1 Introduction↩︎

1.1 Background↩︎

Tropical geometry has found an increasing number of applications in algebraic geometry, providing combinatorial shadows of algebro-geometric objects. Its primary geometric objects are rational polyhedral complexes called tropical varieties, which are often obtained via the tropicalization of algebraic varieties.

Recently, attempts have been made to develop a theory of tropical geometry from an algebraic perspective. The basic algebras used in tropical geometry are the semifields \[\begin{align} &\mathbb{B}=(\{-\infty,0\},\operatorname{max},+),\\ &\mathbb{T}=(\mathbb{R}\cup\{-\infty\},\operatorname{max},+), \end{align}\] and (semi)algebras over these semifields. The set of functions on a tropical variety is considered to form such an algebra. One of the main interests in algebraic tropical geometry is what geometric data these algebras contain and how they can be extracted.

In joo2018prime?, the authors defined prime congruences and studied their properties. A prime congruence on a semiring \(S\) is an equivalence relation on \(S\) which gives a surjective semiring homomorphism from \(S\) to a semiring with suitable properties (see 2 for the details). In particular, they proposed the definition of the Krull dimension of an additively idempotent semiring \(S\) and showed that it works well for some semirings which appear in tropical geometry.

In order to compute the Krull dimension, they showed that any prime congruence on \(\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\) or \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) can be represented by a matrix with coefficients in \(\mathbb{R}\), called an admissible matrix. Furthermore, in song2024geometricinterpretationkrulldimensions? and joó2025varietiesprimetropicalideals?, the authors calculated the Krull dimension of semirings of the form \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]/C\) where \(C\) is a congruence satisfying appropriate conditions, using these matrices.

Admissible matrices are useful for expressing prime congruences as numerical data. However, for a given prime congruence, the corresponding matrix is not uniquely determined. This non-uniqueness makes it difficult to decide inclusion relations between prime congruences.

1.2 Our contribution↩︎

Our aim is to treat the set of prime congruences on semirings as a geometric object. To do this in a way analogous to classical scheme theory, in this paper, we begin with a sufficiently abstract setting. Some notions in our framework not only lead to several new results, but also shed new light on previously known results. Note that our setting is different from the one in giansiracusa2016equations?, where the authors construct schemes by gluing sets of prime ideals of semirings.

For a semiring \(S\), we denote by \(\operatorname{CSpec}S\) the set of all prime congruences on \(S\). Let \(M\) be a free abelian group generated by \(X_1,\dots,X_n\). While previous work used matrices to represent an element of \(\operatorname{CSpec}\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\) or \(\operatorname{CSpec}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\), we employ flags instead:

We give an example of an \((\mathbb{R}\times M)\)-flag of length 2 where the rank \(n\) of \(M\) is equal to 1: \[\begin{align} &H_1 = \{(r,m)\in \mathbb{R}\times M_{\mathbb{R}}\mid r=0.5m\},\\ &H_{1,+}= \{(r,m)\in \mathbb{R}\times M_{\mathbb{R}}\mid r>0.5m\},\\ &H_2 = \{(0,0)\},\\ &H_{2,+}=H_1\cap \{(r,m)\in \mathbb{R}\times M_{\mathbb{R}}\mid r>0\}. \end{align}\] This \((\mathbb{R}\times \mathbb{Z})\)-flag is illustrated in 1 where the "positive part" \((H_{1,+}\cup H_{2,+})\cap (\mathbb{R}\times M)\) is represented by the red region.

The notion of a flag arises naturally from the general framework developed in the first half of this paper. As a consequence, an \((\mathbb{R}\times M)\)-flag provides a unique representation of prime congruences on \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) in the following sense:

Similarly, the set of prime congruences on various tropical algebras, such as

• \(\operatorname{CSpec}\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\)(24),

• \(\operatorname{CSpec}\mathbb{B}[X_1,\dots,X_n]\)(32 ),

• \(\operatorname{CSpec}\mathbb{T}[X_1,\dots,X_n]\)(33 ),

can be identified with sets of flags.

An inclusion relation between prime congruences corresponds to a certain relation between flags (see 28). Thus we can describe the set of prime congruences on several algebras, as shown in 8. In particular, we give alternative proofs of some results in joo2018prime? (see 2 3 6). See 26 for the difference between our approach and computations using admissible matrices.

In friedenberg2025geometric?, the authors have obtained a related correspondence, which classifies the continuous spectrum of a topological semiring using flags of polyhedra. The two approaches share similar underlying ideas. The previous work is developed from the viewpoint of Gröbner theory and adic geometry, whereas our approach is algebraic and scheme-theoretic. Furthermore, in Section 4 of mikami2025tropicalcohomologysmoothalgebraic?, the author gives another interpretation of the set of homomorphisms from \(M\) to totally ordered groups where \(M\) is a free abelian group of rank \(n\).

Following classical scheme theory, we attempt to construct a global object using the space of prime congruences. We use techniques of toric geometry, where the process of gluing is given by the combinatorial data of a fan. For a fan \(\Sigma\) in \(N_{\mathbb{R}}\cong \mathbb{R}^n\), we define a topological space \(X_{\Sigma}^{\operatorname{cong}}\) by \[\begin{align} X_{\Sigma}^{\operatorname{cong}}=\bigcup_{\sigma\in\Sigma}\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\sigma^{\vee}\cap M], \end{align}\] where the spaces are glued along the inclusions induced by face relations of cones. Here, \(\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\sigma^{\vee}\cap M]\) is the subset of all points in \(\operatorname{CSpec}\mathbb{T}[\sigma^{\vee}\cap M]\) which are lying over \(\mathbb{T}\) (see 4).

On the other hand, analogues of toric varieties have already been constructed in Kajiwara2008TropicalToric? and payne2009analytification?. The tropical toric variety associated with a fan \(\Sigma\) is the topological space \[\begin{align} X_{\Sigma}^{\operatorname{trop}}=\bigcup_{\sigma\in\Sigma}\operatorname{Hom}(\mathsf{M}_{\sigma},\mathbb{T}). \end{align}\]

Regarding \(X_{\Sigma}^{\operatorname{trop}}\) as the set of \(\mathbb{T}\)-valued points of \(X_{\Sigma}^{\operatorname{cong}}\), we have the following result.

Note that \(\dim_{P}X_{\Sigma}^{\operatorname{cong}}\) and \(\dim X_{\Sigma}^{\operatorname{cong}}\) are locally equal to the length of maximal chains of prime congruences. This subspace is also related to the separatedness and properness of \(X_{\Sigma}^{\operatorname{cong}}\), although we will not give a precise definition of the notions in this paper. See 36 37 for the details.

Our algebraic framework is also useful to study purely algebraic problems. Somewhat surprisingly, even for standard tropical algebras such as \(\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\) or \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\), many prime congruences fail to be finitely generated. This phenomenon is illustrated by the following theorem.

The monoid \(\mathsf{L}(P)\) appearing in the condition (2) is a submonoid of an ordered group \(\mathbb{R}\times M\) corresponding to \(P\) (see 12 ). The equivalence of (1) and (2) holds in a more general setting (see 15). The conditions (3) and (4) arise from describing the submonoid \(\mathsf{L}(P)\) as a flag. Note that finitely generated congruences play important roles in the Nullstellensatz for congruences in BertramEaston2017TropicalNullstellensatz?. We also note that in ito2026nonfinitegeneratednesscongruencesdefined?, the author gives another criterion for a congruence to be finitely generated.

1.3 Structure of the paper↩︎

We begin in 2 by reviewing some properties of semirings. In ¿sec:section:B[M]?, we introduce a \(\mathbb{B}\)-algebra associated to an ordered monoid, denoted by \(\mathbb{B}[\mathsf{M}]\), which is a key ingredient of this paper. \[\mathbb{B}[\mathsf{M}]\mathrel{\vcenter{:}}= \left(\bigoplus_{m\in\mathsf{M}}\mathbb{B}\right)/\sim.\] This algebra is not merely the direct sum of copies of \(\mathbb{B}\) indexed by \(\mathsf{M}\). That is, the equivalence relation \(\sim\) is non-trivial in general depending on the order of \(\mathsf{M}\). 2 summarizes the hierarchy of some classes of algebras introduced in this paper.

It is important that several tropical algebras such as \(\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\) and \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) can be treated as \(\mathbb{B}\)-algebras \(\mathbb{B}[\mathsf{M}]\) associated to suitable ordered monoids \(\mathsf{M}\) (by forgetting \(\mathbb{T}\)-algebra structure if necessary).

The main assertions are stated in ¿sec:section:B[M]cong? and 5, where we describe the details of the space of prime congruences. In 6, we discuss the relation between our framework and tropical toric varieties, and in 7, we characterize when a prime congruence is finitely generated.

2 Preliminaries↩︎

2.1 Semirings and prime congruences↩︎

We briefly recall some basic properties of semirings. See hebisch1998semirings? and golan2013semirings? for more details. When a specific statement is difficult to locate in the literature, we include a short proof for completeness.

In this paper, a semiring \(S\) is always assumed to be commutative and nontrivial. Namely, a semiring is a nonempty set \(S\) with two binary operations \((+,\cdot)\) satisfying:

• \((S,+)\) is a commutative monoid with the identity \(0_S\),

• \((S,\cdot)\) is a commutative monoid with the identity \(1_S\),

• For any \(a,b,c\in S\), \(a(b+c)=ab+ac\),

• \(0_S\neq1_S\) and \(a\cdot 0_S=0_S\) for any \(a\in S\).

A semiring \(S\) is said to be cancellative if whenever \(ab=ac\) for some \(a,b,c\in S\), then either \(a=0_S\) or \(b=c\). A semiring \(S\) is called a semifield if every nonzero element of \(S\) is multiplicatively invertible. If a subset \(I\subseteq S\) containing \(0_S\) satisfies \(a+b\in I\) and \(sa\in I\) for any \(a,b\in I\) and \(s\in S\), then \(I\) is called an ideal of \(S\), which rarely appears in this paper.

