\mathcal{P}_{\bullet}}$ and
$P^{\prime}=P_{\mathcal{P}_{\bullet}^{\prime}}$ for some flags
$\mathcal{P}_{\bullet}$ and $\mathcal{P}_{\bullet}^{\prime}$ of polyhedra,
then $\mathcal{F}_{P}=\mathcal{F}_{P^{\prime}}$ if and only if
$\mathcal{P}_{\bullet}$ and $\mathcal{P}_{\bullet}^{\prime}$ are locally
equivalent.

###### Proof.

This is immediate from Corollary 3.18 and Theorem 3.20. ∎

Theorem 3.20 allows us to consider examples of primes in
$\operatorname{Cont}_{S}S[M]$ and understand their filters via flags
$\mathcal{P}_{\bullet}$ of polyhedra.

######  Example 3.22.

Let $S=\mathbb{Q}_{\mathrm{max}}$, $\Gamma=\mathbb{Q}$, and
$M=\mathbb{Z}^{2}$.

Let $C=\begin{pmatrix}1&0&0\end{pmatrix}$,
$C^{\prime}=\begin{pmatrix}1&\sqrt{2}&0\end{pmatrix}$, and
$C^{\prime\prime}=\begin{pmatrix}1&\sqrt{2}&\sqrt{3}\end{pmatrix}$, and let
$\mathcal{P}_{\bullet}$, $\mathcal{P}_{\bullet}^{\prime}$, and
$\mathcal{P}_{\bullet}^{\prime\prime}$ be the corresponding flags of polyhedra
in $N_{\mathbb{R}}=\mathbb{R}^{2}$. Let $P,P^{\prime},$ and $P^{\prime\prime}$
be the corresponding prime congruences in
$\operatorname{Cont}_{\mathbb{Q}_{\mathrm{max}}}\mathbb{Q}_{\mathrm{max}}[x^{\pm
1},y^{\pm 1}]$, and let $\mathcal{F},\mathcal{F}^{\prime}$, and
$\mathcal{F}^{\prime\prime}$ be the corresponding prime filters on the lattice
of $\Gamma$-rational polyhedral sets. Each of $\mathcal{P}_{\bullet}$,
$\mathcal{P}_{\bullet}^{\prime}$, and $\mathcal{P}_{\bullet}^{\prime\prime}$
consist of a single point whose coordinates are given by the second and third
entries of the matrix. However, the corresponding filters look different.

The filter $\mathcal{F}$ consists of the zero-dimensional polyhedron
$\\{(0,0)\\}$ and every rational polyhedral set containing it. On the other
hand, $\mathcal{F}^{\prime}$ contains no zero-dimensional polyhedron: any
rational polyhedron in $\mathbb{R}^{2}$ containing $(\sqrt{2},0)$ must contain
an open interval on the $x$-axis. Similarly, $\mathcal{F}^{\prime\prime}$
consists only of two-dimensional polyhedral sets.  $\diamondsuit$

######  Example 3.23.

Still in the case where $S=\mathbb{Q}_{\mathrm{max}}$ and $M=\mathbb{Z}^{2}$,
we now consider the matrices $C=\begin{pmatrix}1&\sqrt{2}&0\\\
0&1&\sqrt{3}\end{pmatrix}$ and $C^{\prime}=\begin{pmatrix}1&\sqrt{2}&0\\\
0&0&1\end{pmatrix}$ and let $\mathcal{P}_{\bullet}$ and
$\mathcal{P}_{\bullet}^{\prime}$ be the corresponding flags of polyhedra,
which are depicted in Figure 1. Then the corresponding filters $\mathcal{F}$
and $\mathcal{F}^{\prime}$ are the same: they both consist of those rational
polyhedral sets which contain $(r_{1},r_{2})\times[0,r_{3})$ for some
$r_{1},r_{2},r_{3}\in\mathbb{R}$ with $r_{1}<\sqrt{2}<r_{2}$ and $r_{3}>0$.
One can see this visually depicted in Figures 2 and 3. We will show in
Corollary 3.36 that this implies that $C$ and $C^{\prime}$ define the same
point
$P\in\operatorname{Cont}_{\mathbb{Q}_{\mathrm{max}}}\mathbb{Q}_{\mathrm{max}}[x^{\pm
1},y^{\pm 1}]$. One can also check this by hand, by verifying that they give
the same preorder on $\operatorname{Terms}(\mathbb{Q}_{\mathrm{max}}[x^{\pm
1},y^{\pm 1}])$, even though neither of $C,C^{\prime}$ can be obtained from
the other by downward gaussian elimination as in Lemma 2.7.  $\diamondsuit$

Figure 1. The flags $\mathcal{P}_{\bullet}$ and
$\mathcal{P}_{\bullet}^{\prime}$ of Example 3.23. Figure 2. This picture
illustrates the filters corresponding to $\mathcal{P}$ and
$\mathcal{P}^{\prime}$ by drawing the “infinitesimal neighborhoods” that a
rational polyhedron must contain to be a neighborhood of the flags. The
turquoise represents the condition to be a neighborhood of the truncation
$\mathcal{P}_{\bullet}^{(0)}=(\mathcal{P}_{\bullet}^{\prime})^{(0)}$; the
purple represents the added conditions to be neighborhoods of
$\mathcal{P}_{\bullet}$ and $\mathcal{P}_{\bullet}^{\prime}$. Figure 3. We see
that the filters are equal, so the flags $\mathcal{P}_{\bullet}$ and
$\mathcal{P}_{\bullet}^{\prime}$ are locally equivalent.

######  Example 3.24.

Consider the prime congruences on $\mathbb{Q}_{\mathrm{max}}[x^{\pm 1},y^{\pm
1}]$ given by the matrices
$C_{1}=\begin{pmatrix}1&\sqrt{2}&\sqrt{3}\end{pmatrix}$,
$C_{2}=\begin{pmatrix}1&0&0\\\ 0&1&0\end{pmatrix}$,
$C_{3}=\begin{pmatrix}1&0&0\\\ 0&1&\sqrt{2}\end{pmatrix}$, and
$C_{4}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\end{pmatrix}$. The corresponding
flags of polyhedra and filters are pictured in Figure 4.

Figure 4. The flags of polyhedra and filters from Example 3.24. In these
pictures, turquoise represents “one-dimensional infinitesimal neighborhoods”
and purple represents “two-dimensional infinitesimal neighborhoods”.

$\diamondsuit$

######  Remark 3.25.

We can formalize the notion of the dimension of the “infinitesimal
neighborhoods” in the figures above by considering the smallest possible
dimension of a polyhedron in $\mathcal{F}_{P}$. See Proposition 3.43 for a way
to compute this dimension as an algebraic invariant of $P$.

###  3.5. From filters back to congruences.

We show that the points of $\operatorname{Cont}_{S}{S[M]}$ are in bijection
with prime filters on the lattice of $\Gamma$-rational polyhedral sets, where
the filter contains some polytope.

