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.


Introduction
For a graph À and a fixed vertex v 0 of À, and for a non-negative integer n, the coordination sequence s n is defined as the number of vertices of À at distance n from v 0 . For example, the first few terms of the coordination sequence of the graph in Fig. 1 are s 0 ¼ 1, s 1 ¼ 4, s 2 ¼ 16, s 3 ¼ 24. 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 s n ¼ 8n (n ! 2).
In this paper, we consider a periodic graph À in the following sense: (A) À is a (possibly directed) graph with a free Z n action such that the quotient graph À=Z n 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 À 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 s 1 is nothing but the usual coordination number and the coordination sequence ðs n Þ n 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 (2002Eon ( , 2012, Goodman-Strauss & Sloane (2019), O'Keeffe (1995O'Keeffe ( , 1998, Shutov & Maleev (2018, 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 À=Z n has only one vertex.
Here, we recall the definition of functions of quasipolynomial type.
Definition 1.2. A function p : Z !0 À!Z is called a quasipolynomial if there exist an integer N > 0 and polynomials p 0 ; . . . ; p NÀ1 2 Q½X with the following condition: (i) For any n 2 Z !0 , the equality pðnÞ ¼ p i ðnÞ holds if N divides n À i.
We say that a function f : Z !0 À!Z is of quasi-polynomial type if there exist an integer M ! 0 and a quasi-polynomial p : Z !0 À!Z such that f ðnÞ ¼ pðnÞ holds for any integer n ! M. As a special case, we say that f is of polynomial type when N ¼ 1.
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.

Coordination sequences of graphs with abelian group action
In this paper, a graph À ¼ ðV; EÞ means a directed simple graph, i.e. V is the set of vertices and E & V Â V n fðx; xÞ j x 2 Vg is the set of edges. We say that a graph À ¼ ðV; EÞ 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 À 0 consists of sets V 0 and E 0 & fA & V 0 j #A ¼ 2g, we obtain a graph À from À 0 by setting V ¼ V 0 and E ¼ fðx; yÞ 2 V Â V j fx; yg 2 E 0 g. Here the symbol #A is employed to denote the cardinality of the set A.
For vertices x; y 2 V, we denote by distðx; yÞ 2 Z !0 [ f1g 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 distðx; yÞ is defined as infinite.
Let H be a group. We say that H acts on À ¼ ðV; EÞ 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 À=H :¼ ðV=H; E=HÞ. We say that an H-action on À 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 À ¼ ðV; EÞ be a graph and let v 0 2 V be a vertex. Suppose that an abelian group H acts freely on À and its quotient graph À=H is finite. Then the function Z !0 À!Z; n7 !#fy 2 V j distðv 0 ; yÞ ng is of quasi-polynomial type. Hence, its difference Z !0 À!Z; n7 !#fy 2 V j distðv 0 ; yÞ ¼ ng is also of quasi-polynomial type. In particular, their generating functions are rational. Moreover, both the functions are of polynomial type if À=H 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 V & G as a set. In fact, since the H-action on V is free, we have a bijective map H Â ðV=HÞ ! V and we can identify V ¼ H Â ðV=HÞ as sets. Let C ¼ Z=cZ be the cyclic group with order c ¼ #ðV=HÞ. Then, we can identify V=H ¼ C as sets, and hence we may take G ¼ H Â C ¼ V. Remark 2.4. We note that if H ¼ Z n as in Theorem 1.1, then we may also take G Here we use the symbol F to denote the disjoint union of sets.
We regard the abelian group G as an additive group, and denote by 0 2 G the identity element of G. For subsets A; B & G and an element 2 G, we put Since V & G, we may assume that v 0 ¼ 0 by translation. Since V=H is a finite set, there is a finite subset F of V such that 0 2 F and holds.
(1) For any two elements ; 2 F, we put Since the map P ; À!E=H; x7 !½ð; þ xÞ is injective, the finiteness of the set E=H implies that the set P ; is finite.
For an -type vertex v 2 V and x 2 P ; , the vertex v þ x is of -type and ðv; v þ xÞ 2 E. In other words, P ; is the set of translations from an -type vertex v to a -type vertex connected to v. Therefore, for an 0 -type vertex v 2 V and x 2 Pð 0 ; . . . ; m Þ, the vertex v þ x is of m -type and there is a path from v to v þ x of length m. For a concrete example of P ; and Pð 0 ; . . . ; m Þ, we refer the reader to Example 3.3.  Note that M S admits a graded monoid structure by the first projection M S ! Z !0 , i.e. the degree of ðd; xÞ 2 M S is defined to be d [cf. Definition A9(1)].