A map \(\phi\colon S_1\to S_2\) between semirings is a semiring homomorphism if it satisfies \[\phi(a+b)=\phi(a)+\phi(b),\;\phi(a\cdot b)=\phi(a)\cdot \phi(b),\;\phi(0_{S_1})=0_{S_2}, \text{ and }\phi(1_{S_1})=1_{S_2},\] for any \(a,b\in S_1\). A bijective homomorphism is called an isomorphism.

A congruence \(C\) on a semiring \(S\) is a subsemiring of \(S\times S\) that defines an equivalence relation on \(S\). Note that if \(C\) is not equal to \(S\times S\), the set of all equivalence classes \(S/C\) naturally inherits a semiring structure and the quotient map \(S\to S/C\) is a semiring homomorphism, which is often denoted by \(\pi_C\). We call the diagonal set \(\Delta=\Delta_S\) of \(S\times S\) the diagonal congruence.

Let \(\{C_{\lambda}\}\) be a family of congruences on a semiring \(S\). Since the intersection \(\bigcap_{\lambda}C_{\lambda}\) is also a congruence on \(S\), for any subset of \(\mathcal{A}\) of \(S\times S\), we can define the congruence generated by \(\mathcal{A}\) as the minimal congruence containing \(\mathcal{A}\), denoted by \(\langle\mathcal{A}\rangle\). We denote by \(\sum_{\lambda} C_{\lambda}\) the congruence generated by the union \(\bigcup_{\lambda}C_{\lambda}\). For some subsets \(\mathcal{A}_1,\dots,\mathcal{A}_k\) of \(S\times S\), we often denote a congruence \(\langle\bigcup_i \mathcal{A}_i\rangle\) by \(\langle\mathcal{A}_1,\dots,\mathcal{A}_k\rangle\). A congruence on \(S\) is called finitely generated if it is generated by a finite subset of \(S\times S\).

Let \(\phi\colon S_1\to S_2\) be a semiring homomorphism. The pull-back of a congruence \(C_2\) on \(S_2\) along \(\phi\) is defined by \[\begin{align} \phi^{*}C_2\mathrel{\vcenter{:}}= C_2\cap S_1\mathrel{\vcenter{:}}= \{(a,b)\in (S_1)^2\mid (\phi(a),\phi(b))\in C_2 \}.\label{eq:pull-back} \end{align}\tag{1}\] Note that the notation \(C_2\cap S_1\) is just a symbol and does not mean the set-theoretical intersection. In particular, the pull-back of the diagonal congruence on \(S_2\) is called the kernel (congruence) of \(\phi\), denoted by \(\operatorname{Ker}\phi\).

The push-out of a congruence \(C_1\) on \(S_1\) along \(\phi\) is defined by \[\begin{align} \phi_{*}C_1\mathrel{\vcenter{:}}= C_1\cdot S_2\mathrel{\vcenter{:}}= \langle\{(\phi(a),\phi(b))\in S_2\times S_2\mid (a,b)\in C_1\}\rangle.\label{eq:push-out} \end{align}\tag{2}\] It is straightforward to see that if \(\phi\) is surjective, \(\phi_{*}C_1\) is equal to the set-theoretical image of \(C_1\): \[\begin{align} \phi_{*}C_1=\{(\phi(a),\phi(b))\in S_2\times S_2\mid (a,b)\in C_1\}.\label{eq:surj32push-out} \end{align}\tag{3}\]

Let \(S\) be a semiring. The twisted product of the pairs \(\alpha=(\alpha_1,\alpha_2)\) and \(\beta=(\beta_1,\beta_2)\) of \(S\times S\) is \[\alpha\rtimes \beta=(\alpha_1\beta_1+\alpha_2\beta_2,\alpha_1\beta_2+\alpha_2\beta_1).\] Note that this product is associative and commutative. For two congruences \(C_1,C_2\) on \(S\), define \(C_1\rtimes C_2\) as a congruence generated by \(\{\alpha\rtimes \beta\mid \alpha\in C_1,\beta\in C_2\}\). Note that \(C_1\rtimes C_2\subseteq C_1\cap C_2\) holds.

In this paper, we denote by \(\operatorname{CSpec}S\) the set of all prime congruences on \(S\). For a congruence \(C\) on \(S\), we define the subset \(V(C)\) of \(\operatorname{CSpec}S\) as follows: \[V(C)\mathrel{\vcenter{:}}= \{P\in\operatorname{CSpec}S\mid C\subseteq P\}.\]

By 3, the set \(\operatorname{CSpec}S\) has a topology whose closed sets are represented as \(V(C)\) for some congruence \(C\) on \(S\). We always consider \(\operatorname{CSpec}S\) endowed with the topology defined above.

For a semiring homomorphism \(\phi:S_1\to S_2\), the pull-back \(\phi^{*}P_2\) of a prime congruence \(P_2\) on \(S_2\) is a prime congruence on \(S_1\). So we obtain the map \(\phi^{*}:\operatorname{CSpec}S_2\to\operatorname{CSpec}S_1\). For a closed set \(V(C_1)\) of \(\operatorname{CSpec}S_1\), \[\begin{align} (\phi^{*})^{-1}(V(C)) &= \{P\in\operatorname{CSpec}S_2\mid C\subseteq \phi^{*}P\}\\ &=\{P\in\operatorname{CSpec}S_2\mid \phi_{*}(C)\subseteq P\}\\ &=V(\phi_{*}(C)) \end{align}\] Therefore \(\phi^{*}\) is continuous.

For the discussion in 6, where we glue topological spaces \(\operatorname{CSpec}S\) together, we introduce terminology for topological spaces. A closed subset \(Z\) of a topological space is said to be reducible if \(Z\) can be written as a union \(Z=Z_1\cup Z_2\) where \(Z_1,Z_2\subsetneq Z\) are nonempty closed subsets. A closed subset \(Z\) of \(X\) is said to be irreducible if \(Z\) is not reducible.

Note that the dimension at a point can be calculated in an open neighborhood of the point, that is, \[\begin{align} \label{eq:dim32at32point32open32nbhd} \dim_x X =\dim_x U, \text{ for any open neighborhood }U\text{ of }x. \end{align}\tag{4}\] The following lemma implies that the notions such as height, coheight, and dimension are topological invariant.

In the remainder of this subsection, we collect several technical propositions. A topological embedding is an injective continuous map \(f:X\to Y\) such that the induced map \(X\to f(X)\) is a homeomorphism.

Let \(T\) be a multiplicatively closed subset of a semiring \(S\). In the following, we always assume \(0_S\notin T\) and \(1_S\in S\). We define an equivalence relation \(\sim\) on \(S\times T\) as follows: \[(s,t)\sim(s',t')\overset{\mathrm{def}}{\iff} st'u=s'tu\text{ for some u\in T.}\] We denote by \(\frac{s}{t}\) the equivalence class of \((s,t)\). The set \(T^{-1}S\mathrel{\vcenter{:}}= (S\times T)/\sim\) is a semiring with the following operation. \[\begin{align} &\frac{s}{t}+\frac{s'}{t'}=\frac{st'+s't}{tt'}\\ &\frac{s}{t}\cdot\frac{s'}{t'}=\frac{ss'}{tt'} \end{align}\] There is a canonical homomorphism of semirings \(S\to T^{-1}S,s\mapsto\frac{s}{1}\). We call this homomorphism the localization of \(S\) by \(T\).

We define a subset of \(\operatorname{CSpec}S\) by \[\begin{align} \begin{aligned}\label{eq:D40M41} D(T) &\mathrel{\vcenter{:}}= \{P\in\operatorname{CSpec}S\mid P\cap (T\times \{0_S\})=\emptyset\}\\ &=\{P\in\operatorname{CSpec}S\mid (t,0_S)\not\in P\text{ for all }t\in T\}. \end{aligned} \end{align}\tag{5}\]

2.2 \(\mathbb{B}\)-algebras↩︎

In the following, we focus on additively idempotent semirings. Then the addition is denoted by \(\oplus\), the multiplication is denoted by \(\odot\), and the additive and multiplicative identity elements are often denoted by \(-\infty\) and 0, respectively. Note that we often omit the multiplication symbol \(\odot\).

Recall that the boolean algebra \(\mathbb{B}=(\{-\infty,0\},\oplus,\odot)\) is the additively idempotent semifield which consists of only the identity elements \(-\infty,0\). An additively idempotent semiring is called a \(\mathbb{B}\)-algebra. Note that \(S\) is a \(\mathbb{B}\)-algebra if and only if there exists a semiring homomorphism from \(\mathbb{B}\) to \(S\). Such homomorphism \(\mathbb{B}\to S\) is uniquely determined and injective.

A \(\mathbb{B}\)-algebra has a natural partial order defined by: \[\begin{align} \label{eq:B-alg32poset} a\geq b \overset{\mathrm{def}}{\iff} a\oplus b=a. \end{align}\tag{6}\] If this is a total order, the \(\mathbb{B}\)-algebra is said to be totally ordered.

The following assertion is crucial for the discussion in ¿sec:section:B[M]cong?.

2.3 \(\mathbb{T}\)-algebras↩︎

Recall that \(\mathbb{T}=\mathbb{R}\cup\{-\infty\}\) is an additively idempotent semifield with the addition \(\oplus=\operatorname{max}\) and the multiplication \(\odot=+\). A \(\mathbb{T}\)-algebra is a pair of a semiring \(S\) and an injective homomorphism \(\iota\colon \mathbb{T}\to S\), called the \(\mathbb{T}\)-algebra structure of \(S\). For a \(\mathbb{T}\)-algebra \(S\), we often regard the semifield \(\mathbb{T}\) as a subset of \(S\).

By 7, any geometric congruence is prime. Thus we have \[\operatorname{CSpec}^{\mathrm{geo}}S\subseteq \operatorname{CSpec}_{\mathbb{T}} S\subseteq \operatorname{CSpec}S.\]

It is easy to show that the pull-back (see 1 ) of a prime congruence over \(\mathbb{T}\) by a homomorphism of \(\mathbb{T}\)-algebras is also a prime congruence over \(\mathbb{T}\). Thus a homomorphism \(S_1\to S_2\) of \(\mathbb{T}\)-algebras induces a continuous map \(\operatorname{CSpec}_{\mathbb{T}}S_2\to \operatorname{CSpec}_{\mathbb{T}}S_1\).

