Coordination sequences of crystals are of quasi-polynomial type

It is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889] in their study of coordination sequences of zeolites.

Abstract

The coordination sequence of a graph measures how many vertices the graph has at each distance from a fixed vertex and is a generalization of the coordination number. Here it is proved that the coordination sequence of the graph obtained from a crystal is of quasi-polynomial type, as had been postulated by Grosse-Kunstleve et al. [Acta Cryst. (1996), A52, 879–889].

1. Introduction  

For a graph Γ and a fixed vertex of Γ, and for a non-negative integer n, the coordination sequence is defined as the number of vertices of Γ at distance n from . For example, the first few terms of the coordination sequence of the graph in Fig. 1 ▸ are , , , . That is, the graph has only one point at distance 0 from the origin (by definition), has four vertices at distance 1, and so on. An easy observation shows that in this case we have ().

Figure 1.

Graph with a translation symmetry. The number attached to each vertex represents the graph distance from the origin O.

In this paper, we consider a periodic graph Γ in the following sense:

(A) Γ is a (possibly directed) graph with a free action such that the quotient graph is finite.

This assumption is motivated from the crystallographic viewpoint. Our main example is a graph obtained by a crystal, i.e. the vertices of Γ are the set of atoms of the crystal, and to each atomic bond connecting two atoms u and v, we associate an edge connecting the vertices u and v. Then, the number is nothing but the usual coordination number and the coordination sequence can be thought of as its generalization.

The coordination sequences of periodic graphs are predicted to be of quasi-polynomial type (see Definition 1.2) by Grosse-Kunstleve et al. (1996 ▸). After that, various mathematical methods to calculate coordination sequences have been developed and they are actually calculated in many specific cases as in the work of Conway & Sloane (1997 ▸), Eon (2002 ▸, 2012 ▸), Goodman-Strauss & Sloane (2019 ▸), O’Keeffe (1995 ▸, 1998 ▸), Shutov & Maleev (2018 ▸, 2019 ▸, 2020 ▸).

The purpose of this paper is to give the affirmative answer to the question posed by Grosse-Kunstleve et al. (1996 ▸) (see Theorem 2.2 for the more general statement).

Theorem 1.1

The coordination sequence of a graph satisfying the condition (A) is of quasi-polynomial type. In particular, its generating function is rational. Furthermore, it becomes of polynomial type if the quotient graph has only one vertex.

Here, we recall the definition of functions of quasi-polynomial type.

Definition 1.2

A function is called a quasi-polynomial if there exist an integer and polynomials with the following condition:

(i) For any , the equality holds if N divides .

We say that a function is of quasi-polynomial type if there exist an integer and a quasi-polynomial such that holds for any integer . As a special case, we say that f is of polynomial type when .

The paper is organized as follows: in Section 2, we prove Theorem 1.1 in more general settings (Theorem 2.2). The key ingredient of the proof of Theorem 2.2 is monoid theory in abstract algebra. Readers unfamiliar with monoid theory are advised to read Appendix A first, where we summarize definitions and propositions in monoid theory with typical examples. In Appendix A , we also prove Theorem A12, which is a key to the proof of Theorem 2.2. In Section 3, we explain how to apply Theorem 2.2 to the graph obtained from a crystal, and we give some examples.

2. Coordination sequences of graphs with abelian group action  

In this paper, a graph means a directed simple graph, i.e. V is the set of vertices and is the set of edges. We say that a graph is finite when both V and E are finite sets.

Remark 2.1

A simple undirected graph can be regarded as a graph in the sense above. In fact, since a simple undirected graph consists of sets and , we obtain a graph Γ from by setting and . Here the symbol is employed to denote the cardinality of the set A.

For vertices , we denote by the length of a shortest directed path in E from x to y. If there is no directed path connecting x and y, then the distance is defined as infinite.

Let H be a group. We say that H acts on when H acts on both V and E and these actions are compatible, i.e. we assume that H acts on V and the action preserves the adjacency. We then obtain the quotient graph . We say that an H-action on Γ is free when any element of H, except for the unit, does not fix a vertex. For more detail, we refer the reader to Eon (2012 ▸).

Theorem 2.2

Let be a graph and let be a vertex. Suppose that an abelian group H acts freely on Γ and its quotient graph is finite. Then the function

is of quasi-polynomial type. Hence, its difference

is also of quasi-polynomial type. In particular, their generating functions are rational. Moreover, both the functions are of polynomial type if has only one vertex.