######  Remark 3.26.

One may hope that there is a geometrically meaningful bijection between the
points of $\operatorname{Cont}_{S}{S[M]}$ and prime filters on the lattice of
$\Gamma$-rational polyhedral sets. To see why this is not true, consider the
following example:

$\displaystyle N_{\mathbb{R}}=\mathbb{R}^{2},\text{ and
}\sigma=\\{(0,x)\in\mathbb{R}^{2}:x\geq 0\\}$
$\displaystyle\mathcal{F}=\\{U:\Gamma\text{-rational polyhedral sets, with
}U\supseteq\sigma+(0,y)\text{ for some }y\in\mathbb{R}.\\}$

One can verify that $\mathcal{F}$ is a prime filter, however, it can only
correspond to some point at infinity on the positive $y$-axis, and there is no
such point in $\operatorname{Cont}_{S}{S[M]}$.

The following lemma ensures that the prime filter $\mathcal{F}$ of Remark 3.26
does not occur as $\mathcal{F}_{P}$ for any $P\in\operatorname{Cont}_{S}S[M]$.

######  Lemma 3.27.

For any $P\in\operatorname{Cont}_{S}S[M]$, $\mathcal{F}_{P}$ contains a
polytope.

###### Proof.

Write $P=P_{\mathcal{P}_{\bullet}}$ for some flag $\mathcal{P}_{\bullet}$ of
polyhedra. Let $U$ be any full-dimensional $\Gamma$-rational polytope with the
point $\mathcal{P}_{0}$ in its interior. Then $U$ meets the relative interior
of any polyhedron containing $\mathcal{P}_{0}$, so $U$ is a neighborhood of
$\mathcal{P}_{\bullet}$. By Theorem 3.20, $U\in\mathcal{F}_{P}$. ∎

######  Definition 3.28.

A $\Gamma$-rational polytopal set is a finite union of $\Gamma$-rational
polytopes.

The set of $\Gamma$-rational polytopal sets in $N_{\mathbb{R}}$ forms a
lattice when ordered by inclusion. In this lattice, meet and join are given by
intersection and union, respectively.

######  Proposition 3.29.

There is a bijection between the set of prime filters on the lattice of
$\Gamma$-rational polyhedral sets, where the filter contains some polytope and
the set of prime filters on the lattice of $\Gamma$-rational polytopal sets.

###### Proof.

Given a prime filter $\mathcal{F}$ on the lattice of $\Gamma$-rational
polytopal sets, it generates a filter $\mathcal{F}^{\prime}$ on the lattice of
$\Gamma$-rational polyhedral sets such that $\mathcal{F}^{\prime}$ contains a
polytope, and a polytopal set is in $\mathcal{F}$ if and only if it is in
$\mathcal{F}^{\prime}$. Moreover, $\mathcal{F}^{\prime}$ is also prime.
Indeed, say $U=U_{1}\cup U_{2}\in\mathcal{F}^{\prime}$ with $U_{1}$ and
$U_{2}$ both $\Gamma$-rational. Letting $V\in\mathcal{F}^{\prime}$ be a
polytope we have $U\cap V=(U_{1}\cap V)\cup(U_{2}\cap V)$ with $U\cap V$,
$U_{1}\cap V$, and $U_{2}\cap V$ polytopal and $U\cap V\in\mathcal{F}$. So
either, $U_{1}\cap V\in\mathcal{F}$ or $U_{2}\cap V\in\mathcal{F}$, giving us
that $U_{1}\in\mathcal{F}^{\prime}$ or $U_{2}\in\mathcal{F}^{\prime}$.

It remains only to show that if $\mathcal{F}^{\prime}$ is a filter on the
lattice of $\Gamma$-rational polyhedral sets such that $\mathcal{F}^{\prime}$
contains a polytope, then the restriction of $\mathcal{F}^{\prime}$ to the
lattice of $\Gamma$-rational polytopal sets is a prime filter $\mathcal{F}$.
We check that each of the five conditions for a prime filter is satisfied.

(1) Since every $\Gamma$-rational polyhedron contains a $\Gamma$-rational
polytope and $\mathcal{F}^{\prime}$ is not all of the lattice of
$\Gamma$-rational polyhedral sets, $\mathcal{F}$ is not all of the lattice of
$\Gamma$-rational polytopal sets.

(2) By hypothesis, $\mathcal{F}$ is nonempty.

(3) If $U_{1},U_{2}\in\mathcal{F}$, then $U_{1},U_{2}\in\mathcal{F}^{\prime}$
and $U_{1},U_{2}$ are polytopal, so $U_{1}\cap U_{2}\in\mathcal{F}^{\prime}$
and $U_{1}\cap U_{2}$ is polytopal, implying that $U_{1}\cap
U_{2}\in\mathcal{F}$.

(4) If $U\in\mathcal{F}$ and $V$ is a $\Gamma$-rational polytopal set which
contains $U$, then $V\supseteq U\in\mathcal{F}^{\prime}$, implying that
$V\in\mathcal{F}^{\prime}$. Since $V$ is polytopal this gives
$V\in\mathcal{F}$.

(5) If $U=\cup_{i=1}^{m}U_{i}\in\mathcal{F}$ and each $U_{i}$ is a
$\Gamma$-rational polytopal set, then $U\in\mathcal{F}^{\prime}$ with each
$U_{i}$ a $\Gamma$-rational polyhedral set, so some
$U_{i_{0}}\in\mathcal{F}^{\prime}$. Since $U_{i_{0}}$ is polytopal, we have
$U_{i_{0}}\in\mathcal{F}$. ∎

Given a prime filter $\widetilde{\mathcal{F}}$ on the lattice of
$\Gamma$-admissible fan support sets, we define a relation
$\leq_{\widetilde{\mathcal{F}}}$ on $\operatorname{Terms}(S[M])$ by saying
that $a\chi^{u}\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$ if there is some
$\widetilde{U}\in\widetilde{\mathcal{F}}$ such that $\left\langle
w,\begin{pmatrix}\log(a)\\\ u\end{pmatrix}\right\rangle\leq\left\langle
w,\begin{pmatrix}\log(b)\\\ \lambda\end{pmatrix}\right\rangle$ for all
$w\in\widetilde{U}$, i.e.,
$\widetilde{U}\subseteq\widetilde{R}(b\chi^{\lambda},a\chi^{u})$. Similarly,
if $\mathcal{F}$ is a filter on the lattice of $\Gamma$-rational polyhedral
sets, we have the relation $\leq_{\mathcal{F}}$ on
$\operatorname{Terms}(S[M])$ defined by
$a\chi^{u}\leq_{\mathcal{F}}b\chi^{\lambda}$ if there is a $U\in\mathcal{F}$
such that $\log(a)+\left<\xi,u\right>\leq\log(b)+\left<\xi,\lambda\right>$ for
all $\xi\in U$, that is, $U\subseteq R(b\chi^{\lambda},a\chi^{u})$. Note that,
if $\widetilde{\mathcal{F}}=\\{c(U)\;:\;U\in\mathcal{F}\\}$, then
$\leq_{\widetilde{\mathcal{F}}}$ and $\leq_{\mathcal{F}}$ are the same.

