On pseudo-Frobenius elements of submonoids of $\mathbb{N}^d$
J. I. García-García I. Ojeda J. C. Rosales A. Vigneron-Tenorio





Universidad de CádizDepartamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible)
ignacio.garcia@uca.es


Universidad de ExtremaduraDepartamento de Matemáticas/IMUEx
ojedamc@unex.es


Universidad de GranadaDepartamento de Álgebra
jcrosales@ugr.es


Universidad de CádizDepartamento de Matemáticas/INDESS (Instituto Universitario para el Desarrollo Social Sostenible)
alberto.vigneron@uca.es

The first and the third author were partially supported by Junta de Andalucía research group FQM-366 and by the project MTM2017-84890-P


The second author was partially supported by the research groups FQM-024 (Junta de Extremadura/FEDER funds) and by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and by the project MTM2017-84890-P


The fourth author was partially supported by Junta de Andalucía research group FQM-366, by the project MTM2015-65764-C3-1-P (MINECO/FEDER, UE) and by the project MTM2017-84890-P


[2010]20M14, 13D02






In this paper we study those submonoids of ℕ^d which a non-trivial pseudo-Frobenius set. In the affine case, we prove that they are the affine semigroups whose associated algebra over a field has maximal projective dimension possible. We prove that these semigroups are a natural generalization of numerical semigroups and, consequently, most of their invariants can be generalized. In the last section we introduce a new family of submonoids of ℕ^d and using its pseudo-Frobenius elements we prove that the elements in the family are direct limits of affine semigroups.


On pseudo-Frobenius elements of submonoids of ℕ^d
    A. Vigneron-Tenorio
    December 30, 2023
=================================================

myheadings
J. I. García-García, I. Ojeda, J.C. Rosales A. Vigneron-TenorioOn pseudo-Frobenius elements of submonoids of ℕ^d





§ INTRODUCTION


Throughout this paper ℕ will denote the set of nonnegative integers. Unless otherwise stated, all considered semigroups will be submonoids of ℕ^d.
Finetelly generated submonoids of ℕ^d will be called, affine semigroups, as ussual. If d=1, affine semigroups are called numerical semigroups. Numerical semigroup has been widely studied in the literature (see, for instance, <cit.> and the references therein). It is well-known that S ⊆ℕ is a numerical semigroup if and only if S is a submonoid of ℕ such that ℕ∖ S is a finite set. Clearly, this does not hold for affine semigroups in general. In <cit.>, the affine semigroups whose complementary in the cone that they generate is finite, in a suitable sense, are called 𝒞-semigroups and the authors prove that a Wilf's conjecture can be generalized to this new situation.

In the numerical case, the finiteness of ℕ∖ S implies that there exists at least a positive integers a ∈ℕ∖ S such that a+S ∖{0}⊆ S (provided that S ≠ℕ). These integers are called pseudo-Frobenius numbers and the biggest one is the so-called Frobenius number. In this paper, we consider the semigroups such that there exists at least one element 𝐚∈ℕ^d ∖ S with 𝐚+S ∖{0}⊆ S. By analogy, we call these elements the pseudo-Frobenius elements of S. We emphasize that the semigroups in the family of 𝒞-semigroups have pseudo-Frobenius elements.

One the main results in this paper is Theorem <ref> which states that an affine semigroup, S, has pseudo-Frobenius elements if and only if the length of the minimal free resolution of the semigroup algebra [S], with  being a field, as a module over a polynomial ring is maximal, that is, if [S] has the maximal projective dimension possible (see Section <ref>). For this reason we will say that affine semigroups with pseudo-Frobenius elements are maximal projective dimension semigroups, MPD-semigroups for short. Observe that, as minimal free resolution of semigroup algebras can be effectively computed, our results provides a computable necessary and sufficient condition for an affine semigroup to be a MPD-semigroup. In fact, using the procedure outlined <cit.>, one could theoretically compute the minimal free resolution of [S] from the pseudo-Frobenius elements. To this end, the effective computation of the pseudo-Frobenius is needed. We provide a bound of the pseudo-Frobenius elements in terms of d and the cardinal and size of the minimal generating set of S (see Corollary <ref>).

In order to emulate the maximal property of the Frobenius element of a numerical semigroup, we fix a term order on ℕ^d and define Frobenius elements as the maximal elements for the fixed order of the elements in the integral points in the complementary of an affine semigroup S with respect to its cone. These Frobenius elements (if exist) are necesarily pseudo-Frobenius elements of S for a maximality matter (see Lemma <ref>). So, if Frobenius elements of S exist then [S] has maximal projective dimension, unfortunately the converse it is not true in general. However, there are relevant families of affine semigroups with Frobenius elements as the family of 𝒞-semigroups. In the section devoted to Frobenius elements (Section <ref>), we prove generalizations of well-known results for numerical semigroups such as Selmer's Theorem that relates the Frobenius numbers and Apéry sets (Theorem <ref>) or the characterization of the pseudo Frobenius elements in terms of the Apéry sets (Proposition <ref>). We close this section proving that affine semigroups having Frobenius elements are stable by gluing, by giving a formula for a Frobenius element in the gluing (Theorem <ref>).

In the Section <ref>, we deal with the problem of the irreducibility of the MPD-semigroups. Again, we prove the analogous results for MPD-semigroups than the known-ones for irreducible numerical semigroups. Of special interest is the characterization of the pseudo-Frobenius sets for 𝒞-semigroups (Theorem <ref> and Proposition <ref>). Pedro A. García-Sánchez communicated us that similar results were obtained by C. Cisto, G. Fiolla, C. Peterson and R. Utano in <cit.> for ℕ^d-semigroups, that is, those 𝒞-semigroups whose associated cone is the whole ℕ^d.