Remark 2.3

It is worth emphasizing that we consider not only an undirected graph, but also a directed graph. Hence Theorem 2.2 can be used for a graph whose edges have a direction. See Example 3.5 for a concrete example.

First, we may assume that there exists an abelian group G such that H is a subgroup of G and as a set. In fact, since the H-action on V is free, we have a bijective map and we can identify as sets. Let be the cyclic group with order . Then, we can identify as sets, and hence we may take .

Remark 2.4

We note that if as in Theorem 1.1, then we may also take . Indeed, for , if satisfy for any , then V can be realized as . Here we use the symbol to denote the disjoint union of sets.

We regard the abelian group G as an additive group, and denote by the identity element of G. For subsets and an element , we put

Since , we may assume that by translation. Since is a finite set, there is a finite subset F of V such that and

holds.

Definition 2.5

(1) For any two elements , we put

Since the map is injective, the finiteness of the set implies that the set is finite.

(2) For elements with , we put

By convention, we define when . Note that is also a finite set since each is a finite set.

Remark 2.6

Let . We say that a vertex is of α-type if there is an element such that . For an α-type vertex and , the vertex is of β-type and . In other words, is the set of trans­lations from an α-type vertex v to a β-type vertex connected to v. Therefore, for an -type vertex and , the vertex is of -type and there is a path from v to of length m. For a concrete example of and , we refer the reader to Example 3.3.

Lemma 2.7

For elements with , it follows that

Proof

This lemma is obvious by definition (cf. Remark 2.6).

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For a subset , we define a monoid as a submonoid of . We note that admits a monoid structure by regarding and G as monoids by their addition [cf. Example A3(3)].

Definition 2.8

Let S be a subset of F. We define as the submonoid of generated by the elements in the set

Note that admits a graded monoid structure by the first projection , i.e. the degree of is defined to be d [cf. Definition A9(1)].

Remark 2.9

For a generator of , there exists a path

on Γ of length such that is of -type for some , and . Although the sum of generators is simply defined by the addition of , it is not possible in general to interpret the sum as the procedure to connect paths, even if one is allowed to translate each path and to change the order of segments at will. For example, if , and , then does not always correspond to a path on Γ. See Example 3.3 for a concrete example.

Lemma 2.10

For a non-empty subset S of F, the monoid is finitely generated. More precisely, is generated by elements with degree at most .

Proof

Take an element for some with and . Since , one has for some , and hence

This fact shows that is generated by elements with degree at most . Since there exist only finitely many such elements, is finitely generated.

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Next, for a subset and , we define as a subset of , which is proved to be a finitely generated -module in Lemma 2.13.

Definition 2.11

(1) For any two elements , we define the set by

(2) Let S be a non-empty subset of F and . We define the set by

(3) For a subset and , we define by

Remark 2.12

(1) By definition, for an element , the vertex y is of β-type and there is a path from α to y of length at most d. In other words, the set consists of the β-type vertices whose distance from α is less than or equal to d.

(2) If a β-type vertex y is in , then we have a special path from α to y, namely, there exists a path

of length such that is of -type for some and . Note that this special path must visit γ-type vertices at least once for each γ in S, which plays an essential role in the proof of Lemma 2.13. The set consists of the β-type vertices with such a special path. See Example 3.3 for a concrete example.

Lemma 2.13

Let S be a non-empty subset of F and let . Then, is a finitely generated graded -module, where the graded structure of is induced by the first projection .

Proof

First, we shall prove that is a graded -module, i.e. holds for [cf. Definition A9(2)]. Since each element of can be written as the sum of generators of the form in Definition 2.8, it is sufficient to show that any generator of of the form in Definition 2.8 and any element satisfy . Since is a generator of the form in Definition 2.8, we have for some with and . Moreover, by the definition of , we have for some with satisfying , and .

Since , there exists such that . Then, we obtain

which proves . Hence, we have and thus is a graded -module.

Next we shall see that is generated by elements with degree at most . Let be a sequence with , , and . Since , there exist and a subset such that and holds for any .

Here we claim that there exist with such that .

Let and let be the elements of Λ. Since , for each , we can take with such that δ appears among

Since , we can take

Then and satisfy the claim.

Then the inclusion

shows that an element of of degree larger than can be written by the sum of an element of of lower degree and an element of . Therefore, is generated by elements with degree at most . Since there exist only finitely many such elements, is finitely generated.

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Lemma 2.14

Let be an integer. Then the following claims are valid.