We will see some properties of the subset \(\operatorname{CSpec}_{\mathbb{T}} S\). For a \(\mathbb{T}\)-algebra \(S\), the \(\mathbb{T}\)-algebra structure \(\iota\colon \mathbb{T}\to S\) gives a continuous map \[\begin{align} \label{eq:fiberation} \phi=\iota^{*}\colon\operatorname{CSpec}S\to \operatorname{CSpec}\mathbb{T}. \end{align}\tag{7}\] The base space \(\operatorname{CSpec}\mathbb{T}\) contains two points: the diagonal \(\Delta\), and the kernel \(P_0\) of the surjective map \(\mathbb{T}\to \mathbb{B}\). Note that the singleton \(\{\Delta\}\) is an open subset of \(\operatorname{CSpec}\mathbb{T}\) whose complement is a closed subset \(V(P_0)=\{P_0\}\).

With respect to the fibration \(\operatorname{CSpec}S\to\operatorname{CSpec}\mathbb{T}\), the special fiber \(\phi^{-1}(P_0)\) corresponds to what is often called the recession in tropical geometry (see also 12).

We would like to call the property of 10 the separatedness of \(\operatorname{CSpec}_{\mathbb{T}} S\). In this paper, this is not defined in a precise way, but it is motivated by the valuative criterion for separatedness of usual schemes (see hartshorne2013algebraic? for the details).

3 Tropical algebras with monomial sets {#sec:section:B[M]}↩︎

We now proceed gradually toward a tropical setting. When we consider tropical algebras, additive idempotency is one of the minimal requirements which we should impose on algebras. However, for simplicity, we impose not only additive idempotency but also a monomial structure on algebras.

3.1 Ordered monoids↩︎

Recall that a (commutative) monoid is a set equipped with an associative and commutative binary operation \(+\) and an identity element 0. We write the monoid operation additively, although the operation \(+\) will be identified with the product \(\odot\) of a \(\mathbb{B}\)-algebra in the subsequent discussion.

An ordered monoid is a pair \((\mathsf{M},\geq)\) consisting of a commutative monoid \(\mathsf{M}=(\mathsf{M},+)\) together with a partial order \(\geq\) satisfying \[a\geq b\implies a+c\geq b+c,\] for any \(a,b,c\in \mathsf{M}\). The pair \((\mathsf{M},\geq)\) is often denoted simply by \(\mathsf{M}\). If every element of an ordered monoid \((\mathsf{M},\geq)\) is invertible, then we call it a ordered group. A monoid homomorphism \(\phi:(\mathsf{M}_1,\geq_1)\to (\mathsf{M}_2,\geq_2)\) between ordered monoids is an o-homomorphism if for any \(a,b\in \mathsf{M}_1\), \[a\geq_1 b\implies \phi(a)\geq_2\phi(b).\] If an o-homomorphism \(\phi\) is a bijective map whose inverse map is also an o-homomorphism, then \(\phi\) is called an o-isomorphism. If there exists an o-isomorphism between two ordered monoids \(\mathsf{M}_1\) and \(\mathsf{M}_2\), then \(\mathsf{M}_1\) and \(\mathsf{M}_2\) are said to be o-isomorphic.

The order on \(\mathsf{M}\) is said to be trivial if no two distinct elements of \(\mathsf{M}\) are comparable. We often regard any monoid as an ordered monoid with the trivial order.

For an ordered monoid \((\mathsf{M},\geq)\), every submonoid \(\mathsf{M}'\) of \(\mathsf{M}\) inherits the induced order, also denoted by \(\geq\). Unless otherwise stated, we equip a submonoid of an ordered monoid \(\mathsf{M}\), such as the kernel or image of an o-homomorphism, with the order induced from \(\mathsf{M}\).

We next define the localization of an ordered monoid, in particular, the group completion of an ordered monoid. Let \(\mathsf{M}\) be an ordered monoid and \(T\) be a multiplicatively closed subset of \(\mathsf{M}\). Recall that the localization of \(\mathsf{M}\) by \(T\) is a monoid \(T^{-1}\mathsf{M}=(\mathsf{M}\times T)/\sim\) where the relation \(\sim\) is given as follows: \[(a,b)\sim (c,d) \overset{\mathrm{def}}{\iff} \exists t\in T,\;a+d+t= b+c+t.\] We often denote by \(a-b\) the equivalent class of \((a,b)\). Now we also endow \(T^{-1} \mathsf{M}\) with the order induced by \(\mathsf{M}\): \[a-b\geq c-d \overset{\mathrm{def}}{\iff} \exists t\in T,\;a+d+t\geq b+c+t.\] Note that the natural homomorphism \(\mathsf{M}\to T^{-1}\mathsf{M}\) is an o-homomorphism. In particular, we denote \(\mathsf{M}^{-1}\mathsf{M}\) by \(\mathsf{M}^{\mathrm{gp}}\), called the group completion of \(\mathsf{M}\).

Before giving examples, we introduce some terminology. Let \(X_1,\dots,X_n\) be formal variables. Throughout this paper, we use the following notation. \[\begin{align} \begin{aligned}\label{monomial32notation} &M_{\geq 0}=\{u_1X_1+u_2X_2+\dots +u_nX_n\mid u_1,\dots,u_n\in \mathbb{Z}_{\geq 0}\},\\ &M=\{u_1X_1+u_2X_2+\dots +u_nX_n\mid u_1,\dots,u_n\in \mathbb{Z}\},\\ &\mathbb{R}\times M_{\geq 0}=\{(r,u_1X_1+u_2X_2+\dots +u_nX_n)\mid r\in\mathbb{R},\;u_1,\dots,u_n\in \mathbb{Z}_{\geq 0}\},\\ &\mathbb{R}\times M=\{(r,u_1X_1+u_2X_2+\dots +u_nX_n)\mid r\in\mathbb{R},\;u_1,\dots,u_n\in \mathbb{Z}\}. \end{aligned} \end{align}\tag{8}\] By abuse of notation, we often write \(u_1X_1+\dots+u_nX_n\) as \(u\cdot X\) or \(X^u\). Similarly, we write \((r,u_1X_1+\dots +u_nX_n)\) as \(r+u\cdot X\) or \(rX^u\). While we equip \(M_{\geq 0}\) and \(M\) with the trivial order, we define orders on \(\mathbb{R}\times M_{\geq 0}\) and \(\mathbb{R}\times M\) as follows: \[\begin{align} rX^{u}\geq r'X^{u'}\overset{\mathrm{def}}{\iff} r\geq r' \text{ and }u=u'. \end{align}\] These ordered monoids serve as the guiding examples throughout the paper.

3.2 \(\mathbb{B}\)-algebras associated to ordered monoids {#sec:subsection:B[M]}↩︎

We work under the setting of 2.2: an additively idempotent semiring is called a \(\mathbb{B}\)-algebra and its operations are denoted by \(\oplus\) and \(\odot\). By the following definition, we construct a \(\mathbb{B}\)-algebra from an ordered monoid \(\mathsf{M}=(\mathsf{M},+)\), identifying the monoid operation \(+\) with the product \(\odot\) of the \(\mathbb{B}\)-algebra.

For an element \(f=\sum_{m\in A}m\in\mathbb{B}[\mathsf{M}]\), the set of all maximal elements of \(A\) is called the support of \(f\), denoted \(\mathop{\mathrm{supp}}(f)\). Then \(\mathop{\mathrm{supp}}(f)\) is uniquely determined, and two distinct elements of \(\mathbb{B}[\mathsf{M}]\) have distinct supports. In other words, \(\mathop{\mathrm{supp}}\) gives the following bijection: \[\begin{align} \label{eq:bij} \mathbb{B}[\mathsf{M}]\xrightarrow{1:1}\{A\subseteq \mathsf{M}\mid A\text{ is finite and any two elements in A are not comparable}\},\, f\mapsto\mathop{\mathrm{supp}}(f). \end{align}\tag{9}\] In particular, there is an injective map \(\mathsf{M}\to \mathbb{B}[\mathsf{M}]\). So we often treat \(\mathsf{M}\) as a submonoid of \(\mathbb{B}[\mathsf{M}]\). Recall that a \(\mathbb{B}\)-algebra has a natural partial order (see 6 ). Note that the original order of \(\mathsf{M}\) and the order induced by that of \(\mathbb{B}[\mathsf{M}]\) are equivalent. We also note that any o-homomorphism \(\mathsf{M}_1\to \mathsf{M}_2\) induces a semiring homomorphism \(\mathbb{B}[\mathsf{M}_1]\to\mathbb{B}[\mathsf{M}_2]\).

We next introduce a basic invariant of prime congruences on \(\mathbb{B}[\mathsf{M}]\), which we call the mobile face.

Thus, if a face \(\mathsf{F}\subseteq \mathsf{M}\) is given, then we obtain a closed subset \(V(\Gamma_{\mathsf{F}})\) of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\). Note that for faces \(\mathsf{F}_1,\mathsf{F}_2\) of \(\mathsf{M}\), the following are equivalent:

• \(\mathsf{F}_1\subseteq \mathsf{F}_2\),

• \(\Gamma_{\mathsf{F}_2}\subseteq \Gamma_{\mathsf{F}_1}\),

• \(V(\Gamma_{\mathsf{F}_1})\subseteq V(\Gamma_{\mathsf{F}_2})\).

The space \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) can be written as a disjoint union: \[\begin{align} \label{eq:B-str} \operatorname{CSpec}\mathbb{B}[\mathsf{M}]=\coprod_{\mathsf{F}}\{P\mid \text{P is of mobile face \mathsf{F}}\}, \end{align}\tag{10}\] where \(\mathsf{F}\) runs over the set of all faces of \(\mathsf{M}\). We call this the stratification of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\). In ¿sec:section:B[M]cong?, we give a precise description of the stratification.

If an ordered monoid \(\mathsf{M}\) is totally ordered, then \(\mathbb{B}[\mathsf{M}]\) is identified with \(\mathsf{M}\sqcup \{-\infty\}\). Using this identification, we can simply express the quotient of \(\mathbb{B}[\mathsf{M}]\) by its prime congruence as follows.