Since $\widetilde{\mathcal{F}}$ is a filter, we have that
$a\chi^{u}\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$ if and only
$\widetilde{R}(b\chi^{\lambda},a\chi^{u})\in\widetilde{\mathcal{F}}$.
Similarly, $a\chi^{u}\leq_{\mathcal{F}}b\chi^{\lambda}$ if and only if
$R(b\chi^{\lambda},a\chi^{u})\in\mathcal{F}$.

######  Proposition 3.30.

For any prime filter $\widetilde{\mathcal{F}}$ on the lattice of
$\Gamma$-admissible fan support sets, $\leq_{\widetilde{\mathcal{F}}}$ is
$\leq_{P}$ for some prime congruence $P$ on $S[M]$. For any prime filter
$\mathcal{F}$ on the lattice of $\Gamma$-rational polyhedral sets such that
$\mathcal{F}$ contains a polytope, the relation $\leq_{\mathcal{F}}$ is
$\leq_{P}$ for some $P\in\operatorname{Cont}_{S}{S[M]}$.

###### Proof.

We begin by verifying that $\leq_{\widetilde{\mathcal{F}}}$ satisfies
conditions (1), (2), and (3) in Proposition 2.9.

(1) ($\leq_{\widetilde{\mathcal{F}}}$ is a total preorder) The relation is
clearly reflexive. We need to show transitivity. Suppose
$a\chi^{u}\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$ and
$b\chi^{\lambda}\leq_{\widetilde{\mathcal{F}}}c\chi^{\mu}$, so there are
$\widetilde{U}_{1},\widetilde{U}_{2}\in\mathcal{F}$ such that

$\displaystyle\left\langle w,\begin{pmatrix}\log(a)\\\
u\end{pmatrix}\right\rangle\leq\left\langle w,\begin{pmatrix}\log(b)\\\
\lambda\end{pmatrix}\right\rangle\text{ for all }w\in\widetilde{U}_{1}\text{
and }$ $\displaystyle\left\langle w,\begin{pmatrix}\log(b)\\\
\lambda\end{pmatrix}\right\rangle\leq\left\langle w,\begin{pmatrix}\log(c)\\\
\mu\end{pmatrix}\right\rangle\text{ for all }w\in\widetilde{U}_{2}.$ Then
$U_{1}\cap U_{2}\in\mathcal{F}$ and $\displaystyle\left\langle
w,\begin{pmatrix}\log(a)\\\ u\end{pmatrix}\right\rangle\leq\left\langle
w,\begin{pmatrix}\log(c)\\\ \mu\end{pmatrix}\right\rangle\text{ for all
}w\in\widetilde{U}_{1}\cap\widetilde{U}_{2}$

so $a\chi^{u}\leq_{\widetilde{\mathcal{F}}}c\chi^{\mu}$. Thus,
$\leq_{\widetilde{\mathcal{F}}}$ is a preorder. To see that it is a total
preorder, suppose that
$a\chi^{u}\not\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$. Then for every
$\widetilde{U}\in\widetilde{\mathcal{F}}$,
$\widetilde{U}\not\subseteq\widetilde{R}(b\chi^{\lambda},a\chi^{u})$. We now
consider two cases: either there is some
$\widetilde{U}\in\widetilde{\mathcal{F}}$ such that
$\widetilde{U}\subseteq\widetilde{R}(a\chi^{u},b\chi^{\lambda})$ or, for all
$\widetilde{U}\in\widetilde{\mathcal{F}}$ both
$\widetilde{U}\cap\widetilde{R}(a\chi^{u},b\chi^{\lambda})$ and
$\widetilde{U}\cap R(b\chi^{\lambda},a\chi^{u})$ are non-empty. In the first
case $a\chi^{u}\geq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$ and we are done.
In the second case, fix any $\widetilde{U}\in\widetilde{\mathcal{F}}$. Since

$\widetilde{U}=(\widetilde{U}\cap\widetilde{R}(a\chi^{u},b\chi^{\lambda}))\cup(\widetilde{U}\cap\widetilde{R}(b\chi^{\lambda},a\chi^{u})),$

we get that either
$\widetilde{U}\cap\widetilde{R}(a\chi^{u},b\chi^{\lambda})\in\widetilde{\mathcal{F}}$
or
$\widetilde{U}\cap\widetilde{R}(b\chi^{\lambda},a\chi^{u})\in\widetilde{\mathcal{F}}$,
because $\widetilde{\mathcal{F}}$ is prime. However, since all
$\widetilde{V}\in\widetilde{\mathcal{F}}$ are not contained in
$\widetilde{R}(b\chi^{\lambda},a\chi^{u})$, we must have
$\widetilde{U}\cap\widetilde{R}(a\chi^{u},b\chi^{\lambda})\in\widetilde{\mathcal{F}}$.
Thus, we have $a\chi^{u}\geq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$.

(2) ($\leq_{\widetilde{\mathcal{F}}}$ is multiplicative) Let
$a\chi^{u},b\chi^{\lambda}$ and $c\chi^{\mu}$ be terms in $S[M]$, with
$a\chi^{u}\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}$. Then there exists
$\widetilde{U}\in\widetilde{\mathcal{F}}$ such that
$\widetilde{U}\subseteq\widetilde{R}(b\chi^{\lambda},a\chi^{u})=\widetilde{R}(b\chi^{\lambda}c\chi^{\mu},a\chi^{u}c\chi^{\mu})$,
so
$a\chi^{u}c\chi^{\mu}\leq_{\widetilde{\mathcal{F}}}b\chi^{\lambda}c\chi^{\mu}$.

(3) ($\leq_{\widetilde{\mathcal{F}}}$ respects the order on $S$) Let $a,b\in
S^{\times}$ such that $a\leq b$. Then every $\widetilde{U}\in\mathcal{F}$ is
in $\widetilde{R}(b,a)=\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$.

Thus, $\leq_{\widetilde{\mathcal{F}}}$ is $\leq_{P}$ for some prime congruence
$P$ on $S[M]$.

Now consider $\leq_{\mathcal{F}}$. Applying the above to
$\widetilde{\mathcal{F}}=\\{c(U)\;:\;U\in\mathcal{F}\\}$, we get that
$\leq_{\mathcal{F}}$, which is the same as $\leq_{\widetilde{\mathcal{F}}}$,
satisfies conditions (1), (2), and (3) in Proposition 2.9. So it suffices to
show that $\leq_{\mathcal{F}}$ satisfies condition (4) from that proposition.

Consider a term $a\chi^{u}\in S[M]$. Fixing a polytope $U\in\mathcal{F}$, the
set $\\{\log(a)+\left<\xi,u\right>\;:\;\xi\in U\\}$ has a minimum value
$\gamma\in\mathbb{R}$. Pick $\beta<\gamma$ in $\Gamma$ and set $b=t^{\beta}\in
S^{\times}$. Then $b<_{\mathcal{F}}a\chi^{u}$. ∎

In light of Proposition 3.30, and because a prime congruence $P$ on $S[M]$ is
determined by the relation $\leq_{P}$ on $\operatorname{Terms}(S[M])$, we can
make the following definitions.