Finally, in the last section of this paper, we introduce a new family of (non-necessarily finitely generated) submonoids of ℕ^d which have pseudo-Frobenius elements, the elements of the family are called PI-monoids. These monoids are a natural generalization of the MED-semigroups (see <cit.>). We conclude the paper by proving that any PI-monoid is direct limit of MPD-semigroups.

The study of ℕ^d-semigroups, which is a subfamily of the MPD-semigroups, is becoming to be an active research area in affine semigroups theory. For instance, in <cit.> algorithms for dealing with ℕ^d-semigroups are given. These algorithms are implemented in the development version site of the GAP (<cit.>)   package (<cit.>):

<https://github.com/gap-packages/numericalsgps>.





§ PSEUDO-FROBENIUS ELEMENTS OF AFFINE SEMIGROUPS


Set 𝒜 = {𝐚_1, …, 𝐚_n}⊂ℕ^d and let S be the submonoid of ℕ^d generated by 𝒜. Consider the cone of S in ℚ^d_≥ 0 
    pos(S) := {∑_i=1^n λ_i 𝐚_i | λ_i ∈ℚ_≥ 0, i = 1, …,n }

and define ℋ(S) := (pos(S) ∖ S) ∩ℕ^d.


An integer vector 𝐚∈ℋ(S) is called a pseudo-Frobenius element of S if 𝐚 + S∖{0}⊆ S. The set of pseudo-Frobenius elements of S is denoted by PF(S).


Observe that the set PF(S) may be empty: indeed, let 
    𝒜={(2,0),(1,1),(0,2)}⊂ℕ^2.
 The semigroup S generated by 𝒜 is the subset of points in ℕ^2 whose sum of coordinates is even. Thus, we has that ℋ(S) + S = ℋ (S). Therefore PF(S) = ∅.

On other hand, PF(S) ≠ℋ(S) in general as the following example shows.


Let S be the submonoid of ℕ^2 generated by the columns of the following matrix

    A =
    (
    [ 3 5 0 1 2; 0 0 1 3 3 ])

In this case, 
    ℋ(S) = {     (1,0),(2,0),(4,0),(1,1),(2,1),(4,1), 
        (1,2),(2,2),(4,2),(7,0),(7,1),(7,2)},

whereas PF(S) = {(7,2)}:


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[color=gray,dash pattern=on 2pt off 2pt, xstep=1.0cm,ystep=1.0cm] (0,0) grid (9.5,4.5);
[->,color=black] (0,0) – (9.5,0);
in 1,2,3,4,5,6,7,8
[shift=(,0),color=black] (0pt,2pt) – (0pt,-2pt) node[below] ;
[->,color=black] (0,0) – (0,4.5);
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[shift=(0,),color=black] (2pt,0pt) – (-2pt,0pt) node[left] ;
[color=black] (0pt,-10pt) node[right] 0;
(-0.5,-0.5) rectangle (9.5,4.5);
[fill=blue,fill opacity=0.1] (0,3) – (10,3) – (10,5) – (0,5) – cycle;
[fill=blue,fill opacity=0.1] (8,0) – (10,0) – (10,3) – (8,3) – cycle;

[color=blue] (3,0) circle (3pt);
[color=blue] (3.3,0.26) node a_1;
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[color=blue] (0,0) circle (2pt);
[color=blue] (0,2) circle (2pt);
[color=blue] (3,1) circle (2pt);
[color=blue] (3,2) circle (2pt);

[->,color=red] (0,0) – (7,2);
[color=red] (4,1.33) node 𝐯;




The elements in S are the blue points and the points in the shadowed blue area. The red vector is the only pseudo-Frobenius element of S. The big blue points correspond to the minimal generators of S.


There are relevant families of numerical semigroups for which the pseudo-Frobenius element exist.


If ℋ(S) is finite, then S is said to be a 𝒞-semigroup, where 𝒞 denotes the cone pos(S).


The 𝒞-semigroups were introduced in <cit.>.


If S is a 𝒞-semigroup different from 𝒞∩ℕ^d, then PF(S) ≠∅.



Let ≼ be a term order on ℕ^d and set 𝐚:=max_≼(ℋ(S)). If 𝐛∈ S ∖{0} is such that 𝐚 + 𝐛∉S, then 𝐚 + 𝐛∈ℋ(S) and 𝐚 + 𝐛≻𝐚 which contradicts the maximality of 𝐚.


The converse of the above proposition is not true, as the following example shows.


Let 𝒜⊂ℕ^2 be the columns of the matrix

    A = ([ 18 18  4 20 23  8 11 11 10 14  7  7;  9  3  1  8 10  3  5  2  3  3  2  3 ])

and let S be the subsemigroup of ℕ^2 generated by 𝒜.
The elements in S are the integer points in an infinite family of homotetic pentagons.



    < g r a p h i c s >



This semigroup is a so-called multiple convex body semigroup (see <cit.> for further details). Clearly, S is not a 𝒞-semigroup, but (13,4)∈PF(S).







§ MAXIMAL PROJECTIVE DIMENSION


As in the previous section, set 𝒜 = {𝐚_1, …, 𝐚_n}⊂ℕ^d and let S be the submonoid of ℕ^d generated by 𝒜. Let  be an arbitrary field.