3.3 \(\mathbb{T}\)-algebras associated to ordered monoids {#sec:subsection:T[M]}↩︎

We define \(\mathbb{T}\)-algebras associated to ordered monoids using [definition:B-alg]. As we have seen in 3, we can already construct some \(\mathbb{T}\)-algebra using suitable ordered monoids. However, in general, even a \(\mathbb{B}\)-algebra of the form \(\mathbb{B}[\mathbb{R}\times \mathsf{M}]\) may admit several \(\mathbb{T}\)-algebra structures. In order to fix a \(\mathbb{T}\)-algebra structure, we introduce the following notation.

::: {#definition:T[M] .definition} Definition 9. For an ordered monoid \(\mathsf{M}\), we define the order of \(\mathbb{R}\times \mathsf{M}\) by \[(r_1,m_1)\geq (r_2,m_2) \overset{\mathrm{def}}{\iff} r_1\geq r_2 \text{ and }m_1\geq m_2.\] Then \(\mathbb{R}\times \mathsf{M}\) is an ordered monoid. An o-homomorphism \[\mathbb{R}\to \mathbb{R}\times \mathsf{M},\;r\mapsto (r,0)\] induces \(\mathbb{T}\to \mathbb{B}[\mathbb{R}\times \mathsf{M}]\). With this homomorphism, we obtain a \(\mathbb{T}\)-algebra \(\mathbb{B}[\mathbb{R}\times \mathsf{M}]\), denoted by \(\mathbb{T}[\mathsf{M}]\). :::

Faces of \(\mathbb{R}\times \mathsf{M}\) take a simple form.

Similarly to 10 , the space \(\operatorname{CSpec}\mathbb{T}[\mathsf{M}]\) can be also stratified: \[\operatorname{CSpec}\mathbb{T}[\mathsf{M}]=\coprod_{\mathsf{F}}\{P\mid \text{P is of mobile face \mathbb{R}\times\mathsf{F}}\},\] where \(\mathsf{F}\) runs over the set of all faces of \(\mathsf{M}\).

Recall that the space of prime congruences on a \(\mathbb{T}\)-algebra \(S\) has a special open subset, denoted by \(\operatorname{CSpec}_{\mathbb{T}}S\) (see 4).

Thus as a set, the space \(\operatorname{CSpec}\mathbb{T}[\mathsf{M}]\) can be identified with a disjoint union of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) and \(\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}]\). We often treat them separately, as in 6.

Recall that a geometric congruence \(P\in\operatorname{CSpec}^{\mathrm{geo}}\mathbb{T}[\mathsf{M}]\) induces a \(\mathbb{T}\)-algebra homomorphism \(\mathbb{T}[\mathsf{M}]\to \mathbb{T}\) (see 4). This map \(\mathbb{T}[\mathsf{M}]\to \mathbb{T}\) is determined by an o-homomorphism \(\mathsf{M}\to \mathbb{T}\) where we regard \(\mathbb{T}\) as a monoid with the operation \(\odot\). In other words, the following map is a bijection: \[\begin{align} \label{eq:top32emb} \operatorname{Hom}(\mathsf{M},\mathbb{T})\to\operatorname{CSpec}^{\mathrm{geo}}\mathbb{T}[\mathsf{M}], \;\phi\mapsto\operatorname{Ker}\overline{\phi}, \end{align}\tag{11}\] where \(\overline{\phi}:\mathbb{T}[\mathsf{M}]\to\mathbb{T}\) is the homomorphism of \(\mathbb{T}\)-algebras induced by \[\mathbb{R}\times \mathsf{M}\to \mathbb{R}, \;(r,m)\mapsto r\odot \phi(m).\] We equip the semifield \(\mathbb{T}\) with the topology such that \[\begin{align} \mathbb{T}\to \mathbb{R}_{\geq 0}, \;r\mapsto \exp(r) \end{align}\] is a homeomorphism. We also equip the set \(\operatorname{Hom}(\mathsf{M},\mathbb{T})\) with the topology of pointwise convergence. We will show that the map 11 induces a topological embedding from \(\operatorname{Hom}(\mathsf{M},\mathbb{T})\) to \(\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}]\).

4 Prime congruences and submonoids {#sec:section:B[M]cong}↩︎

The aim of this section is to understand \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) as a poset. We mainly use 7. In 4.2, we consider the problem in the case where \(\mathsf{M}=G\) is an ordered group. In [subsection:The stratification] [subsection:The stratification T], we consider the problem in the general setting.

4.1 Convex congruences↩︎

In this and the following subsection, we assume that \(\mathsf{M}=G\) is an ordered group. We first give some observations. Let \(C\) be a congruence on \(\mathbb{B}[G]\) and \(\pi_C:\mathbb{B}[G]\to \mathbb{B}[G]/C\) be the quotient homomorphism. Then \[\begin{align} \label{eq:right32arrow} \mathsf{L}(C)\mathrel{\vcenter{:}}= \{m\in G\mid \pi_C(m)\geq 0\} \end{align}\tag{12}\] is a submonoid of \(G\). Since \(\pi_C|_G:G\to\pi_{C}(G)\) is an o-homomorphism, the submonoid \(\mathsf{L}=\mathsf{L}(C)\) satisfies the following conditions. \[\begin{align} &\{m\in \mathsf{M}\mid m\geq 0\}\subseteq \mathsf{L}. \label{L1} \end{align}\tag{13}\]

Conversely, for an arbitrary subset \(A\) of \(G\), we can define a congruence \(\langle A\geq 0\rangle\) on \(\mathbb{B}[G]\) as follows.

For a subset \(A\) of \(G\), let \(\mathsf{L}\) be a submonoid generated by \(A\) and \(\{m\in G\mid m\geq 0\}\). Then we have \[\begin{align} \label{eq:property1} \langle A\geq 0\rangle=\langle\mathsf{L}\geq 0\rangle. \end{align}\tag{14}\] Furthermore, for a family \(\{A_i\}_i\) of subsets of \(G\), we have \[\begin{align} \label{eq:property2} \left\langle\bigcup_i A_i\geq 0\right\rangle=\sum_{i}\langle A_i\geq 0\rangle, \end{align}\tag{15}\] where \(\sum_i\langle A_i\geq 0\rangle\) is the congruence generated by \(\bigcup_i\langle A_i\geq 0\rangle\).

On the other hand, for an arbitrary submonoid \(\mathsf{L}\) of \(G\) satisfying 13 , We define an order of \(G/(\mathsf{L}\cap -\mathsf{L})\) as follows: \[\begin{align} \label{eq:order} \overline{g_1}\geq \overline{g_2} \overset{\mathrm{def}}{\iff} g_1-g_2\in \mathsf{L}, \end{align}\tag{16}\] where \(\overline{g_i}\) is the image of \(g_i\in G\) in \(G/(\mathsf{L}\cap -\mathsf{L})\). Then \(G\to G/(\mathsf{L}\cap -\mathsf{L})\) is an o-homomorphism and induces the homomorphism \[\begin{align} \label{eq:hoge} \pi:\mathbb{B}[G]\to \mathbb{B}[G/(\mathsf{L}\cap -\mathsf{L})]. \end{align}\tag{17}\]

Thus two maps \(C \mapsto \mathsf{L}(C)\) and \(\mathsf{L}\mapsto \langle\mathsf{L}\geq 0\rangle\) give a one-to-one correspondence: \[\begin{align} \label{eq:0-th32bijection} \{C \mid C \text{ is a convex congruence on } \mathbb{B}[G]\} \leftrightarrow \{L \mid L \text{ is a submonoid of } G \text{ satisfying } \eqref{L1}\}. \end{align}\tag{18}\] Note that the correspondence preserves inclusions.

The following claim is crucial in 7.

4.2 The case of ordered groups↩︎

Now we consider the case of prime congruences. Since we assume that \(G\) is an ordered group, 7 can be stated more simply as follows.

If \(P\) is a prime congruence on \(\mathbb{B}[G]\), then \(\mathsf{L}(P)\) satisfies not only 13 but also the following condition: \[\begin{align} &\mathsf{L}\cup -\mathsf{L}=G. \label{L2} \end{align}\tag{20}\] We denote by \(\mathcal{L}_G\) the set of all submonoids of \(G\) satisfying 13 and 20 .

Conversely, if a submonoid \(\mathsf{L}\) of \(G\) satisfies 13 and 20 , then the congruence \(\langle\mathsf{L}\geq 0\rangle\) is prime by 17. Thus we obtain a one-to-one correspondence as a restriction of 18 .

As illustrated in 1 3, for a typical ordered group, submonoids satisfying 13 and 20 necessarily have a very restricted form. See 5 for the details.

We consider the case of a \(\mathbb{T}\)-algebra \(\mathbb{T}[G]=\mathbb{B}[\mathbb{R}\times G]\). By 16, we have an inclusion-preserving one-to-one correspondence as follows: \[\begin{align} \label{eq:TTTT} \operatorname{CSpec}\mathbb{T}[G] \longleftrightarrow \mathcal{L}_{\mathbb{R}\times G}=\{\mathsf{L}\subseteq\mathbb{R}\times G\mid \text{\mathsf{L} is a submonoid satisfying \eqref{L1} and \eqref{L2}} \}. \end{align}\tag{21}\] To describe the structure of \(\operatorname{CSpec}\mathbb{T}[G]\) in more detail, we next decompose an element of \(\mathcal{L}_{\mathbb{R}\times G}\) into two submonoids.

Let \(\mathsf{L}\) be a submonoid of \(\mathbb{R}\times G\) satisfying 13 and 20 . Since the order of \(\mathbb{R}\times G\) is given by 9, the projection \[\Pi:\mathbb{R}\times G\to G\] is an o-homomorphism. We define submonoids as follows: \[\begin{align} \begin{aligned}\label{eq:Lin} &\mathsf{L}^{\mathrm{out}} \mathrel{\vcenter{:}}= \Pi(\mathsf{L}),\\ &\mathsf{L}^{\mathrm{inn}} \mathrel{\vcenter{:}}= \mathsf{L}\cap (\mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})). \end{aligned} \end{align}\tag{22}\] It is straightforward to check that the submonoids \(\mathsf{L}^{\mathrm{out}}\subseteq G\) and \(\mathsf{L}^{\mathrm{inn}}\subseteq \mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})\) both satisfy 13 and 20 .