######  Definition 3.31.

Given a prime filter $\widetilde{\mathcal{F}}$ on the lattice of
$\Gamma$-admissible fan support sets, the _prime congruence
$P_{\widetilde{\mathcal{F}}}$ defined by $\widetilde{\mathcal{F}}$_ is the
unique prime congruence $P$ on $S[M]$ such that
$\leq_{\widetilde{\mathcal{F}}}$ equals $\leq_{P}$. For any prime filter
$\mathcal{F}$ on the lattice of $\Gamma$-rational polyhedral sets such that
$\mathcal{F}$ contains a polytope, the _prime congruence $P_{\mathcal{F}}$
defined by $\mathcal{F}$_ is the unique $P\in\operatorname{Cont}_{S}S[M]$ such
that $\leq_{\mathcal{F}}$ equals $\leq_{P}$.

######  Lemma 3.32.

Consider elements $a,a_{1},a_{2},\ldots,a_{n}\in S^{\times}$ and
$u,u_{1},u_{2},\ldots,u_{n}\in M$, and suppose that
$\displaystyle\bigcap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})$ is nonempty. Then
$\displaystyle\bigcap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})\subseteq
R(1_{S},a\chi^{u})$ if and only if there are
$m_{1},m_{2},\ldots,m_{n}\in\mathbb{Z}_{\geq 0}$, $m\in\mathbb{Z}_{>0}$, and
$b\in S$ with $b\geq 1_{S}$ such that
$b(a\chi^{u})^{m}=\displaystyle\prod_{l=1}^{n}(a_{l}\chi^{u_{l}})^{m_{l}}$.

###### Proof.

Note that
$b(a\chi^{u})^{m}=\displaystyle\prod_{l=1}^{n}(a_{l}\chi^{u_{l}})^{m_{l}}$ is
equivalent to having

$\log b+m\log a=\sum_{l=1}^{n}m_{l}\log a_{l}\text{ and
}mu=\sum_{l=1}^{n}m_{l}u_{l},$

which is in turn equivalent to the existence of nonnegative rational numbers
$r_{1},\ldots,r_{n}$, such that

$\log a\leq\sum_{l=1}^{n}r_{l}\log a_{l}\text{ and
}u=\sum_{l=1}^{n}r_{l}u_{l}.$

By [5, Proposition 2.3] applied with the extension of ordered fields
$\mathbb{R}/\mathbb{Q}$, this happens exactly if $\langle x,-u\rangle\geq\log
a$ is valid for all $x$ in

$\\{x\in N_{\mathbb{R}}\ :\ \langle x,-u_{l}\rangle\geq\log a_{l}\text{, for
}1\leq l\leq n\\}=\displaystyle\bigcap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}}),$

i.e., if $\displaystyle\bigcap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})\subseteq
R(1_{S},a\chi^{u})$. ∎

######  Remark 3.33.

The version of Lemma 3.32 with $\widetilde{R}$ in place of $R$ is false. For
example, let $S=\mathbb{T}$ and $M=\mathbb{Z}$. Then
$\widetilde{R}(1_{\mathbb{T}},t^{1}x^{-1})\cap\widetilde{R}(1_{\mathbb{T}},t^{1})\subseteq\widetilde{R}(1_{\mathbb{T}},t^{1}x^{-2})$,
but there are no $m_{1},m_{2}\in\mathbb{Z}_{\geq 0}$, $m\in\mathbb{Z}_{>0}$,
and $b\in\mathbb{T}$ with $b\geq 1_{\mathbb{T}}$ such that
$b(t^{1}x^{-2})^{m}=(t^{1}x^{-1})^{m_{1}}(t^{1})^{m_{2}}$.

In light of Remark 3.33, the proof of the following proposition requires case-
work.

######  Proposition 3.34.

For any prime congruence $P$ on $S[M]$,
$\widetilde{R}(1_{S},a\chi^{u})\in\widetilde{\mathcal{F}}_{P}$ if and only if
$1_{\kappa(P)}\geq|a\chi^{u}|_{P}$. If $P\in\operatorname{Cont}_{S}S[M]$ then
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P}$ if and only if
$1_{\kappa(P)}\geq|a\chi^{u}|_{P}$.

###### Proof.

The “if” directions in the above statements follow immediately from the
definitions of $\widetilde{\mathcal{F}}_{p}$ and $\mathcal{F}_{P}$. For the
other directions, note that the statements are trivially true if $a=0_{S}$, so
assume $a\neq 0_{S}$.

Suppose $P\in\operatorname{Cont}_{S}S[M]$ and
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P}$. So there are terms
$a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}$ such that
$R(1_{S},a\chi^{u})\supseteq\cap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})$ and
$|a_{l}\chi^{u_{l}}|_{P}\leq 1_{\kappa(P)}$ for each $l$. Lemma 3.13 tells us
that $\cap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})$ is nonempty. So, by Lemma
3.32, there are $m_{1},m_{2},\ldots,m_{n}\in\mathbb{Z}_{\geq 0}$,
$m\in\mathbb{Z}_{>0}$, and $b\in S$ with $b\geq 1_{S}$ such that
$b(a\chi^{u})^{m}=\displaystyle\prod_{l=1}^{n}(a_{l}\chi^{u_{l}})^{m_{l}}$.
Then $b^{-1}\leq 1_{S}$ and

$|a\chi^{u}|_{P}^{m}=|b^{-1}|_{P}\displaystyle\prod_{l=1}^{n}|a_{l}\chi^{u_{l}}|_{P}^{m_{l}}\leq
1_{\kappa(P)},$

so $|a\chi^{u}|_{P}\leq 1_{\kappa(P)}$.

Now consider a general prime congruence $P$ on $S[M]$, and suppose
$\widetilde{R}(1_{S},a\chi^{u})\in\widetilde{\mathcal{F}}_{P}$. There are
terms $a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}$ such that
$\widetilde{R}(1_{S},a\chi^{u})\supseteq\cap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$
and $|a_{l}\chi^{u_{l}}|_{P}\leq 1_{\kappa(P)}$ for each $l$. Let
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\cdots\leq\mathcal{C}_{k})$ be a
simplicial flag of cones such that $P=P_{\mathcal{C}_{\bullet}}$. Theorem 3.20
tells us that there is some $\widetilde{U}\in\widetilde{\mathcal{F}}_{P}$ with
$\widetilde{U}\subseteq\\{0\\}\times N_{\mathbb{R}}$ if and only if
$\mathcal{C}_{k}\subseteq\\{0\\}\times N_{\mathbb{R}}$. We consider two cases,
as to whether this happens or not.

Suppose that $\mathcal{C}_{k}\not\subseteq\\{0\\}\times N_{\mathbb{R}}$. Since
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})=\cap_{l=1}^{n}\widetilde{R}(1_{S},a_{l}\chi^{u_{l}})$
and $\widetilde{R}(1_{S},a\chi^{u})$ are in $\widetilde{\mathcal{F}}_{P}$,
neither of them is contained in $\\{0\\}\times N_{\mathbb{R}}$. Therefore,
$\widetilde{R}(1_{S},a\chi^{u})=c(R(1_{S},a\chi^{u}))$ and
$\widetilde{R}(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})=c(R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}}))$.
So

$R(1_{S},a\chi^{u})\supseteq
R(1_{S},a_{1}\chi^{u_{1}},\ldots,a_{n}\chi^{u_{n}})=\cap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})$

with $\cap_{l=1}^{n}R(1_{S},a_{l}\chi^{u_{l}})$ nonempty. We can now apply
Lemma 3.32 and, as in the previous case, conclude that $|a\chi^{u}|_{P}\leq
1_{\kappa(P)}$.