(1) One has

(2) For any element , one has

Here S runs over the subsets of F containing 0 and α.

Proof

Claim (2) is obvious by definition. We only give a proof of claim (1). The inclusion

follows from Lemma 2.7. Let us show the opposite inclusion. Take an element with . By the definition of the distance, there are edges with and . For each i, the unique element is determined by . Then , and so one has

Since , we conclude .

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Proof of Theorem 2.2

For an element , put and write for its power set. Then Lemma 2.14 and the inclusion–exclusion principle imply

Hence it is enough to show that the function

is of quasi-polynomial type.

Lemma 2.10 and Proposition A6(3) show that is a finitely generated graded monoid. Furthermore, Lemma 2.13 and Theorem A12 show that is a finitely generated graded ()-module. Hence Proposition A11 implies that the function is of quasi-polynomial type, which completes the proof of the first assertion.

By Lemma 2.10, the monoid is generated by elements of degree one if . Therefore if , the functions in Theorem 2.2 turn out to be actually of polynomial type, which completes the proof of the second assertion.

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Remark 2.15

It is natural to ask how to calculate the quasi-polynomials in Theorem 2.2 from the graph Γ in some concrete situations.

By the argument in this section, in order to determine the quasi-polynomials, it is sufficient to determine the Hilbert polynomial of for each Λ. If we know generators of the monoid and the module , then the Hilbert polynomial of can be calculated in principle via a free resolution (cf. Bruns & Herzog, 1993 ▸, Lemma 4.1.13), and can be computed by the standard computational commutative algebra packages Singular and Macaulay2. Actually in Example 3.1, Example 3.2 and the face-centered cubic system in Example 3.4, generators of and are easily computed. Therefore, in these cases, it is not hard to calculate their Hilbert polynomials from the material in this section. It should be noted, however, that in many cases, even when generators are known, it is easier to calculate their coordination sequences by predicting the region that consists of the vertices with distance at most n and proving it by induction.

Computing generators of and is a more difficult problem in general situations. Indeed, Theorem A12 or even its proof does not give a procedure to compute their generators. For example, as we will see later, in the case of Example 3.3, computing generators of and by hand is not easy at all, whereas the graph itself looks simple.

3. Examples  

In this section, we explain using some examples how to apply Theorem 2.2 to the graph obtained from crystals. In Example 3.3, we see the complicated notations defined in the previous section. It should be noted in advance, however, that although Theorem 2.2 guarantees that the coordination sequence of a crystal is of quasi-polynomial type, it is not practical in general to concretely calculate the whole sequence through the theorem or its proof (see Remark 2.15). Throughout this section, we take G as a Euclidean space so that the reader can easily visualize examples.

Example 3.1

One of the simplest examples of crystal structure is the square tiling. Let , and let

where and . It is obvious that this graph satisfies all the assumptions of Theorem 2.2. An easy observation shows that the coordination sequence of this graph from the origin is , the general term of which can be written as , ().

Let us briefly look at the notations in the previous section. In this case, the finite set F used in Section 2 consists of the origin O. To be precise, since , we have . Hence, the is the graded monoid generated by , and , which coincides with the since .

In contrast, in the case where F consists of two or more points, calculating the coordination sequence by this procedure is much more complicated as seen in Remark 2.15 and Example 3.3.

Example 3.2

Let be the graph corresponding to the hexagonal tiling as in Fig. 2 ▸. Let be the vectors corresponding to the edges from O, respectively. Note that these vectors satisfy . Let and . Then, V and E are H-invariant and the quotient graph is finite. Since , we have . An easy observation shows that the coordination sequence of this graph is , the general term of which can be written as , ().

Figure 2.

Hexagonal tiling. The number attached to each vertex represents the graph distance from the origin O.

Example 3.3

Let us see the notations used in Section 2 on an example. Let and . Let

and let

where

Then, the graph is as in Fig. 3 ▸.

The group H acts freely on Γ as translations and the quotient graph has three vertices. The F in Section 2 can be taken as , where , , . Then, the for each in Definition 2.5 is

and, for example, one can compute

Next, let us see generators of the monoids and the modules in Definition 2.8 and Definition 2.11, respectively. For each subset satisfying , a generator of the monoid can be computed as

i.e. for example, one can take

as a generator of the monoid . For each subset satisfying and for each , a generator of the -module can be taken as

For example, contains , which corresponds to the path

of length 6. Indeed, decomposes as

where and is contained in the generator described above.