Remark 2.9. For a generator ðd; xÞ of M S , there exists a path y 0 ! y 1 ! Á Á Á ! y mÀ1 ! y m on À of length m d such that y i is of i -type for some i 2 S, 0 ¼ m and x ¼ y m À y 0 . Although the sum of generators is simply defined by the addition of Z !0 Â G, 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 S ¼ f 0 ; 1 ; 2 ; 3 g, x 2 Pð 0 ; 1 ; 0 Þ and y 2 Pð 2 ; 3 ; 2 Þ, then x þ y does not always correspond to a path on À. See Example 3.3 for a concrete example.
Lemma 2.10. For a non-empty subset S of F, the monoid M S is finitely generated. More precisely, M S is generated by elements with degree at most #S.
Proof. Take an element x 2 Pð 0 ; 1 ; . . . ; m Þ for some 0 ; . . . ; m 2 S with m > #S and 0 ¼ m . Since m > #S, one has i ¼ j for some 0 i < j < m, and hence This fact shows that M S is generated by elements with degree at most #S. Since there exist only finitely many such elements, M S is finitely generated. & Next, for a subset S & F and ; 2 S, we define X ; S as a subset of Z !0 Â V, which is proved to be a finitely generated M S -module in Lemma 2.13. Definition 2.11. (1) For any two elements ; 2 F, we define the set X ; & Z !0 Â V by . . . ; m Þ for some 0 ; . . . ; m 2 F with 0 m d; 0 ¼ ; and m ¼ : (2) Let S be a non-empty subset of F and ; 2 S. We define the set X ; (1) By definition, for an element ðd; yÞ 2 X ; , the vertex y is of -type and there is a path from to y of length at most d. In other words, the set ðX ; Þ d consists of the -type vertices whose distance from is less than or equal to d.
(2) If a -type vertex y is in ðX ; S Þ d , then we have a special path from to y, namely, there exists a path Note that this special path must visit -type vertices at least once for each in S, which plays an essential role in the proof of Lemma 2.13. The set ðX ; S Þ d consists of the -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 ; 2 S. Then, X ; S is a finitely generated graded M S -module, where the graded structure of X ; S is induced by the first projection X ; S À!Z !0 ; ðd; yÞ7 !d.
Proof. First, we shall prove that X ; (2)]. Since each element of M S can be written as the sum of generators of the form in Definition 2.8, it is sufficient to show that any generator ðd; xÞ of M S of the form in Definition 2.8 and any element ðd 0 ; þ yÞ 2 X ; S . Since ðd; xÞ is a generator of the form in Definition 2.8, we have x 2 Pð 0 ; . . . ; m Þ for some 0 ; . . . ; m 2 S with 0 m d and 0 ¼ m . Moreover, by the definition of X ; Next we shall see that X ; S is generated by elements with degree at most ð#SÞ 2 À 1.
. . . ; 0 m 0 g, for each 2 S n fg, we can take k with 1 k c À 1 such that appears among . . . ; c À 1g n fk j 2 S n fgg: Then j :¼ j ' and j 0 :¼ j 'þ1 satisfy the claim. Then the inclusion shows that an element of X ; S of degree larger than ð#SÞ 2 À 1 can be written by the sum of an element of X ; S of lower degree and an element of M S . Therefore, X ; S is generated by elements with degree at most ð#SÞ 2 À 1. Since there exist only finitely many such elements, X ; S is finitely generated. & Lemma 2.14. Let d ! 0 be an integer. Then the following claims are valid.
(1) One has (2) For any element 2 F, one has Here S runs over the subsets of F containing 0 and .
Proof. Claim (2) is obvious by definition. We only give a proof of claim (1). The inclusion For an element 2 F, put Ã :¼ fS & F j 0; 2 Sg and write PðÃ Þ 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 T S2Ã M S is a finitely generated graded monoid. Furthermore, Lemma 2.13 and Theorem A12 show that T S2Ã X 0; S is a finitely generated graded ( T S2Ã M S )-module. Hence Proposition A11 implies that the function ' ;Ã is of quasi-polynomial type, which completes the proof of the first assertion.