Finally, suppose that $\mathcal{C}_{k}\subseteq\\{0\\}\times N_{\mathbb{R}}$.
Then, by the definition of $P_{\mathcal{C}_{\bullet}}$,
$|b|_{P}=1_{\kappa(P)}$ for any $b\in S^{\times}$. Note that
$\widetilde{R}(1_{S},a\chi^{u})\cap(\\{0\\}\times
N_{\mathbb{R}})=\\{0\\}\times u^{\vee}$, where $u^{\vee}=\\{\zeta\in
N_{\mathbb{R}}\;:\;\left\langle\zeta,u\right\rangle\leq 0_{\mathbb{R}}\\}$.
Similarly, $\cap_{l=1}^{n}\widetilde{R}(1_{S},a\chi^{u})\cap(\\{0\\}\times
N_{\mathbb{R}})=\\{0\\}\times\\{u_{1},\ldots,u_{n}\\}^{\vee}$, so we have that
$\\{u_{1},\ldots,u_{n}\\}^{\vee}\subseteq u^{\vee}$. So, by the
$\mathbb{Q}$-version of [19, Proposition 1.9], there are
$m_{1},m_{2},\ldots,m_{n}\in\mathbb{Z}_{\geq 0}$ and $m\in\mathbb{Z}_{>0}$
such that $mu=\sum_{l=1}^{n}m_{l}u_{l}$. Thus,

$|a\chi^{u}|_{P}^{m}=|\chi^{u}|_{P}^{m}=\displaystyle\prod_{l=1}^{n}|\chi^{u_{l}}|_{P}^{m_{l}}=\displaystyle\prod_{l=1}^{n}|a_{l}\chi^{u_{l}}|_{P}^{m_{l}}\leq
1_{\kappa(P)},$

so $|a\chi^{u}|_{P}\leq 1_{\kappa(P)}$. ∎

######  Theorem 3.35.

The map $P\mapsto\widetilde{\mathcal{F}}_{P}$ gives a bijection from the set
of prime congruences on $S[M]$ to the set of prime filters on the lattice of
$\Gamma$-admissible fan support sets. The inverse map is given by
$\widetilde{\mathcal{F}}\mapsto\widetilde{\mathcal{F}}_{P}$.

The map $P\mapsto\mathcal{F}_{P}$ gives a bijection from
$\operatorname{Cont}_{S}{S[M]}$ to the set of prime filters on the lattice of
$\Gamma$-rational polyhedral sets where the filter contains some polytope. The
inverse map is given by $\mathcal{F}\mapsto P_{\mathcal{F}}$.

###### Proof.

Note that, by Theorem 3.17, Lemma 3.27, and Definition 3.31, all of these maps
are well-defined.

We show that $P\mapsto\mathcal{F}_{P}$ and $\mathcal{F}\mapsto
P_{\mathcal{F}}$ are inverses. The proof for
$P\mapsto\widetilde{\mathcal{F}}_{P}$ and $\widetilde{\mathcal{F}}\mapsto
P_{\widetilde{\mathcal{F}}}$ is analogous.

To show that $P=P_{\mathcal{F}_{P}}$ for any
$P\in\operatorname{Cont}_{S}S[M]$, it suffices to show that
$a\chi^{u}\leq_{P}b\chi^{\lambda}$ if and only if
$a\chi^{u}\leq_{P_{\mathcal{F}_{P}}}b\chi^{\lambda}$ for any terms
$a\chi^{u},b\chi^{\lambda}\in S[M]$. Since $b\chi^{\lambda}$ is a unit in
$S[M]$, we can divide both inequalities by $b\chi^{\lambda}$, and so we may
assume without loss of generality that $b\chi^{\lambda}=1_{S}$. By Proposition
3.34, we know that $a\chi^{u}\leq_{P}1_{S}$ if and only if
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P}$. By the definitions of
$P_{\mathcal{F}_{P}}$ and $\leq_{\mathcal{F}_{P}}$,
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P}$ if and only if
$a\chi^{u}\leq_{\mathcal{F}_{P}}1_{S}$ which, in turn, is the same as having
$a\chi^{u}\leq_{P_{\mathcal{F}_{P}}}1_{S}$.

Now fix a prime filter $\mathcal{F}$ on the lattice of $\Gamma$-rational
polyhedral sets such that $\mathcal{F}$ contains a polytope; we wish to show
that $\mathcal{F}=\mathcal{F}_{P_{\mathcal{F}}}$. Since $\mathcal{F}$ and
$\mathcal{F}_{P_{\mathcal{F}}}$ are prime filters on the lattice of
$\Gamma$-rational polyhedral sets, they are determined by which
$\Gamma$-rational polyhedra they contain. Using the fact that $\mathcal{F}$
and $\mathcal{F}_{P_{\mathcal{F}}}$ are both filters, we then see that they
are determined by which $\Gamma$-rational half-spaces they contain. But every
$\Gamma$-rational half-space can be written as $R(1_{S},a\chi^{u})$ for some
term $a\chi^{u}\in S[M]$, so it suffices to show that
$R(1_{S},a\chi^{u})\in\mathcal{F}$ if and only if
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P_{\mathcal{F}}}$. By Proposition 3.34,
$R(1_{S},a\chi^{u})\in\mathcal{F}_{P_{\mathcal{F}}}$ if and only if
$a\chi^{u}\leq_{P_{\mathcal{F}}}1_{S}$. By the definitions of
$P_{\mathcal{F}}$ and $\leq_{\mathcal{F}}$, this happens if and only if
$R(1_{S},a\chi^{u})\in\mathcal{F}$. ∎

######  Corollary 3.36.

Let $\mathcal{C}_{\bullet}$ and $\mathcal{C}_{\bullet}^{\prime}$ be simplicial
flags of cones. Then
$P_{\mathcal{C}_{\bullet}}=P_{\mathcal{C}_{\bullet}^{\prime}}$ if and only if
$\mathcal{C}_{\bullet}$ and $\mathcal{C}_{\bullet}^{\prime}$ are locally
equivalent.

If $P=P_{\mathcal{P}_{\bullet}}$ and
$P^{\prime}=P_{\mathcal{P}_{\bullet}^{\prime}}$ for some flags
$\mathcal{P}_{\bullet}$ and $\mathcal{P}_{\bullet}^{\prime}$ of polyhedra,
then $P=P^{\prime}$ if and only if $\mathcal{P}_{\bullet}$ and
$\mathcal{P}_{\bullet}^{\prime}$ are locally equivalent.

###### Proof.

By Theorem 3.35, $P=P^{\prime}$ if and only if
$\widetilde{\mathcal{F}}_{P}=\widetilde{\mathcal{F}}_{P^{\prime}}$. Corollary
3.21 tells us that this happens exactly if $\mathcal{C}_{\bullet}$ and
$\mathcal{C}_{\bullet}^{\prime}$ are locally equivalent.