The surjective -algebra morphism 
    φ_0 : R:=[x_1, …, x_n]
    ⟶[S] := ⊕_𝐚∈ S χ^𝐚; x_i ⟼χ^𝐚_i
 is
S-graded, thus, the ideal I_S := (φ_0) is a
S-homogeneous ideal called the ideal of S. Notice that I_S is
a toric ideal generated by 
    {𝐱^𝐮 - 𝐱^𝐯 :
    ∑_i=1^n u_i 𝐚_i = ∑_i=1^n v_i 𝐚_i }.


Now, by using the S-graded Nakayama's lemma recursively, we may construct S-graded
-algebra homomorphism 
    φ_j+1 : R^s_j+1⟶ R^s_j,
 corresponding to a choice of a minimal
set of S-homogeneous generators for each module of syzygies N_j
:= (φ_j),j ≥ 0 (see <cit.> and the references therein). Notice that N_0 = I_S. Thus, we obtain a
minimal free S-graded resolution for the R-module
[S] of the form

    …⟶ R^s_j+1φ_j+1⟶ R^s_j⟶…⟶ R^s_2φ_2⟶
    R^s_1φ_1⟶ R
    φ_0⟶[S]
    ⟶ 0,

where s_j+1 := ∑_𝐛∈ Sdim_ V_j(𝐛),
with V_j(𝐛) := (N_j)_𝐛/(𝔪 N_j)_𝐛, is the so-called
(j+1)th Betti number of [S], where 𝔪= ⟨ x_1, …, x_n ⟩ is the irrelevant maximal ideal.

Observe that the dimension of V_j(𝐛) is
the number of generators of degree 𝐛 in a minimal system of
generators of the jth module of syzygies N_j (i.e. the
multigraded Betti number s_j,𝐛). The 𝐛∈ S such that V_j(𝐛) ≠ 0 are called S-degrees of the j-minimal syzygy of [S]. So, by the Noetherian property
of R, s_j+1 is finite. Moreover, by the Hilbert's syzygy theorem and the Auslander-Buchsbaum's
formula, it follows that s_j = 0 for j > p = n - depth_R
[S] and s_p ≠ 0. Such integer p is called the projective dimension of S.

Since depth_R [S] ≥ 1, the projective dimension of S is lesser than or equal to n-1.
We will say that S is a maximal projective dimension semigroup (MPD-semigroup, for short) if its projective dimension is n-1, equivalently, if depth_R [S] = 1.

Recall that S is said to be Cohen-Macaulay if depth_R [S] = ([S]). So, if S is a MPD-semigroup, then S is Cohen-Macaulay if and only if [S] is the coordinate ring of a monomial curve; equivalently, S is a numerical semigroup.


The necessary and sufficient condition for S to be a MPD-semigroup is that PF(S) ≠∅.
In this case, PF(S) has finite cardinality.



By definition, S is a MPD-semigroup if and only if V_n-2(𝐛) ≠∅ for some 𝐛∈ S. By <cit.>, given 𝐛∈ S, V_n-2(𝐛) ≠∅ if and only if 𝐛 - ∑_i=1^n 𝐚_i ∉S and 𝐛 - ∑_i ∈ F𝐚_i ∈ S, for every F ⊊{1, …, n}. Clearly, if PF(S) ≠∅, there exists 𝐚∈ℋ(S) such that 𝐚 + S ∖{0}⊆ S. So, by taking 𝐛 = 𝐚 + ∑_i=1^n 𝐚_i, one has that 𝐛 - ∑_i=1^n 𝐚_i ∉S and 𝐛 - ∑_i ∈ F𝐚_i ∈ S, for every F ⊊{1, …, n}, and we conclude S is a MPD-semigroup. Conversely, if there exists 𝐛∈ S such that V_n-2(𝐛) ≠∅, then we have that 𝐚 := 𝐛 - ∑_i=1^n 𝐚_i ∈ℤ^d ∖ S. Clearly,
𝐚∈ℤ^d ∖ S and 𝐚 + S ∖{0}⊆ S, because 𝐚 + 𝐚_j ∈ S, for every j ∈{1, …, n}. Thus,  in order to see that 𝐚∈ℋ(S) it suffices to prove that 𝐚∈pos(S). Without loss of generality, we assume that {a_1, …, a_ℓ} is a minimal set of generators of pos(S).
By Farkas' Lemma, pos(S) is a rational convex polyhedral cone. Then, for each j ∈{1, …, ℓ} there exists 𝐜_j ∈ℝ^d such that 𝐚_j ·𝐜_j = 0 and
𝐚_i ·𝐜_j ≥ 0, i ≠ j, where · denotes the usual inner product on ℝ^n. Now, since 𝐚 + 𝐚_j ∈ S, one has that 𝐚 + 𝐚_j = ∑_i=1^n u_ij𝐚_i, for some u_ij∈ℕ. Therefore 
    𝐚·𝐜_j = (𝐚 + 𝐚_j) ·𝐜_j = (∑_i=1^n u_ij𝐚_i) ·𝐜_j = ∑_i=1^n u_ij (𝐚_i ·𝐜_j) ≥ 0,
 for every j ∈{1, …, ℓ}. That is to say 𝐚∈pos(S).

Finally, if PF(S) ≠∅, the finiteness of PF(S) follows from the finiteness of the Betti numbers.


Observe that from the arguments in the proof of Theorem <ref> it follows that s_p is the cardinality of PF(S). In fact, we have proved the following fact:


If S is a MPD-semigroup, then 𝐛∈ S is the S-degree of the (n-2)th minimal syzygy of [S] if and only if 𝐛∈{𝐚 + ∑_i=1^n 𝐚_i, 𝐚∈PF(S)}.