By translating the first coordinate, a group \(\mathbb{R}\) acts on \(\mathbb{R}\times G\), and thus also on \(\mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})\). Any orbit of the action on \(\mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})\) is written as \(\Pi^{-1}(g)\) for some \(g\in \mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}}\). Then the submonoid \(\mathsf{L}^{\mathrm{inn}}\) in 22 satisfies the following condition: \[\begin{align} \text{Every orbit intersects \mathsf{L}^{\mathrm{inn}}.} \label{L3} \end{align}\tag{23}\]

Conversely, we take arbitrary submonoids \(\mathsf{L}^{\mathrm{out}}\in \mathcal{L}_{G}\) and \(\mathsf{L}^{\mathrm{inn}}\in \mathcal{L}_{\mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})}\) where \(\mathsf{L}^{\mathrm{inn}}\) satisfies the condition 23 . Then we define the submonoid of \(\mathbb{R}\times G\) as follows: \[\begin{align} \label{eq:Lin32conv} \mathsf{L}\mathrel{\vcenter{:}}= (\mathbb{R}\times (\mathsf{L}^{\mathrm{out}}\setminus (\mathsf{L}^{\mathrm{out}}\cap -\mathsf{L}^{\mathrm{out}})))\cup \mathsf{L}^{\mathrm{inn}}. \end{align}\tag{24}\]

The construction 22 and 24 yield the following one-to-one correspondence:

Under this decomposition, \(\mathsf{L}^{\mathrm{out}}\) and \(\mathsf{L}^{\mathrm{inn}}\) are called the outer part and the inner part of \(\mathsf{L}\), respectively. The decomposition of 18 is particularly useful for prime congruences lying over \(\mathbb{T}\).

Therefore the bijection of 18 preserves inclusions in the following sense:

4.3 The stratification of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\)↩︎

Now we focus again on \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\), where \(\mathsf{M}\) is an ordered monoid, not necessarily an ordered group. Recall that the mobile face of \(P\in\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) is the face of \(\mathsf{M}\) given by \[\mathsf{F}=\{m\in\mathsf{M}\mid (m,-\infty)\notin P\}.\] We will see that \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) has the canonical stratification along the mobile face of prime congruences.

A face \(\mathsf{F}\) of \(\mathsf{M}\) can be seen as a multiplicatively closed subset of \(\mathbb{B}[\mathsf{M}]\). Recall that the localization \(\mathbb{B}[\mathsf{M}]\to\mathsf{F}^{-1}\mathbb{B}[\mathsf{M}]\) induces an inclusion \(\operatorname{CSpec}\mathsf{F}^{-1}\mathbb{B}[\mathsf{M}]\to\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) (See 5). As in the notation of 5 , we denote the image of the inclusion by \(D(\mathsf{F})\). Namely, we define \[\begin{align} \label{eq:UM} D(\mathsf{F})=\{P\in\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\mid (m,-\infty)\notin P \text{ for all }m\in\mathsf{F}\}. \end{align}\tag{25}\] We also recall that for a face \(\mathsf{F}\) of \(\mathsf{M}\), the symbol \(O_{\mathsf{F}}\) denotes the set of prime congruences on \(\mathbb{B}[\mathsf{M}]\) of mobile face \(\mathsf{F}\).

From 6, we have \[O_{\mathsf{F}}\cong \operatorname{CSpec}(\pi_{\Gamma_{\mathsf{F}}}(\mathsf{F}))^{-1}(\mathbb{B}[\mathsf{M}]/\Gamma_{\mathsf{F}}).\] From 11 and 6, we have a commutative diagram below. \[\begin{tikzcd} \mathbb{B}[\mathsf{F}]\ar[rr,"\cong"] \ar[dr,hook] \ar[ddd] & & \mathbb{B}[\mathsf{M}]/\Gamma_{\mathsf{F}} \ar[ddd]\\ & \mathbb{B}[\mathsf{M}]\ar[ur,two heads] \ar[d]& \\ & \mathsf{F}^{-1}\mathbb{B}[\mathsf{M}] \ar[dr,two heads] & \\ \mathbb{B}[\mathsf{F}^{\mathrm{gp}}]\ar[rr,"\cong"] \ar[ur,hook] & & \pi_{\Gamma_{\mathsf{F}}}(\mathsf{F})^{-1}(\mathbb{B}[\mathsf{M}]/\Gamma_{\mathsf{F}}) \end{tikzcd}\] Therefore we obtain the stratification of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\): \[\begin{align} \label{eq:process} \begin{aligned} &\operatorname{CSpec}\mathbb{B}[\mathsf{M}]=\coprod_{\mathsf{F}\subseteq \mathsf{M}}O_{\mathsf{F}},\\ &O_{\mathsf{F}}\xrightarrow{\cong} \operatorname{CSpec}\mathbb{B}[\mathsf{F}^{\mathrm{gp}}],\;P\mapsto \tilde{P}=(P\cap\mathbb{B}[\mathsf{F}])\cdot\mathbb{B}[\mathsf{F}^{\mathrm{gp}}]. \end{aligned} \end{align}\tag{26}\] Note that \(\tilde{P}\) satisfies \[P=\langle(\tilde{P}\cap \mathbb{B}[\mathsf{F}])\cdot \mathbb{B}[\mathsf{M}],\Gamma_{\mathsf{F}}\rangle.\] The symbol \(\tilde{P}\) is used in the sense above in the rest of this paper.

By 16, there is a one-to-one correspondence: \[\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\longleftrightarrow \coprod_{\mathsf{F}\subseteq \mathsf{M}}\mathcal{L}_{\mathsf{F}^{\mathrm{gp}}}.\] To understand how these strata \(O_{\mathsf{F}}\) form a whole space, we have to see when a prime congruence contains another in a distinct stratum.

For a while, we fix \(P_1,P_2\) to be points in \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) whose mobile faces are \(\mathsf{F}_1,\mathsf{F}_2\), respectively.

Thus we obtain a way to compare two points belonging to different strata.

4.4 The stratification of \(\operatorname{CSpec}\mathbb{T}[\mathsf{M}]\)↩︎

Next we consider a \(\mathbb{T}\)-algebra \(\mathbb{T}[\mathsf{M}]\). Recalling 13 and replacing \(\mathsf{M}\) with \(\mathbb{R}\times \mathsf{M}\) in 26 , we have \[\begin{align} \begin{aligned}\label{Tprocess} &\operatorname{CSpec}\mathbb{T}[\mathsf{M}] = \coprod_{\mathsf{F}\subseteq \mathsf{M}}O_{\mathbb{R}\times \mathsf{F}},\\ &O_{\mathbb{R}\times \mathsf{F}} \xrightarrow{\cong} \operatorname{CSpec}\mathbb{T}[\mathsf{F}^{\mathrm{gp}}],\;P\mapsto \tilde{P}=(P\cap\mathbb{T}[\mathsf{F}])\cdot\mathbb{T}[\mathsf{F}^{\mathrm{gp}}]. \end{aligned} \end{align}\tag{27}\]

By 22, the stratification 27 is refined as follows: \[\begin{align} \begin{aligned}\label{Tprocess2} &\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}]=\coprod_{\mathsf{F}\subseteq \mathsf{M}}(O_{\mathbb{R}\times \mathsf{F}}\cap \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}]),\\ &O_{\mathbb{R}\times \mathsf{F}}\cap \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}] \xrightarrow{\cong} \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{F}^{\mathrm{gp}}],\;P\mapsto \tilde{P}=(P\cap\mathbb{T}[\mathsf{F}])\cdot\mathbb{T}[\mathsf{F}^{\mathrm{gp}}]. \end{aligned} \end{align}\tag{28}\]

Note that by 10, any prime congruence \(P\) on \(\mathbb{T}[\mathsf{M}]\) is contained in at most one geometric congruence.

5 Representation of submonoids↩︎

In this section, we examine the set of prime congruences on typical tropical algebras, such as \[\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}],\;\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}],\;\mathbb{B}[X_1,\dots,X_n],\;\mathbb{T}[X_1,\dots,X_n].\]

5.1 The Laurent polynomial semirings over \(\mathbb{B}\)↩︎

In this subsection, we consider the set of prime congruences on the Laurent polynomial semiring \(\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\). It is represented as an algebra introduced in [definition:B-alg]: \[\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]=\mathbb{B}[M].\] Applying \(G=M\) to 16, we obtain a one-to-one correspondence: \[\begin{align} \operatorname{CSpec}\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\longleftrightarrow \mathcal{L}_M. \end{align}\] Since \(M\) is equipped with the trivial order, we may impose only the condition 20 on submonoids of \(M\). Namely, \[\mathcal{L}_M=\{\mathsf{L}\subseteq M\mid \mathsf{L}\cup -\mathsf{L}=M\}.\] As we show in this subsection, such a submonoid \(\mathsf{L}\) admits a lexicographic stratification described by an \(M\)-flag.

Let \(V\) be an \(\mathbb{R}\)-linear space and \(W=\{v\in V\mid l(v)=0\}\) be a linear hyperplane. An orientation of \(W\) is defined as either the set \(\{v\in V\mid l(v)> 0\}\) or \(\{v\in V\mid l(v)< 0\}\).

It follows directly that the codimension of \(H_i\) in \(M_{\mathbb{R}}\) is greater than or equal to \(i\). Thus we have \(l\leq n\).