By Lemma 2.10, the monoid M S is generated by elements of degree one if #S ¼ 1. Therefore if #F ¼ 1, the functions in Theorem 2.2 turn out to be actually of polynomial type, which completes the proof of the second assertion. & Remark 2.15. It is natural to ask how to calculate the quasipolynomials in Theorem 2.2 from the graph À 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 T S2Ã ðX 0; S Þ for each Ã. If we know generators of the monoid T S2Ã M S and the module T S2Ã ðX 0; S Þ, then the Hilbert polynomial of T S2Ã ðX 0; S Þ 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 T S2Ã M S and T S2Ã ðX 0; S Þ 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 T S2Ã M S and T S2Ã ðX 0; S Þ 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 T S2Ã M S and T S2Ã ðX 0; S Þ by hand is not easy at all, whereas the graph itself looks simple.

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.
where e 1 ¼ ð1; 0Þ and e 2 ¼ ð0; 1Þ. 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 1; 4; 8; 12; . . . , the general term of which can be written as s 0 ¼ 1, s n ¼ 4n (n ! 1). 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 F ¼ fOg, we have P O;O ¼ fAEe 1 ; AEe 2 g. Hence, the M F & Z !0 Â Z 2 is the graded monoid generated by ð1; ð0; 0ÞÞ, ð1; AEe 1 Þ and ð1; AEe 2 Þ, which coincides with the X F since #F ¼ 1.
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 À ¼ ðV; EÞ be the graph corresponding to the hexagonal tiling as in Fig. 2. Let v 1 ; v 2 ; v 3 be the vectors corresponding to the edges from O, respectively. Note that these vectors satisfy Then, V and E are H-invariant and the quotient graph À=H is finite. Since V ¼ H t ðv 1 þ HÞ, we have F ¼ fO; v 1 g. An easy observation shows that the coordination sequence of this graph is 1; 3; 6; 9; . . . , the general term of which can be written as s 0 ¼ 1, s n ¼ 3n (n ! 1).
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)     which is not of polynomial type but of quasi-polynomial type.
Note that À has another simple realization on R 2 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.
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.
where e 1 ¼ ð1; 0; 0Þ, e 2 ¼ ð0; 1; 0Þ, e 1 ¼ ð0; 0; 1Þ and the signs in the definition of E are arbitrary. Then, the graph À ¼ ðV; EÞ corresponds to the face-centered cubic system. It is clear by definition that H acts on À 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 H ¼ Ze 1 þ Ze 2 þ Ze 3 when one considers crystal structure. Even in that case, the graph À 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 where e 1 ¼ ð1; 0Þ and e 2 ¼ ð0; 1Þ. An easy observation shows that the coordination sequence of this graph from the origin is 1; 2; 3; 4; . . . , the general term of which can be written as s n ¼ n þ 1 (n ! 0). In this case, the finite set F used in Section 2 consists of the origin O. To be precise, since F ¼ fOg, we have P O;O ¼ fe 1 ; e 2 g. Hence, the M F & Z !0 Â Z 2 is the graded monoid generated by ð1; ð0; 0ÞÞ, ð1; e 1 Þ and ð1; e 2 Þ, which coincides with the X F since #F ¼ 1.
The graph À also admits a Z=2Z-action as follows. Let Z=2Z ¼ fe; g, where e is the identity element of Z=2Z. We define a Z=2Z-action on V by the map : VÀ!V satisfying ðx; yÞ ¼ ðx þ 1; y þ 1Þ ðif x is evenÞ; ðx À 1; y À 1Þ ðif x is oddÞ: & Then, it is easy to see that the set E of edges is stable under the Z=2Z-action, and that commutes with the H 1 -action. Therefore the product H :¼ H 1 Â ðZ=2ZÞ acts on À. Then, À ¼ ðV; EÞ and H also satisfy all the assumptions of Theorem 2.2, and we have #ðV=HÞ ¼ 1 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.

Conclusion
In this paper, we proved that if a graph À has a free Z n -action such that the quotient À=Z n is finite, then the coordination sequence of À 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 quasipolynomial by just calculating the first ðdegðpÞ þ 1ÞN þ M À 1 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 ]. 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. (2) A monoid M is called integral when it has the cancelation property, i.e. when x þ y ¼ x þ z implies y ¼ z for all x; y; z 2 M. (5) Let F be a subset of a monoid M. We say that F generates M if M ¼ Z !0 F holds, i.e. any element m can be written by m ¼ n 1 m 1 þ Á Á Á þ n r m r for some r ! 1, n i 2 Z !0 and m i 2 F. 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 M gp by M gp :¼ ðM Â MÞ= $ with the following relation $: (i) ða; bÞ $ ðc; dÞ if and only if a þ d ¼ b þ c. Note that M gp becomes an abelian group by the addition ½ða; bÞ þ ½ðc; dÞ ¼ ½ða þ c; b þ dÞ. Furthermore, we may consider M as a submonoid of M gp by the injective homomorphism MÀ!M gp ; m7 !½ðm; 0Þ.