In <cit.>, it is outlined a procedure for a partial computation of the minimal free resolution of S starting from a set of S-degrees of the jth minimal syzygy of [S], for some j. This procedure gives a whole free resolution if one knows
all the S-degrees of the pth minimal syzygies of [S], where p is the projective dimension of S. Therefore, by Corollary <ref>, if S is a MPD-semigroup, we can use the proposed method in <cit.> to compute the minimal free resolution of [S], provided that we were able to compute PF(S). However, this is not easy at all, for this reason it is highly interesting to given bounds for the elements in PF(S).

Given 𝐮 = (u_1, …, u_n) ∈ℕ^n, let ℓ(u) be the length of 𝐮 that is, ℓ(𝐮) = ∑_i=1^n u_i, and for an d × n-integer matrix B=(𝐛_1|⋯ | 𝐛_n), we will write || B || _∞ for _i ∑_j=1^n |b_ij|. In <cit.>, the authors provide an explicit bound for the S-degrees of the minimal generators of N_j, for every j ∈{1, …, p}.

Let A ∈ℕ^d × n be the matrix whose ith column is 𝐚_i, i = 1, …, n.

<cit.> 
If 𝐛∈ S is an S-degree of a minimal j-syzygy of [S], then 𝐛 = A 𝐮 with 𝐮∈ℕ^n such that 
    ℓ(𝐮) ≤ (1 + 4 || A || _∞)^d(d_j-1) + (j+1)d_j -1,
 where d_j = nj+1.


By Corollary <ref>, this bound can be particularized for j=n-2 as follows.


Let S be a MPD-semigroup. If 𝐚∈PF(S), then 𝐚 = A 𝐯 for some 𝐯∈ℕ^n satisfying 
    ℓ(𝐯) ≤ (1 + 4 || A || _∞)^d(n-1) + n^2-1.




Assuming 𝐚 is a pseudo-Frobenius element of S, by Corollary <ref>, there exists 𝐛=A 𝐮∈ S for some 𝐮∈ℕ^n, with V_n-2(𝐛) ≠∅ and such that 𝐚=𝐛-∑_i=1^n 𝐚_i; in particular 𝐚= A 𝐮-A 1=A(𝐮-1). Consider 𝐯=𝐮-1. Note that ||𝐯|| _1≤||𝐮|| _1 + ||1|| _1= ||𝐮|| _1 +n. By Theorem <ref>, ||𝐮|| _1≤ (1 + 4 || A || _∞)^d(d_j-1) + (j+1)d_j -1 where j=n-2 and d_n-2=n. So,

    ℓ(𝐯)    ≤  (1 + 4 || A || _∞)^d(n-1) + (n-1)n-1+n
        = (1 + 4 || A || _∞)^d(n-1) + n^2-1,

as claimed.


Note that, given any affine semigroup and the graded minimal free resolution of its associated algebra over a field, Theorem <ref> and Corollary <ref> allow us to check if the semigroup has pseudo-Frobenius elements and, in affirmative case, to compute them. Thus, the combination of both results provide an algorithm for the computation of the pseudo-Frobenius elements of a affine semigroup, provided that they exist (i.e. if the depth of the algebra is one). The following example illustrates this fact:


Let S be the multiple convex body semigroup associated to the convex hull 𝒫 of the set {(1.2, .35), (1.4, 0), (1.5, 0), (1.4, 1)}, that is to say, 
    S=⋃_k ∈ℕ  k 𝒫∩ℕ^2.




    < g r a p h i c s >



Using the  package  introduced in <cit.>, we obtain that the minimal generating system of S is the set of columns of the following matrix

    A=(
    [ 3 4 4 5 7 7 7 7 8 9; 0 1 2 2 0 3 4 5 1 2;   ]).

Now, we can easily check that S is not a 𝒞-semigroup, because (3,2)+λ (7,5) ∈ℋ(S), for every λ∈ℕ. Moreover, we can compute the S-graded minimal free resolution of [S] using Singular (<cit.>) as follows:

Finally, using the command , we obtain that the degrees of the minimal generators of the 9-th syzygy module are (72,20) and (73,21). So, S has two pseudo-Frobenius elements: (11,0)=(72,20)- (61,20) and (12,1)=(73,21)- (61,20).







§ ON FROBENIUS ELEMENTS OF MPD-SEMIGROUPS


Throughout this section, S will be a MPD-semigroup generated by 𝒜 = {𝐚_1, …, 𝐚_n}⊂ℕ^d.


We say that 𝐟∈ℋ(S) is a Frobenius element of S if
𝐟 = max_≺ℋ(S) for some term order ≺ on ℕ^d. Let us write F(S) for the set of Frobenius elements of S.


Frobenius elements of S may not exist. However, if S is a 𝒞-semigroup, then it has Frobenius elements because ℋ(S) is finite.


Every Frobenius element of S is a pseudo-Frobenius element of S, in symbols: F(S) ⊆PF(S).



If 𝐟∈F(S), then there is a term order ≺ on ℕ^d such that 𝐟 = max_≺ℋ(S). If there exists 𝐚∈ S ∖{0} such that 𝐟 + 𝐚∉S, then 𝐟≺𝐟 + 𝐚∈ℋ(S), in contradiction to the maximality of 𝐟. Therefore 𝐟∈PF(S).


The following notion of Frobenius vectors was introduced in <cit.>: we say that S has a Frobenius vector if there exists 𝐟∈ G(𝒜)∖ S such that 
    𝐟+ relint(pos(S)) ∩ G(𝒜)⊆ S∖{0}⊆ S,
 where G(𝒜) denotes the group generated by 𝒜 in ℤ^d and relint(pos(S)) the relative interior of the cone pos(S).