First, for an \(M\)-flag \(H_{\bullet}\), we define a submonoid of \(M\): \[\Psi_1(H_{\bullet})\mathrel{\vcenter{:}}= \left(\bigcup_{i=1}^l H_{i,+}\cup H_{l}\right)\cap M.\]

Thus we obtain a map \[\Psi_1\colon \mathcal{F}_M \to \{\mathsf{L}\subseteq M\mid \text{\mathsf{L} is a submonoid satisfying \eqref{L2}}\}.\]

Next, we construct the inverse map of \(\Psi_1\). Let \(\mathsf{L}\) be a submonoid of \(M\) satisfying 20 . We denote by \(\overline{\operatorname{cone}(\mathsf{L})}\) the closure of the convex cone spanned by \(\mathsf{L}\) in \(M_{\mathbb{R}}\). The subset \(\overline{\operatorname{cone}(\mathsf{L})}\cap-\overline{\operatorname{cone}(\mathsf{L})}\) is the linear subspace of \(M_{\mathbb{R}}\).

If the codimension of \(H=\overline{\operatorname{cone}(\mathsf{L})}\cap-\overline{\operatorname{cone}(\mathsf{L})}\) is 1, we obtain a new submonoid \[\mathsf{L}'\mathrel{\vcenter{:}}= \mathsf{L}\cap H.\] It also satisfies 20 as a submonoid of \(M\cap H\). Furthermore, the hyperplane \(H\) of \(M_{\mathbb{R}}\) has a natural orientation \(\overline{\operatorname{cone}(\mathsf{L})}\setminus H\).

Repeating this procedure yields a sequence of subspaces. That is, for a submonoid \(\mathsf{L}\) of \(M\) satisfying 20 , we set \[\begin{align} &H_0=M_{\mathbb{R}},\\ &H_{i+1}=\overline{\operatorname{cone}(H_i\cap \mathsf{L})}\cap -\overline{\operatorname{cone}(H_i\cap \mathsf{L})},\\ &H_{i+1,+}=\overline{\operatorname{cone}(H_i\cap \mathsf{L})}\setminus H_{i+1}. \end{align}\] Since the dimension of \(M_{\mathbb{R}}\) is finite, there exists an integer \(i\) such that \[\begin{align} &H_i \text{ is a hyperplane of }\mathbb{R}(H_{i-1}\cap M) \text{ if i\leq l,}\\ &H_i=\mathbb{R}(H_{i-1}\cap M)\text{ if i> l.} \end{align}\] Then we obtain a sequence \(\{H_i\}_{i=0}^l\), which is actually an \(M\)-flag. Thus we obtain the map \[\Psi_2\colon\{\mathsf{L}\subseteq M\mid \text{\mathsf{L} is a submonoid satisfying \eqref{L2}}\}\to\mathcal{F}_M.\]

We often denote by \(H(P)_{\bullet}\) the \(M\)-flag corresponding to \(P\): \[\begin{align} {2} \operatorname{CSpec}\mathbb{B}[M] &\xrightarrow{1:1} &\mathcal{L}_M \xrightarrow{1:1} &\mathcal{F}_M,\\ P &\longmapsto &\mathsf{L}(P) \longmapsto &H(P)_{\bullet}. \end{align}\] Note that by definition of \(\Psi_1\), we have \[\mathsf{L}(P)\cap-\mathsf{L}(P)=H(P)_l\cap M.\]

For an \(M\)-flag \(H_{\bullet }=\{H_i\}_{i=0}^{l}\) and \(l'\in \{0,\dots,l\}\), the sequence \(H'_{\bullet}=\{H_i\}_{i=0}^{l'}\) is also an \(M\)-flag. We call \(H'_{\bullet}\) a truncation of \(H_{\bullet}\).

We give an alternative proof of the following proposition, which was established joo2018prime?.

5.2 The Laurent polynomial semirings over \(\mathbb{T}\)↩︎

In this subsection, we consider the set of prime congruences on the Laurent polynomial semiring \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]=\mathbb{B}[\mathbb{R}\times M]\). Applying \(G=\mathbb{R}\times M\) to 16, we obtain a one-to-one correspondence: \[\begin{align} \operatorname{CSpec}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\longleftrightarrow \mathcal{L}_{\mathbb{R}\times M}. \end{align}\] In a manner similar to 24, the space \(\operatorname{CSpec}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) can be described by the sequence of linear spaces. However since \(\mathbb{R}\times M\) has the non-trivial order, its non-negative cone is not zero. For this reason, we have to impose a condition on the choice of each orientation.

As in the case of \(M\)-flags, we often denote by \(H(P)_{\bullet}\) the \((\mathbb{R}\times M)\)-flag corresponding to \(P\): \[\begin{align} {2} \operatorname{CSpec}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]&\xrightarrow{1:1} &\mathcal{L}_{\mathbb{R}\times M} \xrightarrow{1:1} &\mathcal{F}_{\mathbb{R}\times M},\\ P &\longmapsto &\mathsf{L}(P) \longmapsto &H(P)_{\bullet}. \end{align}\] Note that \(P\) is obtained by defining the order of \((\mathbb{R}\times M)/(H_l\cap(\mathbb{R}\times M))\) as follows: \[\begin{align} \label{eq:order32by32flag32over32T} \overline{m_1}\geq \overline{m_2} \overset{\mathrm{def}}{\iff} m_1-m_2\in \bigcup_{i=1}^l H_{i,+}\cup H_l. \end{align}\tag{30}\]

Truncation of an \((\mathbb{R}\times M)\)-flag is defined in the same way as those of an \(M\)-flag. The following proposition and corollary are proved in a similar way to 25 and 2.

As the submonoid \(\mathsf{L}(P)\) is decomposed in 18, an \((\mathbb{R}\times M)\)-flag we obtain another description of \(\operatorname{CSpec}\mathbb{T}[M]\).

By 19, the decomposition of 29 is refined as follows:

Thus we obtain a corollary.

We will describe the decomposition of flags explicitly. Let \(\Pi\colon \mathbb{R}\times M\to M\) be the projection. For an \((\mathbb{R}\times M)\)-flag \(H_{\bullet}\) of length \(l\), let \(l'\geq 0\) be the largest integer such that \(\mathbb{R}\times \{0\}\subseteq H_{l'}\). For the following lemma, we temporarily define an \(M\)-flag \(H_{\bullet}^{\mathrm{(out)}}\) by \[\begin{align} &H_i^{\mathrm{(out)}}=\Pi(H_i)\text{ for }i=0,\dots,l'\\ &H_{i,+}^{\mathrm{(out)}}=\Pi(H_{i,+})\text{ for }i=1,\dots,l', \end{align}\] and an \((H_{l'}\cap(\mathbb{R}\times M))\)-flag \(H_{\bullet}^{\mathrm{(inn)}}\) by \[\begin{align} &H_i^{\mathrm{(inn)}}= \begin{cases*} H_{l'}\cap(\mathbb{R}\times M) \text{ if }i=0,\\ H_{l'+i} \text{ if }i=1,\dots,l-l', \end{cases*}\\ &H_{i,+}^{\mathrm{(inn)}}=H_{l'+i}\text{ for }i=1,\dots,l-l'. \end{align}\]

Then the bijection of 29 is described as follows: \[\begin{align} \begin{aligned}\label{eq:bij32inn32out32flag} H_{\bullet}=\{H_i\}_{i=0}^l &\longmapsto (H_{\bullet}^{\mathrm{(out)}},H_{\bullet}^{\mathrm{(inn)}})=(\{\Pi(H_i)\}_{i=0}^{l'},\{H_{l'+i}\}_{i=0}^{l''}),\\ [\mathbb{R}\times H_0^{\mathrm{out}}\supsetneq \dots\supsetneq \mathbb{R}\times H_{l'}^{\mathrm{out}}\supsetneq H_1^{\mathrm{inn}}\supsetneq \dots\supsetneq H_{l''}^{\mathrm{inn}}] &\longmapsfrom (H_{\bullet}^{\mathrm{out}},H_{\bullet}^{\mathrm{inn}}). \end{aligned} \end{align}\tag{31}\] Note that via the bijection, the length \(l\) of \(H_{\bullet}\) is equal to \(l'+l''\).

5.3 The polynomial semirings over \(\mathbb{B}\) and \(\mathbb{T}\)↩︎

In this subsection, we consider the set of prime congruences on the polynomial semiring \(\mathbb{B}[X_1,\dots,X_n]\) and \(\mathbb{T}[X_1,\dots,X_n]\). By the notation 8 , we have \[\mathbb{B}[X_1,\dots,X_n]=\mathbb{B}[M_{\geq 0}],\;\mathbb{T}[X_1,\dots,X_n]=\mathbb{B}[\mathbb{R}\times M_{\geq 0}].\]

For \(I\subseteq \{1,\dots,n\}\), we define a submonoid \(\mathsf{F}_I\) of \(M_{\geq 0}\) as follows: \[\mathsf{F}=\{uX=u_1X_1+\dots+u_nX_n \mid u_i=0 \text{ for all }i\notin I\},\] which is isomorphic to \((\mathbb{Z}_{\geq 0})^{I}\).

The stratification 26 is described as follows: For \(\mathsf{M}=M_{\geq 0}\), \[\begin{align} \begin{aligned}\label{eq:flag32corr32monoid32B} &\operatorname{CSpec}\mathbb{B}[X_1,\dots,X_n]=\coprod_{I\subseteq \{1,\dots,n\}}O_{\mathsf{F}_I},\\ &O_{\mathsf{F}_I}\xrightarrow{\cong} \operatorname{CSpec}\mathbb{B}[\mathsf{F}_I^{\mathrm{gp}}],\,P\mapsto \tilde{P}=(P\cap \mathbb{B}[\mathsf{F}_I])\cdot \mathbb{B}[\mathsf{F}_I^{\mathrm{gp}}],\\ &\operatorname{CSpec}\mathbb{B}[\mathsf{F}_I^{\mathrm{gp}}]=\operatorname{CSpec}\mathbb{B}[X_i^{\pm}\mid i\in I]\longleftrightarrow \mathcal{F}_{\mathbb{Z}^I}, \end{aligned} \end{align}\tag{32}\] and for \(\mathsf{M}=\mathbb{R}\times M_{\geq 0}\), \[\begin{align} \begin{aligned}\label{eq:flag32corr32monoid32T} &\operatorname{CSpec}\mathbb{T}[X_1,\dots,X_n]=\coprod_{I\subseteq \{1,\dots,n\}}O_{\mathbb{R}\times\mathsf{F}_I},\\ &O_{\mathbb{R}\times \mathsf{F}_I}\xrightarrow{\cong} \operatorname{CSpec}\mathbb{T}[\mathsf{F}_I^{\mathrm{gp}}],\,P\mapsto \tilde{P}=(P\cap \mathbb{T}[\mathsf{F}_I])\cdot \mathbb{T}[\mathsf{F}_I^{\mathrm{gp}}],\\ &\operatorname{CSpec}\mathbb{T}[\mathsf{F}_I^{\mathrm{gp}}]=\operatorname{CSpec}\mathbb{T}[X_i^{\pm}\mid i\in I]\longleftrightarrow \mathcal{F}_{\mathbb{R}\times \mathbb{Z}^I}, \end{aligned} \end{align}\tag{33}\]

24 provides the complete description of each stratum \(O_{\mathsf{F}_I}\) and \(O_{\mathbb{R}\times\mathsf{F}_I}\). Moreover, by 21, we know precisely how these strata form the whole space.