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 k½M (see Bruns & Gubeladze, 2009, p. 51 for detail). Then the finite generation of M is equivalent to the finite generation of k½M as a k-algebra (cf. Bruns & Gubeladze, 2009, Proposition 2.7).
(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 Z !0 of non-negative integers is a typical example of a monoid that is not a group.
(4) Using the notion of polyhedral convex geometry, we can obtain various non-trivial examples of monoids. A cone C in R n 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 R n , i.e. it is a set of the form fðx 1 ; . . . ; x n Þ 2 R n j P i a i x i ! 0g where a i 2 R. Then C \ Z n is a submonoid of Z n .
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 P i a i x i with rational coefficients a i 2 Q. In this case, C \ Z n is known to be a finitely generated monoid [see Proposition A8(1) below].
For example, if M 1 is finitely generated, but M 2 is not. M 1 is actually generated by ð1; 0Þ, ð1; 1Þ and ð1; 2Þ. Furthermore, we have M gp 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 M Â XÀ!X, which is written as +, satisfying the following conditions for all m 1 ; m 2 2 M and x 2 X: (2) Let F be a subset of an M-module X. The module X is said to be generated by The module X is said to be finitely generated if some finite subset F & X 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 M þ X & X 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 F & N such that N ¼ M þ F. Furthermore, since X is finitely generated as an N-module, there exists a finite set E & X such that X ¼ N þ E. Then we have Since F þ E is a finite set, X is finitely generated as an M-module.
(3) A polyhedron P in R n 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 R n , i.e. it is a set of the form where H i is an affine halfspace. Then the recession cone recðPÞ of P is defined by i denotes the linear closed halfspace parallel to H i . Then we have recðPÞ þ P & P since x þ y 2 H i holds for x 2 H 0 i and y 2 H i for each i. Since recðPÞ þ P & P, we have ðrecðPÞ \ Z n Þ þ ðP \ Z n Þ & P \ Z n : Therefore, P \ Z n is a ðrecðPÞ \ Z n Þ-module. We say that a polyhedron P ¼ H 1 \ Á Á Á \ H c is rational if each halfspace H i is defined by an affine function b þ P i a i x i of rational coefficients b; a i 2 Q. In this case, P \ Z n is known to be a finitely generated ðrecðPÞ \ Z n Þ-module [see Proposition A8(2) below].
We list some properties on finite generation.
(1) , 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.
(3) (cf. Bruns & Gubeladze, 2009, Corollary 2.11) Let N be an integral monoid and let M 1 and M 2 be submonoids of N. If M 1 and M 2 are finitely generated, then M 1 \ M 2 is also a finitely generated monoid.
Remark A7. Via the correspondence between a monoid M and a monoid ring k½M (cf. , 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 f : M ! N and g : P ! N be monoid homomorphisms. If the monoids M and P are finitely generated and N is integral, then their fiber product M Â N P is a finitely generated monoid.
We note that the fiber product is defined as M Â N P :¼ fðm; pÞ 2 M Â P j f ðmÞ ¼ gðpÞg; and it is easy to see that M Â N P is a submonoid of M Â P. (2) can be seen as a special case of (4) by setting g as the inclusion map N 0 ! N. We note that in Bruns & Gubeladze (2009, Corollary 2.11 is proved only when N ¼ R n . In general cases, the assertion follows from (2), applying it to the inclusion map M ¼ M 1 ! N and N 0 ¼ M 2 . We also note that in , a monoid M is called affine when M is finitely generated and is isomorphic to a submonoid of Z n .
If a cone and a polyhedron are rational [see Example A3 (4) and Example A5 (3)], then their restrictions to Z n give a finitely generated monoid M and a finitely generated M-module. In (3), we denote by R !0 M the cone which consists of the elements of the form P c i¼1 r i m i 2 R n with c 2 Z > 0 , r i 2 R !0 and m i 2 M. Proposition A8. (1) , Lemma 2.9) For a rational cone C & R n , the monoid C \ Z n is finitely generated. generated ðM 1 \ M 2 Þ-module by Proposition A6(1), which completes the proof.