Every Frobenius element of S is a Frobenius vector of S.



Let 𝐟∈F(S). Since 𝐚 := 𝐟 + 𝐚_1 ∈ S, then 𝐟 = 𝐚 - 𝐚_1 ∈ G(𝒜) ∖ S. Let 𝐛∈relint(pos(S)) ∩ G(𝒜). If 𝐛∈ S,  then  𝐟+𝐛∈ S by Lemma <ref>. If 𝐛∉S,  then 𝐛∈ℋ(S) = (pos(S) ∖ S) ∩ℕ^d and therefore either 𝐟+𝐛∈ S or 𝐟+𝐛∈ℋ(S). However, since 𝐟+𝐛 is greater than 𝐟 for every term order on ℕ^d, we are done.


As a consequence of the above result, we have that the set of 𝒞-semigroups is a new family of affine semigroups for which Frobenius vectors exist.

Although Frobenius vectors may not exist in general, there are families of submonoids of ℕ^d with Frobenius vectors that are not MPD-semigroups (see <cit.>). However, even for MPD-semigroups, the converse of the above proposition does not hold in general, for instance, the MPD-semigroup in Example <ref> has a Frobenius vector which is not a Frobenius element.


Now, similarly to the numerical case (n=1), we can give a Selmer's formula type (see <cit.>) for 𝒞-semigroups. To do this, we need to recall the notion of Apéry set.


The Apéry set of a submonoid S of ℕ^d relative to 𝐛∈ S ∖{0} is defined as Ap(S,𝐛) = {𝐚∈ S | 𝐚-𝐛∈pos(S) ∖ S}.


Clearly Ap(S,𝐛) - 𝐛⊆ℋ(S); in particular, if S is 𝒞-semigroup, we have that Ap(S,𝐛) is finite for every 𝐛∈ S∖{0}.


Let S be a submonoid of ℕ^d and 𝐛∈ S ∖{0}. For each 𝐚∈ S there exists an unique (k, 𝐜) ∈ℕ×Ap(S, 𝐛) such that 𝐚 = k 𝐛 + 𝐜. In particular, (Ap(S, 𝐛) ∖{0}) ∪{𝐛} is a system of generators of S.



If suffices to take k as the highest non-negative integer such that 𝐛 - k 𝐚∈ S.


The following result is the generalization of <cit.>.


If 𝐟∈F(S), there exists a term order ≺ on ℕ^d such that 
    𝐟 = max_≺Ap(S,𝐛) - 𝐛,
 for every 𝐛∈ S ∖{0}.



By definition, there exists a term order ≺ on ℕ^d such that 𝐟 =  max_≺ℋ(S). By Lemma <ref>, 𝐟 + 𝐛∈ S and clearly (𝐟 + 𝐛) - 𝐛 = 𝐟∉pos(S) ∖ S. Thus, 𝐟 + 𝐛∈Ap(S,𝐛). Now, suppose that there exist 𝐚∈Ap(S,𝐛) such that 𝐟 + 𝐛≺𝐚. In this case, 𝐚 - 𝐛∈ℋ(S), so 𝐚 - 𝐛≼𝐟 and therefore 𝐚≺𝐟 + 𝐛 which contradicts the anti-symmetry property of ≺.


Next result generalizes <cit.>.


Let S be a submonoid of ℕ^d and 𝐛∈ S ∖{0}. Then PF(S) ≠∅ if and only if maximals_≼_SAp(S,𝐛) ≠∅. In this case, 
    PF(S) = {𝐚 - 𝐛|𝐚∈maximals_≼_SAp(S,𝐛) }.




Suppose that there exists 𝐛' ∈PF(S) and let 𝐚 = 𝐛' + 𝐛. Clearly 𝐚∈Ap(S,𝐛), and we claim that 𝐚∈maximals_≼_SAp(S,𝐛). Otherwise, there exists 𝐚' ∈Ap(S,𝐛) such that 𝐚' - 𝐚∈ S, then 
    𝐚' - 𝐛 = (𝐚' - 𝐛) + 𝐛' - 𝐛'= 𝐛' + (𝐚' - (𝐛' + 𝐛)) = 𝐛' + (𝐚' - 𝐚) ∈ S,
 which contradicts the definition of Apéry set of S relative to 𝐛. Therefore, 𝐛' = 𝐚 - 𝐛 with 𝐚∈maximals_≼_SAp(S,𝐛). Consider now 𝐚”∈maximals_≼_SAp(S,𝐛) and let 𝐛” = 𝐚”-𝐛. If 𝐛”∉PF(S), then 𝐛” + 𝐚_i ∉S, for some i ∈{1, …, n}, that is to say, 𝐚”+𝐚_i ∈Ap(S,𝐛) which is not possible by the maximality of 𝐚” in Ap(S,𝐛) with respect to ≼_S.


Observe that (<ref>) holds for every MPD-semigroup.

We end this section by proving that MPD-semigroups are stable by gluing. First of all, let us recall the notion of gluing of affine semigroups.

Given an affine semigroup S⊆ℕ^d, denote by G(S) the group spanned by S, that is,

    G(S)={𝐳∈ℤ^m |𝐳= 𝐚 - 𝐛, 𝐚, 𝐛∈ S }.