We introduce restriction of flags, which is a basic operation on flags on a par with truncation. For a prime congruence \(P\) on \(\mathbb{B}[M]\), let \(P'\) be the pull-back of \(P\) by the inclusion \(\mathbb{B}[M']\subseteq \mathbb{B}[M]\), where \(M'\) is a subgroup of \(M\) satisfying \(M'_{\mathbb{R}}\cap M=M'\). We compare the \(M\)-flag \(H_{\bullet}\) corresponding to \(P\) and the \(M'\)-flag corresponding to \(P'\).

By 17, we have \(\mathsf{L}(P')=\mathsf{L}(P)\cap M'\). \[\begin{align} \label{eq:restriction} \mathsf{L}(P')\cap M'=\left(\bigcup_{i=1}^{l}H_{i,+}\cup H_l \right)\cap M'=\left(\bigcup_{i=1}^{l}(H_{i,+}\cap {M'_{\mathbb{R}}})\cup (H_l\cap M'_{\mathbb{R}}) \right)\cap M'. \end{align}\tag{34}\] Although the sequence \(\{H_{i}\cap M'_{\mathbb{R}}\}_{i=0}^l\) is not an \(M'\)-flag in general, the codimension of \(H_i\cap M'_{\mathbb{R}}\) in \(\mathbb{R}((H_{i-1}\cap M'_{\mathbb{R}})\cap M')\) is at most 1 for \(i=1,\dots,l\). Then we obtain an \(M'\)-flag by eliminating \(H_i\cap M'_{\mathbb{R}}\) such that \[H_i=\mathbb{R}((H_{i-1}\cap M'_{\mathbb{R}})\cap M')\] from the sequence \(\{H_{i}\cap M'_{\mathbb{R}}\}_{i=0}^l\).

By the assumption that \(M'_{\mathbb{R}}\cap M=M'\) and 34 , the \(M'\)-flag \(H_{\bullet}\cap^{*}M'\) corresponds to the submonoid \(\mathsf{L}\cap M'\) and the prime congruence \(P'=P\cap \mathbb{B}[\mathsf{M}']\). An \(M'\)-flag \(H_{\bullet}\cap^{*}M'\) has length at most the length of \(H_{\bullet}\). In particular, if \(M'\subsetneq M\), then the inequality is strict.

For an \((\mathbb{R}\times M)\)-flag \(H_{\bullet}\) and a subgroup \(M'\) of \(M\) satisfying \(M'_{\mathbb{R}}\cap M=M'\), we define an \((\mathbb{R}\times M')\)-flag \(H_{\bullet}\cap^{*}(\mathbb{R}\times M')\) similarly.

In fact, these assertions are generalized as follows: \[\begin{align} \dim\mathbb{B}[\mathsf{M}_{\sigma}]=n,\;\dim\mathbb{T}[\mathsf{M}_{\sigma}]=n+1, \end{align}\] where \(\sigma\) is a cone in \(N\) and \(\mathsf{M}_{\sigma}=M\cap \sigma^{\vee}\) (see 6 for details). In particular, properties of the open subset \(\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\subseteq \operatorname{CSpec}\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) are examined in 6.1.

Here we present a concrete example.

6 Tropical toric schemes↩︎

In this section, we construct tropical toric schemes using sets of prime congruences. Most of the discussion below generalizes that of 5.

First we recall the usual setting in toric geometry. Let \(M\) and \(N\) be free abelian groups of rank \(n\) with a pairing \[\langle\;\cdot\;,\;\cdot\;\rangle\colon M\times N\to \mathbb{Z}, \, (m,n)\mapsto \langle m,n\rangle.\] Set \(M_{\mathbb{R}}=M\otimes_{\mathbb{Z}} \mathbb{R}\) and \(N_{\mathbb{R}}=N\otimes_{\mathbb{Z}} \mathbb{R}\). A (convex) cone \(\sigma\) is a subset of \(N_{\mathbb{R}}\) represented by \[\sigma=\left\{\sum_{v\in S}a_v v\; \middle|\; a_v\in\mathbb{R}_{\geq0}\right\},\] for some finite subset \(S\subseteq N\), and its dimension is the dimension of the linear span of \(\sigma\) in \(N_{\mathbb{R}}\). A cone \(\sigma\) is strongly convex if \(\sigma\cap -\sigma=\{0\}\). A subset \(\tau\) of \(\sigma\) is called a face of \(\sigma\) if the following condition holds for any \(v_1,v_2\in \sigma\): \[v_1+v_2\in \tau\implies v_1,v_2\in \tau.\] We denote it by \(\tau\prec\sigma\). Note that if we regard cones as ordered monoids with the trivial order, 7 agrees with the present definition.

A fan \(\Sigma\) is a polyhedral complex in \(N_{\mathbb{R}}\) consisting of strongly convex cones. For a cone \(\sigma\subseteq N_{\mathbb{R}}\), define subsets of \(M\): \[\begin{align} &\sigma^{\perp}\mathrel{\vcenter{:}}= \{m\in M_{\mathbb{R}}\mid \langle m,n\rangle= 0 \text{ for all }n\in \sigma\},\\ &\sigma^{\vee}\mathrel{\vcenter{:}}= \{m\in M_{\mathbb{R}}\mid \langle m,n\rangle\geq 0 \text{ for all }n\in \sigma\},\\ &\mathsf{M}_{\sigma}\mathrel{\vcenter{:}}= \sigma^{\vee}\cap M. \end{align}\]

6.1 Tropical affine toric schemes↩︎

As in standard toric geometry, we construct local models of tropical toric schemes from cones in \(N_{\mathbb{R}}\). For a cone \(\sigma\), we define \[U_{\sigma}^{\operatorname{cong}}\mathrel{\vcenter{:}}= \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}_{\sigma}]=\operatorname{CSpec}_{\mathbb{T}}\mathbb{B}[\mathbb{R}\times \mathsf{M}_{\sigma}].\]

By 28 and 31, we obtain the stratification of \(U_{\sigma}^{\operatorname{cong}}\): \[\begin{align} \label{eq:toric32process} \begin{aligned} &U_{\sigma}^{\operatorname{cong}} = \coprod_{\tau\prec \sigma}(O_{\mathbb{R}\times(\mathsf{M}_{\sigma}\cap\tau^{\perp})}\cap \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}_{\sigma}])\\ &O_{\mathbb{R}\times (\mathsf{M}_{\sigma}\cap\tau^{\perp})}\cap \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}_{\sigma}] \cong \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\tau^{\perp}\cap M] \end{aligned} \end{align}\tag{35}\] We write each stratum \(O_{\mathbb{R}\times (\mathsf{M}_{\sigma}\cap\tau^{\perp})}\cap \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}_{\sigma}]\) briefly as \(N^{\operatorname{cong}}(\tau)\).

Each element of \(N^{\operatorname{cong}}(\tau)\) corresponds to a pair of two flags by 29: \[\begin{align} \label{eq:toric32process2} \begin{aligned} &N^{\operatorname{cong}}(\tau) \longleftrightarrow \left\{(H_{\bullet}^{\mathrm{out}},H_{\bullet}^{\mathrm{inn}})=(\{H_{i}^{\mathrm{out}}\}_{i=0}^{l'},\{H_{i}^{\mathrm{inn}}\}_{i=0}^{l''})\;\middle|\; \begin{align} &H_{\bullet}^{\mathrm{out}}\in \mathcal{F}_{\tau^{\perp}\cap M}, H_{\bullet}^{\mathrm{inn}}\in \mathcal{F}_{\mathbb{R}\times (H_{l'}^{\mathrm{out}}\cap M)},\\ &\text{l''>0 and \mathbb{R}\times \{0\}\not\subseteq H_1^{\mathrm{inn}} holds.} \end{aligned} \right\} \end{align} \end{align}\tag{36}\]

To characterize points of maximal height, we provide a way to construct a sequence of prime congruences.

Let \(P\) be a point of a stratum \(N^{\operatorname{cong}}(\tau)\) where \(\tau\) is a face of \(\sigma\). We assume that \(P\) is geometric. By 22 31, \(\tilde{P}\) is geometric and corresponds to an \((\mathbb{R}\times (\tau^{\perp}\cap M))\)-flag of length 1: \[\begin{align} &H(P)_0=\mathbb{R}\times (\tau^{\perp}\cap M)\supsetneq H(P)_1, \end{align}\] where \(H(P)_1\) is a hyperplane of \(\mathbb{R}\times \tau^{\perp}\) satisfying \(\mathbb{R}\times \{0\}\not\subseteq H(P)_1\).