The proof of the second statement is analogous. ∎

######  Corollary 3.37.

Let $S$ be a subsemifield of $\mathbb{T}$ and let $M\cong\mathbb{Z}^{n}$.
There are explicit bijections between

  1. (1)

$\operatorname{Cont}_{S}S[M]$,

  2. (2)

the set of flags $\mathcal{P}_{\bullet}$ of polyhedra in $N_{\mathbb{R}}$
modulo local equivalence,

  3. (3)

the set of prime filters on the lattice of $\Gamma$-rational polyhedral sets
such that the filter contains a polytope, and

  4. (4)

the set of prime filters on the lattice of $\Gamma$-rational polytopal sets.

These bijections are given as follows. The map from (2) to (1) sends a flag
$\mathcal{P}_{\bullet}$ to the prime congruence
$P_{\mathcal{P}_{\bullet}^{\prime}}$ where $\mathcal{P}_{\bullet}^{\prime}$ is
any flag of polyhedra which is locally equivalent to $\mathcal{P}_{\bullet}$
and for which $c(\mathcal{P}_{\bullet}^{\prime})$ is simplicial. The map from
(1) to (3) is $P\mapsto\mathcal{F}_{P}$. The map from (3) to (4) sends
$\mathcal{F}$ to the filter $\\{U\in\mathcal{F}\;:\;U\text{ is polytopal}\\}$.

###### Proof.

This follows immediately from Proposition 3.9, Proposition 3.29, Theorem 3.35,
and Corollary 3.36. ∎

######  Corollary 3.38.

Let $S$ be a subsemifield of $\mathbb{T}$ and let $M\cong\mathbb{Z}^{n}$.
There are explicit bijections between

  1. (1)

the set of all prime congruences on $S[M]$,

  2. (2)

the set of flags $\mathcal{C}_{\bullet}$ of cones in $\mathbb{R}_{\geq
0}\times N_{\mathbb{R}}$ modulo local equivalence,

  3. (3)

the set of prime filters on the lattice of $\Gamma$-admissible fan support
sets.

These bijections are given as follows. The map from (2) to (1) sends a flag
$\mathcal{C}_{\bullet}$ to the prime congruence
$P_{\mathcal{C}_{\bullet}^{\prime}}$, where $\mathcal{C}_{\bullet}^{\prime}$
is any simplicial flag of cones which is locally equivalent to
$\mathcal{C}_{\bullet}$. The map from (1) to (3) is
$P\mapsto\widetilde{\mathcal{F}}_{P}$.

###### Proof.

This follows immediately from Proposition 3.9, Theorem 3.35, and Corollary
3.36. ∎

######  Corollary 3.39.

There is a bijection from the set of prime congruences $P$ on $S[M]$ such that
the map $S\to S[M]\to\kappa(P)$ is injective and the set of prime filters on
the lattice of $\Gamma$-rational polyhedral sets. This bijection extends the
bijection $P\mapsto\mathcal{F}_{P}$ from $\operatorname{Cont}_{S}S[M]$ to the
set of prime filters on the lattice of $\Gamma$-rational polyhedral sets such
that the filter contains a polytope.

###### Proof.

Note that the map $\mathcal{F}\mapsto\\{c(U)\;:\;U\in\mathcal{F}\\}$ is a
bijection from the set of prime filters on the lattice of $\Gamma$-rational
polyhedral sets to the set of prime filters $\widetilde{\mathcal{F}}$ on the
lattice of $\Gamma$-admissible fan support sets such that every
$\widetilde{U}\in\widetilde{\mathcal{F}}$ is not contained in $\\{0\\}\times
N_{\mathbb{R}}$. So it suffices to show that, if $P$ is a prime congruence on
$S[M]$, the map $S\to S[M]\to\kappa(P)$ is injective if and only if every
$\widetilde{U}\in\widetilde{\mathcal{F}}_{P}$ is not contained in
$\\{0\\}\times N_{\mathbb{R}}$.

Pick a simplicial flag
$\mathcal{C}_{\bullet}=(\mathcal{C}_{0}\leq\cdots\leq\mathcal{C}_{k})$ of
cones such that $P=P_{\mathcal{C}_{\bullet}}$ and fix any
$w_{i}\in\operatorname{rel.int.}\mathcal{C}_{i}\subseteq\mathcal{C}_{i}\backslash\mathcal{C}_{i-1}$
for $0\leq i\leq k$. Thus, $\begin{pmatrix}w_{0}\\\ \vdots\\\
w_{k}\end{pmatrix}$ is a defining matrix for $P$, and so the map $S\to
S[M]\to\kappa(P)$ is injective if and only if the first column of this matrix
is not all zero, i.e., if $w_{i}\notin\\{0\\}\times N_{\mathbb{R}}$ for some
$i$. Since $w_{i}\in\operatorname{rel.int.}\mathcal{C}_{i}$,
$w_{i}\notin\\{0\\}\times N_{\mathbb{R}}$ if and only if
$\mathcal{C}_{i}\not\subseteq\\{0\\}\times N_{\mathbb{R}}$. So $S\to
S[M]\to\kappa(P)$ is injective if and only if
$\mathcal{C}_{k}\not\subseteq\\{0\\}\times N_{\mathbb{R}}$.

But, by Theorem 3.20, $\mathcal{C}_{k}\not\subseteq\\{0\\}\times
N_{\mathbb{R}}$ if and only if every
$\widetilde{U}\in\widetilde{\mathcal{F}}_{P}$ is not contained in
$\\{0\\}\times N_{\mathbb{R}}$. ∎

######  Remark 3.40.

Under the bijection of Corollary 3.39, the prime filter given in Remark 3.26
corresponds to the prime congruence on $\mathbb{T}[x^{\pm 1},y^{\pm 1}]$ with
defining matrix $\begin{pmatrix}0&0&1\\\ 1&0&0\end{pmatrix}$.

######  Corollary 3.41.

Let $M\cong\mathbb{Z}^{n}$. There are explicit bijections between

  1. (1)

the set of all prime congruences on $\mathbb{B}[M]$,

  2. (2)

the set of flags $\mathcal{C}_{\bullet}$ of cones in $N_{\mathbb{R}}$ modulo
rational local equivalence,

  3. (3)

the set of prime filters on the lattice of rational fan support sets in
$N_{\mathbb{R}}$,

  4. (4)

the set of all monomial preorders on a Laurent polynomial ring in $n$
variables.

The bijections between (1), (2), and (3) are given by applying the bijections
of Corollary 3.38 as follows. Given a prime congruence $P$ on $\mathbb{B}[M]$,
view it as a prime congruence on $\mathbb{T}[M]$ by pulling back along the the
map $\mathbb{T}[M]\to\mathbb{B}[M]$. Given a flag of cones in
$N_{\mathbb{R}}$, consider it as a flag of cones in $\mathbb{R}_{\geq 0}\times
N_{\mathbb{R}}$ by adding a first coordinate $0$. Given a prime filter on the
lattice of rational fan support sets in $N_{\mathbb{R}}$, push it forward to a
filter base on the lattice of $\mathbb{R}$-admissible fan support sets in
$\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ via the map
$N_{\mathbb{R}}\to\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ adjoining a first
coordinate $0$, and let this filter base generate a filter. The map from (1)
to (4) is given by sending a prime congruence $P$ to the preorder $<_{P}$ on
$\operatorname{Terms}(\mathbb{B}[M])=M$.