Assume that S is finitely generated. Let 𝒜 be the minimal generating system of S and 𝒜= 𝒜_1∪𝒜_2 be a nontrivial partition of 𝒜. Let S_i be the submonoid of ℕ^d generated by A_i, i∈{1,2}. Then S=S_1+S_2. We say that S is the gluing of S_1 and S_2 by 𝐝 if


  * 𝐝∈ S_1∩ S_2 and,

  * G(S_1)∩ G(S_2) = 𝐝ℤ.

We will denote this fact by S=S_1+_𝐝 S_2.


Let S be an finitely generated submonoid of ℕ^d.
Assume that S=S_1+_𝐝 S_2. If S_1 and S_2 are MPD-semigroups, and 𝐛_i ∈PF(S_i), i = 1,2, then 𝐛_1 + 𝐛_2 + 𝐝∈PF(S). In particular, S is a MPD-semigroup.



Let 𝐛 := 𝐛_1 + 𝐛_2 + 𝐝. Since 𝐛_i ∈pos(S_i), i = 1,2, and 𝐝∈ S_1 ∩ S_2, we conclude that 𝐛∈pos(S). If 𝐛∈ S, then there exist 𝐛'_i ∈ S_i, i = 1,2, such that 𝐛 = 𝐛'_1 + 𝐛'_2. Then 𝐛_1 + 𝐝 - 𝐛'_1 = 𝐛'_2 - 𝐛_2 ∈ G(S_1)∩ G(S_2) = 𝐝ℤ. So, there exist k ∈ℤ such that 𝐛_1 + 𝐝 - 𝐛'_1 = 𝐛'_2 - 𝐛_2 = k 𝐝. If k ≤ 0, then 𝐛_2 = 𝐛'_2 - k 𝐝∈ S_2, which is impossible. If k > 0, then 𝐛_1 = 𝐛'_1 + (k-1) 𝐝∈ S_1, which is also impossible. All this prove that 𝐛∈ℋ(S).

Now, let 𝐚∈ S ∖{0}. Again there exist 𝐛'_i ∈ S_i, i = 1,2, such that 𝐚 = 𝐛'_1 + 𝐛'_2. Since 𝐝∈ S_1 ∩ S_2 ⊂ S. We have that 𝐛_1 + 𝐛'_1 ∈ S_1 and 𝐛_2 + 𝐛'_2 + 𝐝∈ S_2. Thus 𝐛 + 𝐚∈ S, and we are done.



Let S_1={(x,y,z)∈ℕ^3| z=0}∖{(1,0,0)} and S_2={(x,y,z)∈ℕ^3| x=y}∖{(0,0,1)}. Clearly, (1,0,0)∈PF(S_1) and (0,0,1)∈PF(S_2). They are minimally generated by { (2,0,0),(3,0,0),(0,1,0),(1,1,0)}  and { (1,1,0),(1,1,1),(0,0,2),(0,0,3)}, respectively. The set G(S_1)∩ G(S_2) is equal to (1,1,0)ℤ and S_1+S_2 is generated by 
    {(2,0,0),(3,0,0),(0,1,0),(1,1,0),(1,1,1),(0,0,2),(0,0,3)}
 By Theorem <ref>,  (1,0,0)+(0,0,1)+(1,1,0)=(2,1,1) belongs to PF(S_1+S_2).






§ ON THE IRREDUCIBILITY OF MPD-SEMIGROUPS


Now, let us study the irreducibility of MPD-semigroups with special emphasis in the 𝒞-semigroups case. Recall that a submonoid of ℕ^d is irreducible if cannot be expressed as an intersection of two submonoids of ℕ^n containing it properly.


Let 𝐚∈PF(S). If 2 𝐚∈ S, then S ∪{𝐚} is the submonoid of ℕ^d generated by 𝒜∪{𝐚}. Moreover,


  * if 𝐚∈F(S) and PF(S) ≠{𝐚} then S ∪{𝐚} is a MPD-semigroup;

  * if S is a 𝒞-semigroup, then S ∪{𝐚} is a 𝒞-semigroup.




By definition, 𝐚 + 𝐛∈ S ⊂ S ∪{𝐚}, for every 𝐛∈ S, and by hypothesis 2 𝐚∈ S, so k 𝐚∈ S for every k ∈ℕ; thus, S ∪{𝐚} is the submonoid of ℕ^d generated by 𝒜∪{𝐚}.

Suppose now that 𝐚 is a Frobenius element of S and that PF(S) ≠{𝐚}. Let 𝐛∈PF(S) ∖{𝐚}. Clearly, 𝐚 + 𝐛∈ S, because 𝐚 + 𝐛 is greater than 𝐚 for every term order on ℕ^d, therefore 𝐛 + (S ∪{𝐚}) ∖{0}⊂ S ⊂ S ∪{𝐚} and we conclude that PF(S ∪{𝐚}) ≠∅ which proves (a).

Finally, since 𝐚∈ℋ(S), we have that pos(S ∪{𝐚}) = pos(S). Therefore ℋ(S ∪{𝐚}) ⊂ℋ(S), that is, S ∪{𝐚} is a 𝒞-semigroup if S it so as claimed in (b).



If S has a Frobenius element, 𝐟, and is irreducible then


  * S is maximal among all the submonoids of ℕ^d having 𝐟 as a Frobenius element.

  * S has an unique Frobenius element.




Suppose that S is irreducible and let S' be a submonoid of ℕ^d having 𝐟 as Frobenius element. Since, by Lemma <ref>, S ∪{𝐟} is a MPD-semigroup and S = (S ∪{𝐟}) ∩ S', we conclude that S = S'. Finally, if S has two Frobenius elements, say 𝐟_1 and 𝐟_2, then S = (S ∪{𝐟_1}) ∩ (S ∪{𝐟_2}) which contradicts the irreducibility of S.