We will show that there exists a sequence of prime congruences of length \(n\) contained in \(P\). Take a sequence \[\{0\}=\tau_{0}\prec \tau_{1}\prec \dots\prec \tau_{\dim\tau}=\tau,\;\dim\tau_i=i,\] of faces in \(\tau\). Then we obtain a sequence of linear subspaces of \(M_{\mathbb{R}}\) \[M_{\mathbb{R}}=\tau_{0}^{\perp}\supsetneq \tau_{1}^{\perp}\supsetneq \dots\supsetneq \tau_{\dim\tau-1}^{\perp} \supsetneq \tau^{\perp}.\] We define an \(M\)-flag by \[\begin{align} {2} &H_i^{\mathrm{out}}\mathrel{\vcenter{:}}=\tau_i^{\perp} &\text{ for }i=0,\dots,\dim\tau,\\ &H_{i,+}^{\mathrm{out}}\mathrel{\vcenter{:}}=\tau_{i-1}^{\perp}\cap(-\tau_i^{\vee}\setminus \tau_i^{\perp}) &\text{ for }i=1,\dots,\dim\tau, \end{align}\] For each \(j=0,\dots,\dim\tau\), the restriction \(H_{\bullet}^{\mathrm{out}}\cap^{*}(\tau_{\dim\tau-j}^{\perp}\cap M)\) is of the form \[\tau_{\dim\tau-j}^{\perp}\supsetneq \dots\supsetneq \tau^{\perp}\] and has length \(j\).

On the other hand, since the rank of \(H(P)_1\cap (\mathbb{R}\times M)=\tau^{\perp}\cap M\) is equal to \(\dim\tau^{\perp}\), the flag \(H(\tilde{P})_{\bullet}\) can be extended to one of length \(1+\dim\tau^{\perp}\) arbitrarily. That is, there exists an \((\mathbb{R}\times (\tau^{\perp}\cap M))\)-flag \(H^{\mathrm{inn}}_{\bullet}\) of length \(1+\dim\tau^{\perp}\) such that \(H^{\mathrm{inn}}_1=H(\tilde{P})_1\). In particular, for each \(k=1,\dots,\dim\tau^{\perp}+1\), we obtain an \((\mathbb{R}\times (\tau_k^{\perp}\cap M))\)-flag \(\{H^{\mathrm{inn}}_{i}\}_{i=0}^{k}\) of length \(k\).

Thus we obtain a family of pairs of flags \[(H_{\bullet}^{\mathrm{out}}\cap^{*}(\tau_{\dim\tau-j}^{\perp}\cap M), \{H^{\mathrm{inn}}_{i}\}_{i=0}^{k}), \text{ for }j=0,\dots,\dim\tau, \text{ and }k=1,\dots,\dim\tau^{\perp}+1.\] Let \(P_{j,k}\) be the corresponding point in \(N^{\operatorname{cong}}(\tau_j)\) under the correspondence 36 . Note that the mobile face of \(P_{j,k}\) is \(\mathbb{R}\times (\tau_{\dim\tau-j}^{\perp}\cap \mathsf{M}_{\sigma})\).

We provide a technical lemma.

Note that by 10, for any point \(P\in U_{\sigma}\), its closure \(\overline{\{P\}}\) contains at most one geometric congruence.

6.2 Tropical toric schemes and their properties↩︎

We now turn to the study of global objects. The usual tropical toric varieties are introduced by Kajiwara2008TropicalToric? and payne2009analytification?. We follow the definition of maclagan2015introduction?.

To give this a topology, we define the space of all homomorphisms of monoids from \((\sigma^{\vee}\cap M,+)\) to \((\mathbb{T},\odot)\): \[U_{\sigma}^{\operatorname{trop}}=\operatorname{Hom}(\mathsf{M}_{\sigma},\mathbb{T}),\] where \(\mathsf{M}_{\sigma}=\sigma^{\vee}\cap M\). We equip \(U_{\sigma}^{\operatorname{trop}}\) with the topology of pointwise convergence and \(X_{\Sigma}^{\operatorname{trop}}\) with the topology induced by the covering \(X_{\Sigma}^{\operatorname{trop}}=\bigcup_{\sigma\in\Sigma}U_{\sigma}^{\operatorname{trop}}\).

Now we construct the tropical toric scheme \(X_{\Sigma}^{\operatorname{cong}}\) from spaces of prime congruences. As defined in the previous subsection, define \(U_{\sigma}^{\operatorname{cong}}\) for each cone \(\sigma\) in \(\Sigma\): \[U_{\sigma}^{\operatorname{cong}}=\operatorname{CSpec}_{\mathbb{T}} \mathbb{T}[\sigma^{\vee}\cap M].\] If \(\tau\) is a face of \(\sigma\), we can find \(m\in \sigma^{\vee}\cap M\) such that \(\tau =m^{\perp}\cap \sigma\) where \(m^{\perp}=\{n\in N_{\mathbb{R}}\mid \langle m,n\rangle=0\}\). Then \(\mathsf{M}_{\sigma}+\mathbb{Z}(-m)=\mathsf{M}_{\tau}\) holds. The localization induces \[\mathbb{T}[\mathsf{M}_{\sigma}]\to\{0,m,2m,\dots\}^{-1}\mathbb{T}[\mathsf{M}_{\sigma}]=\mathbb{T}[\mathsf{M}_{\tau}].\] It induces an open immersion \(U_{\tau}^{\operatorname{cong}}\hookrightarrow U_{\sigma}^{\operatorname{cong}}\). Using these maps, we obtain a topological space \(X_{\Sigma}^{\operatorname{cong}}\), called a tropical toric scheme associated to a fan \(\Sigma\).

As in the definition of \(X_{\Sigma}^{\operatorname{trop}}\), a tropical toric scheme \(X_{\Sigma}^{\operatorname{cong}}\) also has the stratification. Recall that affine toric schemes admit a stratification given by 35 . If \(\tau\) is a face of \(\sigma_1\) and \(\sigma_2\), the construction of \(X_{\Sigma}^{\operatorname{cong}}\) identifies two strata \(N^{\operatorname{cong}}(\tau_i)\) for \(i=1,2\), denoted by \(N^{\operatorname{cong}}(\tau)\). Then we have the stratification: \[\begin{align} \label{eq:strati32toric} \begin{aligned} &X_{\Sigma}^{\operatorname{cong}} = \coprod_{\tau\in\Sigma}N^{\operatorname{cong}}(\tau),\\ &N^{\operatorname{cong}}(\tau) \xrightarrow{\cong} \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\tau^{\perp}\cap M], \;P\mapsto \tilde{P}. \end{aligned} \end{align}\tag{37}\] Thus we obtain a stratification of \(X_{\Sigma}^{\operatorname{cong}}\) analogous to that of \(X_{\Sigma}^{\operatorname{trop}}\). By 36 , \(X_{\Sigma}^{\operatorname{cong}}\) can also be seen as a set of flags.

Recall the map 11 : \[\iota_{\sigma}\colon\operatorname{Hom}(\mathsf{M}_{\sigma},\mathbb{T})\to \operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[\mathsf{M}_{\sigma}].\] By 13 and 32, the image of \(\iota_{\sigma}\) is the set of points of height \(n\). The morphisms \(\{\iota_{\sigma}\}_{\sigma\in\Sigma}\) glue together to give an injective map \(\iota:X_{\Sigma}^{\operatorname{trop}}\to X_{\Sigma}^{\operatorname{cong}}\). Recalling 3, the map \(\iota\) satisfies the following theorem.

In what follows, we regard \(X_{\Sigma}^{\operatorname{trop}}\) as a subspace of \(X_{\Sigma}^{\operatorname{cong}}\). A point \(P\) of \(X_{\Sigma}^{\operatorname{cong}}\) is called a closed point if a singleton \(\{P\}\) is a closed subset. Since \(\dim X_{\Sigma}^{\operatorname{cong}}\) is equal to \(n\), any point \(P\) in \(X_{\Sigma}^{\operatorname{trop}}\) is closed. However, a closed point is not of height \(n\) in general. Indeed, in 8, a point \(P_{+}\) is closed in \(\operatorname{CSpec}_{\mathbb{T}}\mathbb{T}[X^{\pm}]\) but of height 0. Next, we examine when the two notions coincide, that is, when every closed point is of height \(n\).

For \(m\in M\), we define \(m^{\perp}=\{n\in N_{\mathbb{R}}\mid \langle m,n\rangle=0\}\).

7 Finitely generated prime congruences↩︎

Finally, we consider a purely algebraic problem: determining which prime congruences are finitely generated. First we refer to a general result.

Let \(\mathsf{M}\) be an ordered monoid and \(P\) a point of \(\operatorname{CSpec}\mathbb{B}[\mathsf{M}]\) whose mobile face is \(\mathsf{F}\subseteq \mathsf{M}\). Recall the relation of \(P\) and \(\tilde{P}\): \[\begin{align} &\tilde{P}=(P\cap \mathbb{B}[\mathsf{F}])\cdot \mathbb{B}[\mathsf{F}^{\mathrm{gp}}],\\ &P=\langle(\tilde{P}\cap \mathbb{B}[\mathsf{F}])\cdot \mathbb{B}[\mathsf{M}],\Gamma_{\mathsf{F}}\rangle. \end{align}\] In general, the congruence \(\Gamma_{\mathsf{F}}\) is not finitely generated. However, in the case of \(\mathsf{M}=\mathsf{M}_{\sigma}\) or \(\mathbb{R}\times \mathsf{M}_{\sigma}\) (see 6), \(\Gamma_{\mathsf{F}}\) is finitely generated. Indeed, the quotient of \(\mathsf{M}\) by any face is a finitely generated monoid.

By 38, it is enough to consider the case of \(\mathbb{B}[X_1^{\pm},\dots,X_n^{\pm}]\) and \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\).

The situation is similar in the case of \(G=\mathbb{R}\times M\). Note that in this case, \(\{m\in G \mid m\geq 0\}\) is equal to \(\{(r,0)\mid r\geq 0\}=\mathbb{Z}_{\geq 0}\langle(1,0)\rangle\). Recall that a geometric congruence on \(\mathbb{T}[X_1^{\pm},\dots,X_n^{\pm}]\) is a prime congruence associated to a point \(x\in\mathbb{R}^n\).

Note that congruences in (4) of 39 are not lying over \(\mathbb{T}\) except for geometric congruences. Thus the following corollary is immediate.

The following claim follows from 38 and 8.

Thus we have obtained another characterization of the subset \(U^{\operatorname{trop}}_{\sigma}\) of \(U^{\operatorname{cong}}_{\sigma}\).