###### Proof.

The primes on $\mathbb{T}[M]$ which are pullbacks of primes on $\mathbb{B}[M]$
are those that are given by matrices with first column zero. This shows that
the primes on $\mathbb{B}[M]$ are in bijection with local equivalence classes
of flags of cones contained in $N_{\mathbb{R}}\times\\{0\\}$ modulo local
equivalence, i.e., flags of cones in $N_{\mathbb{R}}$ modulo rational local
equivalence. By Corollaries 3.38 and 3.39, these are in bijection with the set
of those prime filters on the lattice of $\mathbb{R}$-admissible fan support
sets whose restriction to $N_{\mathbb{R}}\times\\{1\\}$ is not a prime filter
because this restriction contains the empty set. The only way this can occur
is if the filter contains a set that is contained in
$N_{\mathbb{R}}\times\\{0\\}$. This, in turn, happens if and only if the
filter is generated by the pushforward of a prime filter on the lattice of
rational fan support sets in $N_{\mathbb{R}}$ along the inclusion map
$N_{\mathbb{R}}\to\mathbb{R}_{\geq 0}\times N_{\mathbb{R}}$ of the zero-slice.
∎

The following corollary is an application of the above results. It provides an
explicit criterion for when two matrices define the same prime.

######  Corollary 3.42.

Let $C$ and $C^{\prime}$ be real matrices of sizes $k\times n$ and
$k^{\prime}\times n$, respectively. Suppose that $C$ and $C^{\prime}$ define
prime congruences on $S[x_{1}^{\pm 1},\ldots,x_{n}^{\pm 1}]$, i.e., the first
columns of $C$ and $C^{\prime}$ are lexicographically at least the zero
vector. Use downward gaussian elimination and removal of rows of zeros on $C$
and $C^{\prime}$ to obtain matrices $\widetilde{C}$ and
$\widetilde{C}^{\prime}$, respectively, whose first columns have all entries
non-negative and whose rows are linearly independent. Then $C$ and
$C^{\prime}$ define the same prime congruence on $S[x_{1}^{\pm
1},\ldots,x_{n}^{\pm 1}]$ if and only if the flags
$\mathcal{C}_{\bullet}(\widetilde{C})$ and
$\mathcal{C}_{\bullet}(\widetilde{C}^{\prime})$ of cones are locally
equivalent.

###### Proof.

By Lemma 2.7, the prime congruences that $C$ and $C^{\prime}$ define are
$P_{\mathcal{C}_{\bullet}(\widetilde{C})}$ and
$P_{\mathcal{C}_{\bullet}(\widetilde{C}^{\prime})}$, respectively. By
Corollary 3.36, these are equal if and only if
$\mathcal{C}_{\bullet}(\widetilde{C})$ and
$\mathcal{C}_{\bullet}(\widetilde{C}^{\prime})$ are locally equivalent. ∎

We now use Proposition 3.34 to show that the smallest dimension of of a
polyhedron in $\mathcal{F}_{P}$ is an algebraic invariant of
$P\in\operatorname{Cont}_{S}{S[M]}$.

######  Proposition 3.43.

For any $P\in\operatorname{Cont}_{S}{S[M]}$,

$\min\\{\dim
U:U\in\mathcal{F}_{P}\\}=\operatorname{rk}(\kappa(P)^{\times}/S^{\times})=\operatorname{rk}(M)-\operatorname{ht}(P)=\operatorname{tr.deg}(\kappa(P)/S).$

###### Proof.

Lemma 2.13 gives us the second equality in the proposition. By [6, Proposition
B.11], we get the equality with $\operatorname{tr.deg}(\kappa(P)/S)$.

Note that $\min\\{\dim U:U\in\mathcal{F}_{P}\\}$ is attained for some
$\Gamma$-rational polyhedron $U\in\mathcal{F}_{P}$, for which $\dim U$ is the
dimension of the smallest $\Gamma$-rational affine space that contains $U$. In
particular, if we let $\mathcal{H}$ be the subgroup of $\Gamma\times M$
consisting of those $(\alpha,u)$ such that the hyperplane $\\{\xi\in
N_{\mathbb{R}}\;:\;\alpha+\left\langle\xi,u\right\rangle=0\\}$ is in
$\mathcal{F}_{P}$, then $\min\\{\dim U:U\in\mathcal{F}_{P}\\}=\dim
N_{\mathbb{R}}-\operatorname{rk}\mathcal{H}$. Note that, for any term
$a\chi^{u}\in S[M]$, the hyperplane $\\{\xi\in
N_{\mathbb{R}}\;:\;\log(a)+\left\langle\xi,u\right\rangle=0\\}$ is in
$\mathcal{H}$ exactly if $R(1_{S},a\chi^{u})$ and $R(1_{S},a^{-1}\chi^{-u})$
are both in $\mathcal{F}_{P}$. By Proposition 3.34, this happens exactly if
$1_{\kappa(P)}=|a\chi^{u}|_{P}$.

Choose a defining matrix for $P\in\operatorname{Cont}_{S}{S[M]}$ of the form

$\begin{pmatrix}1&\xi_{0}\\\ 0&\xi_{1}\\\ \vdots&\vdots\\\
0&\xi_{k}\end{pmatrix}$

for some $\xi_{0},\ldots,\xi_{k}\in N_{\mathbb{R}}$. Then, looking at the
conditions (3.3.0),…,(3.3.$k$) from Section 3.3, we see that
$1_{\kappa(P)}=|a\chi^{u}|_{P}$ if and only if

$log(a)+\left<\xi_{0},u\right>=\left<\xi_{1},u\right>=\cdots=\left<\xi_{k-1},u\right>=\left<\xi_{k},u\right>=0.$

We now see that the projection $\Gamma\times M\rightarrow M$ maps
$\mathcal{H}$ isomorphically onto its image. Indeed, the values of $\xi_{0}$
and $u$ uniquely determine $\log(a)$. So

$\displaystyle\operatorname{rk}(\mathcal{H})$
$\displaystyle=\operatorname{rk}\left\\{u\in
M\;:\;\left<\xi_{0},u\right>\in\Gamma\text{ and
}\left<\xi_{i},u\right>=0,\text{ for }1\leq i\leq k\right\\}$
$\displaystyle=\operatorname{rk}(\ker(M\rightarrow\kappa(P)^{\times}/S^{\times}))=\operatorname{rk}M-\operatorname{rk}(\kappa(P)^{\times}/S^{\times}),$

where the final equality holds because the map
$M\to\kappa(P)^{\times}/S^{\times}$ is surjective. Thus,

$\displaystyle\min\\{\dim U\;:\;U\in\mathcal{F}_{P}\\}$ $\displaystyle=\dim
N_{\mathbb{R}}-\operatorname{rk}\mathcal{H}=\operatorname{rk}M-\operatorname{rk}(\ker(M\rightarrow\kappa(P)^{\times}/S^{\times}))$
$\displaystyle=\operatorname{rk}\kappa(P)^{\times}/S^{\times}.$

∎

The previous result finally allows us to prove the geometric criterion of when
$\leq_{P}$ is a valuated monomial _order_. A flag
$\mathcal{P}_{\bullet}=(\mathcal{P}_{0}\leq\mathcal{P}_{1}\leq\cdots\leq\mathcal{P}_{k})$
of polyhedra in $N_{\mathbb{R}}$ is called _complete_ if $k=\dim
N_{\mathbb{R}}$.