Notice that the submonoids in condition (a) are necessarily MPD-semi­groups; indeed the existence of a Frobenius elements in a submonoid of ℕ^d implies that the submonoid is a MPD-semigroup by Lemma <ref>.


If S has a Frobenius element, 𝐟, and is irreducible, then either PF(S) = {𝐟} or PF(S) = {𝐟, 𝐟/2}.



Suppose that PF(S) ≠{𝐟}. Now, since PF(S) has cardinality greater than or equal to two, there exists 𝐚∈PF(S) different from 𝐟. If 2𝐚∈ S, then, by Lemma <ref>, S ∪{𝐚} are S ∪{𝐟} are a submonoid of ℕ^d whose intersection is S, in contradiction with irreducibility of S. Therefore, we may assume that 2𝐚∉S which implies 2 𝐚∈PF(S). Then 𝐟 + 𝐮 = 2 𝐚 for some 𝐮∈ℕ^d, because 𝐟 is greater than or equal to 𝐛 for every term order on ℕ^d. Notice that 4 𝐚 = 𝐟 + (𝐟 + 2 𝐮 ) ∈ S; so, by Lemma <ref>, 𝐒∪{2 𝐚} is a MPD-semigroup. Now, if 𝐮≠ 0, then S = (𝐒∪{2 𝐚}) ∩ (𝐒∪{𝐟}), in contradiction with irreducibility of S. Therefore 2 𝐚 = 𝐟 and we are done.



Let S be the MPD-semigroup of Example <ref>. Let us see that S is irreducible. If S is not irreducible, there exist two submonoids, S_1 and S_2,  of ℕ^2 such that S = S_1 ∩ S_2. Since pos(S) = ℕ^2, we have that pos(S_1) = pos(S_2) = ℕ^2 and it follows that ℋ(S_i) ⊆ℋ(S), i = 1,2. On other hand, since (7,2) ∉S, then (7,2) ∉S_1 or (7,2) ∉ S_2. Therefore, S_1 or S_2 is a submonoid of ℕ^2
such that 𝐟∈F(S_1) or 𝐟∈F(S_2), respectively. Now, by Proposition <ref>, we conclude that S = S_1 or S=S_2, that is, S is irreducible.


If n=1, the converse Theorem <ref> is also true (see <cit.>). Let us see that this is also happen if we fix the cone.


If S is 𝒞-semigroup, we say that S is 𝒞-irreducible if cannot be expressed as an intersection of two finitely generated submonoids S_1 and S_2 of ℕ^d with pos(S_1) = pos(S_2) = pos(S) containing it properly.



If S is a 𝒞-semigroup such that PF(S) = {𝐟} or
PF(S) = {𝐟, 𝐟/2},
then S is 𝒞-irreducible.



Suppose there exist two finitely generated submonoids S_1 and S_2 of ℕ^d with  pos(S_1) = pos(S_2) = pos(S) such that S = S_1 ∩ S_2; in particular, S_1 and S_2 are 𝒞-semigroups. For i=1,2, we take 𝐛_i ∈maximals_≼_S S_i ∖ S. Since S_i ∖ S is finite, 𝐛_i is well-defined for i = 1,2. By maximality, 𝐛_i + 𝐚∈ S for every 𝐚∈ S ∖{0}, i = 1,2, that is to say, 𝐛_i ∈PF(S). Therefore, 𝐛_i = 𝐟, i = 1,2 or 𝐛_i = 𝐟 and 𝐟_j = 𝐟/2, {i,j} = {1,2}. In the first case, we obtaim 𝐛_1 = 𝐛_2 which is not possible because 𝐛_i ∉S, i = 1,2. In the second case, we obtain that 𝐟∈ S_1 ∩ S_2 = S which obviously is impossible. Therefore, 𝐛_1 or 𝐛_2 does not exist and we conclude that S = S_1 or S = S_2.






§ PI-MONOIDS


Let ≼_ℕ^d be the usual partial order in ℕ^d, that is, 𝐚 = (a_1, …, a_d) ≼_ℕ^d𝐛 = (b_1, …, b_d) if and only if a_i ≤ b_i, i ∈{1, …, d}.


If S is a submonoid of ℕ^d, we define the multiplicity of S as m(S) := inf_≼_ℕ^d (S ∖{0}).


If d=1, the notion of multiplicity defined above agrees with the notion of multipliciy of a numerical semigroup (see <cit.>) Let us introduce a new family of submonoids of ℕ^d, that we have called principal ideal monoids, or PI-monoids for short. This family generalizes the notion of MED-semigroups (see <cit.> for d > 1.


A submonoid S of ℕ^d is said to be a PI-monoid if there exist a submonoid T of ℕ^d and 𝐚∈ T ∖{0} such that S = (𝐚 + T ) ∪{0}.


Clearly, PI-monoids are not always affine semigroups, since they are not necessarily finitely generated. We will explicitly provide a minimal generating system of any PI-monoid later on, first let us explore some its properties.


In 𝐍^2, an example of finitely generated PI-monoid is S_1=(2,2)+⟨  (1,1) ⟩=⟨ (2,2),(3,3)⟩.

To obtain a non-finitely generated PI-monoid of 𝐍^2, consider T=𝐍^2 and a=(1,1). The PI-monoid S_2=(1,1)+𝐍^2 is equal to {(x,y)| x≥ 1,  y≥ 1}∪{(0,0)} which is not a finitely generated submonoid of 𝐍^2.