Using these data, it is possible in principle to calculate the coordination sequence of the graph. The actual calculation is, however, extremely laborious and shall be omitted in this paper. The coordination sequence of this graph is obtained by Wakatsuki (2018 ▸) as

which is not of polynomial type but of quasi-polynomial type.

Note that Γ has another simple realization on as in Fig. 4 ▸. Since the coordination sequence depends only on its abstract graph structure, these two graphs have the same coordination sequence. As seen in this example, the choice of a realization of a graph is not relevant to Theorem 1.1.

Figure 3.

Graph with a -translation symmetry. The number attached to each vertex represents the graph distance from the origin O. The color of each vertex represents the equivalence class in .

Figure 4.

Another realization of the graph in Fig. 3 ▸. This graph can be obtained by adding vertices and edges to the graph in Fig. 2 ▸. If is a vertex of a hexagon, then the graph distance between and O is exactly the same as that in Example 3.2.

Example 3.4

Let us consider the graphs corresponding to crystal structures of dimension 3. These graphs have such a translation symmetry that the corresponding quotient graph is finite. Therefore, it follows from Theorem 1.1 that the coordination sequences of these graphs are of quasi-polynomial type. It should be stressed that, by definition, any crystal structure of any dimension falls into this category.

Let us consider as an example the graph corresponding to the face-centered cubic system. Let = , = = and

where , , and the signs in the definition of E are arbitrary. Then, the graph corresponds to the face-centered cubic system. It is clear by definition that H acts on Γ and the quotient is finite. Note that while the H defined above is the largest translation symmetry of this system, it might be more natural to take when one considers crystal structure. Even in that case, the graph Γ has an H-action and the quotient is finite. As seen in this example, the choice of a unit cell is not relevant to Theorem 1.1.

Example 3.5

We give one of the simplest examples of a periodic directed graph (one should compare this with Example 3.1). Let , and let

where and . An easy observation shows that the coordination sequence of this graph from the origin is , the general term of which can be written as ().

In this case, the finite set F used in Section 2 consists of the origin O. To be precise, since , we have . Hence, the is the graded monoid generated by , and , which coincides with the since .

Example 3.6

We give one of the simplest examples of a periodic graph with H having a torsion. Let and . Let

and let

where

Then, it is obvious that and satisfy all the assumptions of Theorem 2.2. In this case, we have and .

The graph Γ also admits a -action as follows. Let , where e is the identity element of . We define a -action on V by the map satisfying

Then, it is easy to see that the set E of edges is stable under the -action, and that ι commutes with the -action. Therefore the product acts on Γ. Then, and H also satisfy all the assumptions of Theorem 2.2, and we have in this case. Therefore, according to Theorem 2.2, the coordination sequence of this graph should not only be of quasi-polynomial type, but also of polynomial type.

Actually, an easy observation shows that the coordination sequence of this graph from the origin is , the general term of which can be written as , , ().

4. Conclusion  

In this paper, we proved that if a graph Γ has a free -action such that the quotient is finite, then the coordination sequence of Γ must be of quasi-polynomial type (Theorem 1.1 and Theorem 2.2). As we mentioned in Example 3.4, Theorem 2.2 can be applied to all crystals, which by definition have such a translation symmetry. It should be noted, however, that except for some simple cases, Theorem 2.2 or even its proof does not give a specific procedure to concretely calculate coordination sequences (Remark 2.15). Establishing a systematic method to calculate coordination sequences from an algebraic perspective is left for a future work. The first step would be to determine the period N and the number M in Definition 1.2. Once that is done, we can determine the quasi-polynomial by just calculating the first terms.

Appendix A. On finitely generated monoids

In this appendix, we first recall some basic definitions and results on finitely generated monoids [for more detail we refer the reader to Bruns & Gubeladze (2009 ▸)]. Then, we will prove Theorem A12, which plays an essential role in the proof of Theorem 2.2.

A1. Monoids and modules  

In this paper, all the monoids considered are commutative, that is, a monoid is a commutative semi-group with the unit element 0, and the operation is written additively.

Definition A1

(1) A monoid is a set M with a binary operation with the following three conditions:

(i) holds for .

(ii) holds for any .

(iii) There exists an element such that holds for any .

(2) A monoid M is called integral when it has the cancelation property, i.e. when implies for all .

(3) A submonoid of a monoid M is a subset such that N contains the unit of M and is closed under the additive operation of M.

(4) A map between two monoids is called a homomorphism when and hold for any .