######  Proposition 3.44.

For any $P\in\operatorname{Cont}_{S}{S[M]}$, $\leq_{P}$ is an _order_ on
$\operatorname{Terms}(S[M])$ if and only if we can write
$P=P_{\mathcal{P}_{\bullet}}$ with $\mathcal{P}_{\bullet}$ a complete flag in
$N_{\mathbb{R}}$.

###### Proof.

If $\mathcal{P}_{\bullet}$ is complete, then any $\Gamma$-rational
neighborhood of $\mathcal{P}_{\bullet}$ must be full-dimensional because it
contains $\dim N_{\mathbb{R}}+1$ affinely independent points. So
$\dim_{N_{\mathbb{R}}}=\min\\{\dim
U:U\in\mathcal{F}_{P}\\}=\operatorname{rk}(M)-\operatorname{ht}(P)$. Thus
$\operatorname{ht}(P)=0$ and so, by [6, Corollary 9.15] $\leq_{P}$ is an order
on $\operatorname{Terms}(S[M])$.

For the other direction, suppose that $\leq_{P}$ is an order on
$\operatorname{Terms}(S[M])$. Let
$\mathcal{P}_{\bullet}=(\mathcal{P}_{0}\leq\mathcal{P}_{1}\leq\cdots\leq\mathcal{P}_{k})$
be a flag of polyhedra in $N_{\mathbb{R}}$ such that
$P=P_{\mathcal{P}_{\bullet}}$ and $k$ is as large as possible. Now assume, for
contradiction, that $k<\dim N_{\mathbb{R}}$. For $0\leq i\leq k$ pick
$\xi_{i}\in\mathcal{P}_{i}\backslash\mathcal{P}_{i-1}$, so
$C=\begin{pmatrix}1&\xi_{0}\\\ 1&\xi_{1}\\\ \vdots&\vdots\\\
1&\xi_{k}\end{pmatrix}$ is a defining matrix for $P$. Since $k<\dim
N_{\mathbb{R}}$, there is a point $\xi_{k+1}$ which is affinely independent of
$\xi_{0},\ldots,\xi_{k}$. So the matrix
$C^{\prime}=\left(\begin{array}[]{cc}\lx@intercol\hfil C\hfil\lx@intercol\\\
\hline\cr 1&\xi_{k+1}\end{array}\right)$ defines a flag
$\mathcal{P}_{\bullet}^{\prime}=(\mathcal{P}_{0}^{\prime}\leq\mathcal{P}_{1}^{\prime}\leq\cdots\leq\mathcal{P}_{k+1}^{\prime})$
of simplices in $N_{\mathbb{R}}$. For any two exponent vectors
$\mathfrak{u}_{1}$ and $\mathfrak{u}_{2}$ we have
$C^{\prime}\mathfrak{u}_{1}\leq_{\mathrm{lex}}C^{\prime}\mathfrak{u}_{2}$ if
and only if $C\mathfrak{u}_{1}\leq_{\mathrm{lex}}C\mathfrak{u}_{2}$ because
$\leq_{P}$ is an order. Thus $P_{\mathcal{P}_{\bullet}^{\prime}}=P$,
contradicting the maximality of $k$. ∎

######  Remark 3.45.

The same proof shows that the statement of Proposition 3.44 is also true when
$M$ is any toric monoid.

## References

  * [1] O. Amini, H. Iriarte _Geometry of higher rank valuations_ , arXiv preprint arXiv:2208.06237 (2022)
  * [2] A. Chan, D. Maclagan, _Groebner bases over fields with valuations_ , Mathematics of Computation, Vol 88, Number 315, 467–-483, http://dx.doi.org/10.1090/mcom/3321, (2019).
  * [3] A. Connes and C. Consani, _The Arithmetic Site_ , C. R. Math. Acad. Sci. Paris 352 (2014), no. 12, 971–975. https://doi.org/10.1016/j.crma.2014.07.009.
  * [4] A. Connes, C. Consani, _Geometry of the Scaling Site_ , Selecta Math. 23, no.3, 1803–-1850, (2017) DOI: s00029-017-0313-y.
  * [5] N. Friedenberg _Normal completions of toric varieties over rank one valuation rings and completions of $\Gamma$-admissible fans_ arXiv pre-print arXiv:1908.00064 (2019).
  * [6] N. Friedenberg, K. Mincheva, _Tropical Adic Spaces I: The continuous spectrum of a topological semiring_ , arXiv pre-print 2209.15116.
  * [7] J. Giansiracusa, N. Giansiracusa, _Equations of tropical varieties_ , Duke Math. J. 165, no. 18 (2016), 3379-3433; DOI: 00127094-3645544.
  * [8] J. Golan, _The theory of semirings with applications in mathematics and theoretical computer science_ , Longman Sci & Tech., 54, (1992).
  * [9] Z. Izhakian, _Commutative $\nu$-algebra and supertropical algebraic geometry_, arXiv preprint arXiv:1901.08032 (2019).
  * [10] Z. Izhakian, L. Rowen, _Supertropical algebra_ , Advances in Mathematics, Volume 225, Issue 4, 2222–2286, (2010), https://doi.org/10.1016/j.aim.2010.04.007.
  * [11] D. Joó, K. Mincheva, _Prime congruences of idempotent semirings and a Nullstellensatz for tropical polynomials_ , Sel. Math. New Ser., doi:10.1007/s00029-017-0322-x, (2017).
  * [12] O. Lorscheid _Scheme theoretic tropicalization_ arXiv preprint arXiv: 1508.07949, (2015).
  * [13] D. Maclagan, F. Rincón, _Tropical ideals_ , Compositio Mathematica, Volume 154, Issue 3, 640 – 670, (2018) DOI: S0010437X17008004.
  * [14] M. van der Put, P. Schneider, _Points and topologies in rigid geometry_ , Math. Ann. 302, 81–103 (1995)
  * [15] J. Rabinoff, _Tropical analytic geometry, Newton polygons, and tropical intersections_ , Advances in Mathematics vol. 229, issue 6, 3192–3255, (2012).
  * [16] L. Robbiano, _Term orderings on the polynomial ring_ , EUROCAL 85, Vol. 2 (Linz, 1985), Lecture Notes in Comput. Sci. 204, 513-517, Springer, Berlin (1985)
  * [17] L. Rowen _An informal overview of triples and systems_ , arXiv preprint ArXiv:1709.03174 (2018).
  * [18] T. Vaccon, T. Verron _Universal Analytic Gröbner Bases and Tropical Geometry_ , forthcoming.
  * [19] G. Ziegler, _Lectures on Polytopes_ , Graduate Texts in Mathematics (GTM, volume 152), Springer New York, (1995), DOI: 978-1-4613-8431-1.