(5) Let F be a subset of a monoid M. We say that F generates M if holds, i.e. any element m can be written by

for some , and . A monoid M is said to be finitely generated when M is generated by a finite subset of M.

(6) For an integral monoid M, we define the group of differences by with the following relation ∼:

(i) if and only if .

Note that becomes an abelian group by the addition . Furthermore, we may consider M as a submonoid of by the injective homomorphism .

Remark A2

The notions in (3), (4) and (5) correspond to a subgroup, a group homomorphism and a finitely generated group in group theory, respectively. For a monoid M and a field k, we can associate the monoid ring (see Bruns & Gubeladze, 2009 ▸, p. 51 for detail). Then the finite generation of M is equivalent to the finite generation of as a k-algebra (cf. Bruns & Gubeladze, 2009 ▸, Proposition 2.7).

Example A3

(1) If each element of a monoid M has an inverse, then M is an abelian group. In particular, any abelian group is an integral monoid. All the monoids appearing in Section 2 are submonoids of abelian groups. Hence they are always integral.

(2) The set of non-negative integers is a typical example of a monoid that is not a group.

(3) For monoids M and N, their Cartesian product admits a monoid structure by for .

(4) Using the notion of polyhedral convex geometry, we can obtain various non-trivial examples of monoids. A cone C in is the intersection of finitely many linear closed halfspaces (cf. Bruns & Gubeladze, 2009 ▸, Definition 1.14). A linear closed halfspace here means a closed halfspace which is defined by a linear function on , i.e. it is a set of the form where . Then is a submonoid of .

We say that a cone C is rational if C is the intersection of finitely many linear closed rational halfspaces, i.e. linear closed halfspaces defined by linear functions with rational coefficients . In this case, is known to be a finitely generated monoid [see Proposition A8(1) below].

For example, if

is finitely generated, but is not. is actually generated by , and . Furthermore, we have = = .

Next we introduce the notation of M-modules. It corresponds to the notation of R-modules for a commutative ring R in ring theory.

Definition A4

Let M be a monoid.

(1) An M-module is a set X equipped with an operation , which is written as +, satisfying the following conditions for all and :

(i) ,

(ii) .

(2) Let F be a subset of an M-module X. The module X is said to be generated by F if holds, where we set

The module X is said to be finitely generated if some finite subset generates X.

Example A5

(1) In Section 2, we mainly consider the following situation. M is a submonoid of an abelian group G, and X is a subset of G. Then X is an M-module (by the addition of G) if and only if holds.

(2) Let M be a submonoid of a monoid N. Then N and N-modules can be thought of as M-modules. Suppose that N is finitely generated as an M-module and X is a finitely generated N-module. Then X is finitely generated as an M-module.

We prove this claim below. Since N is finitely generated as an M-module, there exists a finite subset such that . Furthermore, since X is finitely generated as an N-module, there exists a finite set such that . Then we have

Since is a finite set, X is finitely generated as an M-module.

(3) A polyhedron P in is the (possibly unbounded) intersection of finitely many affine closed halfspaces (cf. Bruns & Gubeladze, 2009 ▸, Definition 1.1). An affine closed halfspace here means a closed halfspace which is defined by an affine function on , i.e. it is a set of the form

for some . Let be a polyhedron, where is an affine halfspace. Then the recession cone of P is defined by

where denotes the linear closed halfspace parallel to . Then we have since holds for and for each i. Since , we have

Therefore, is a -module.

We say that a polyhedron is rational if each halfspace is defined by an affine function of rational coefficients . In this case, is known to be a finitely generated -module [see Proposition A8(2) below].

For example, if

then we have

Therefore we have

It is easy to see that is generated by (1,3), (2,2) and (3,1) as a -module.

We list some properties on finite generation.

Proposition A6

(1) (Bruns & Gubeladze, 2009 ▸, Proposition 2.8) Let M be a finitely generated monoid and let X be a finitely generated M-module. Then any M-submodule of X is finitely generated as an M-module.

(2) (cf. Ogus, 2018 ▸, ch. I, Theorem 2.1.17.6) Let be a homomorphism between integral monoids and let be a submonoid. If M and are finitely generated, then so is .

(3) (cf. Bruns & Gubeladze, 2009 ▸, Corollary 2.11) Let N be an integral monoid and let and be submonoids of N. If and are finitely generated, then is also a finitely generated monoid.

Remark A7

Via the correspondence between a monoid M and a monoid ring (cf. Bruns & Gubeladze, 2009 ▸, Proposition 2.7), (1) is a corollary of the Hilbert basis theorem in ring theory, which states that a finitely generated k-algebra is Noetherian (cf. Eisenbud, 1995 ▸, ch. I, Section 1.4).

In Ogus (2018 ▸, ch. I, Theorem 2.1.17.6), (2) is proved in a more general setting:

(4) Let and be monoid homomorphisms. If the monoids M and P are finitely generated and N is integral, then their fiber product is a finitely generated monoid.

We note that the fiber product is defined as

and it is easy to see that is a submonoid of . (2) can be seen as a special case of (4) by setting g as the inclusion map .

We note that in Bruns & Gubeladze (2009 ▸, Corollary 2.11), (3) is proved only when . In general cases, the assertion follows from (2), applying it to the inclusion map and . We also note that in Bruns & Gubeladze (2009 ▸), a monoid M is called affine when M is finitely generated and is isomorphic to a submonoid of .

If a cone and a polyhedron are rational [see Example A3(4) and Example A5(3)], then their restrictions to give a finitely generated monoid M and a finitely generated M-module. In (3), we denote by the cone which consists of the elements of the form with , and .

Proposition A8

(1) (Bruns & Gubeladze, 2009 ▸, Lemma 2.9) For a rational cone , the monoid is finitely generated.

(2) (Bruns & Gubeladze, 2009 ▸, Theorem 2.12) For a rational polyhedron , the set is a finitely generated ()-module.

(3) (Bruns & Gubeladze, 2009 ▸, Corollary 2.10) If M is a finitely generated submonoid of , then the monoid is finitely generated as an M-module.

(4) [Bruns & Gubeladze, 2009 ▸, Proof of Corollary 2.11(a)] If M and N are submonoids of , then = holds.

In the rest of this section, we summarize the fact on the Hilbert function of graded monoids.

Definition A9

(1) A (positively) graded monoid is a monoid M equipped with a monoid homomorphism . We say that is of degree i if . Furthermore, we write .

(2) Let M be a graded monoid. A graded M-module is an M-module X equipped with a map with the condition for all and , where we set . This condition is equivalent to saying that holds for any and .

Definition A10

Let M be a graded monoid and X a graded M-module with for any . The function

is called the Hilbert function associated with X. Its generating function

is called the Hilbert series of X. The Hilbert function and the Hilbert series depend on the grading of M and X.

Proposition A11

(Bruns & Gubeladze, 2009 ▸, Theorems 6.38 and 6.39; Bruns & Herzog, 1993 ▸, Theorem 4.1.3, Proposition 4.4.1, Theorem 4.4.3) Let M be a graded monoid which is generated by finitely many elements of positive degree. Let X be a finitely generated graded M-module. Then the Hilbert series is a rational function, and the Hilbert function is of quasi-polynomial type. Moreover, if M is generated by elements of degree one, then is of polynomial type.

We note that the assumption in Definition A10 is satisfied under the assumption of Proposition A11.

A2. Finite generation of modules  

The following theorem is the main theorem in this appendix.

Theorem A12

Let and be finitely generated sub­monoids of an integral monoid M and let be a finitely generated -submodule for . Then the -module is finitely generated.

Proof

Since is a finitely generated -module, there is a finite subset of such that , and so one has

Hence by replacing with , we may assume that is the form of for each . Next, replacing M with , we may assume that M is an abelian group. Furthermore, replacing M with the monoid generated by and the elements of , we may assume that M is finitely generated as a monoid. Let be a generator of M. Then, one can take a surjective homomorphism:

Then, the monoid is finitely generated by Proposition A6(2). Take an element satisfying . Since , we have

Hence Proposition A6(1) shows that the -module is finitely generated, and hence its translation is also a finitely generated -module. Moreover, we have

since φ is surjective. Thus, in order to prove that is finitely generated as a -module, it is sufficient to show that is finitely generated as a -module. Since we have

the proof is completed if we show the theorem in the case of .

The set is a rational polyhedron satisfying . Proposition A8(2) implies that the set

is a finitely generated -module. Since and are finitely generated monoids, so is the monoid by Proposition A6(3). Furthermore,

holds by Proposition A8(4), and it is a finitely generated -module by Proposition A8(3). Hence Y is a finitely generated -module [cf. Example A5(2)]. Since is an -submodule, it is a finitely generated -module by Proposition A6(1), which completes the proof.

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Funding Statement

This work was funded by Ministry of Education, Culture, Sports, Science and Technology grants 18K13438 and 18K13384.