If S ⊆ℕ^d is a PI-monoid, then m(S) ∈ S ∖{0}. In particular, m(S) = min_≼_ℕ^d (S ∖{0}).



Since S is a PI-monoid, there exist a submonoid T of ℕ^d and 𝐚∈ T ∖{0} such that S = (𝐚 + T ) ∪{0}. Clearly, 𝐚 = min_≼_ℕ^d (S ∖{0}).


The following result is the generalization of <cit.>.


Let S be a submonoid of ℕ^d. Then, S is a PI-monoid if and only if m(S) ∈ S ∖{0} and (S ∖{0}) - m(S) is a submonoid of ℕ^d.



If S is a PI-monoid, by Lemma <ref>, m(S) = min_≼_ℕ^d (S ∖{0}); moreover, there is a submonoid T of ℕ^d such that S = (m(S) + T) ∪{0}. So, (S ∖{0}) - m(S) = T is a submonoid of ℕ^d. For the converse implication, it suffices to note that S = ( m(S) +T) ∪{0} and that T = (S ∖{0}) - m(S).



If S ⊆ℕ^d is a PI-monoid, then there exist an unique submonoid T of ℕ^d and an unique 𝐚∈ T ∖{0} such that S = (𝐚 + T) ∖{0}



It is clear that 𝐚 must be equal to m(S) and that T must be equal to (S ∖{0}) - m(S).



Given a submonoid S of ℕ^d, we will write PI(S) for the set 
    { (𝐚 + S) ∪{0} |𝐚∈ S ∖{0}}.
 Observe that, as an immediate consequence of Corollary <ref>, we have that
the set {PI(S) | S is a submonoid of ℕ^d} is a partition of the set of all PI-monoids of ℕ^d. Moreover, if 𝒜 denotes the set of all submonoids of ℕ^d, for some d, and 𝒫i denotes the set of all PI-monoids of ℕ^d, for some d, we have an injective map 
    𝒜⟶𝒫i; S ↦ (min_lex(S ∖{0}) + S) ∪{0},
 where lex means the lexicographic term order on ℕ^d.


Recall that a system of generators 𝒜 of a submonoid A of ℕ^d is said to be minimal if no proper subset of 𝒜 generates A. The following result identifies a minimal system of generators of an PI-monoid.


Let S be a submonoid of ℕ^d. Then S is a PI-monoid if and only if 
    (Ap(S,m(S)) ∖{0}) ∪{m(S)}
 is a minimal system of generators of S.



By Lemma <ref>, if S is a PI-monoid, then m(S) ∈ S ∖{0} Moreover, by Proposition <ref>, we have that 𝒜 := (Ap(S,m(S)) ∖{0}) ∪{m(S)} is a system of generators of S. So, it suffices to prove that 𝒜 is minimal. Let us assume the contrary, that is, there exists 𝐚∈𝒜 such that 𝒜∖{𝐚} generates S. By the minimality of m(S), 𝐚≠ m(S). Thus, 𝐚∈Ap(S,m(S)) ∖{0} and there exists 𝐛 and 𝐜∈ S with 𝐚 = 𝐛 + 𝐜. By Proposition <ref>, we know that 𝐛 - m(S) + 𝐜 - m(S) = 𝐝 - m(S) for some 𝐝∈ S ∖{0}. Therefore, 𝐚 = 𝐝 + m(S) ∉Ap(S,m(S)), which is impossible. Conversely, if S is not a PI-monoid, by Proposition <ref>, we have that (S ∖{0}) - m(S) is not a submonoid of ℕ^d. So, there exists 𝐚 and 𝐛∈Ap(S,m(S)) ∖{0} such that 𝐚 - m(S) + 𝐛 - m(S) ∉(S ∖{0}) - m(S). In particular, 𝐚 + 𝐛 - m(S) ∉S and consequently 𝐚 + 𝐛∈Ap(S,m(S)). So, Ap(S,m(S)) is not a minimal system of generators of S.


Now, we will show that PI-monoids have non-trivial infinite pseudo-Frobenius set. Recall that every submonoid S of ℕ^d defines a natural partial order on ℕ^d as follows: 𝐱≼𝐲 if and only if 𝐲 - 𝐱∈ S. As in the previous section this partial order will be denoted as ≼_S.


A submonoid S of ℕ^d is a PI-monoid if and only if m(S) ∈ S ∖{0} and Ap(S,m(S)) ∖{0} = m(S) + PF(S).



If S is a PI-monoid, then m(S) ∈ S by Lemma <ref> and, by Proposition <ref>, (Ap(S,m(S)) ∖{0}) ∪{m(S)} is a minimal system of generators of S. Therefore, Ap(S,m(S)) ∖{0} = maximals_≼_S(Ap(S,m(S)). Now, by Proposition <ref>, we are done. Conversely, let us suppose that m(S) ∈ S and that Ap(S,m(S)) ∖{0} = m(S) + PF(S). By Proposition <ref>, we have that PF(S) = maximals_≼_S(Ap(S,m(S)) - m(S). Therefore, Ap(S,m(S)) ∖{0} = maximals_≼_S(Ap(S,m(S)), that is, ( Ap(S,m(S)) ∖{0}) ∪{m(S)} is a minimal system of generators of S. Now, by Proposition <ref>, we are done.


Putting all this together, we have the following characterization of the PI-monoids.


Let S be a submonoid of ℕ^d. The following conditions are equivalent:


  * S is a PI-monoid.

  * m(S) ∈ S ∖{0} and (S ∖{0}) - m(S) is closed under addition.

  * (Ap(S,m(S)) ∖{0}) ∪{m(S)} is a minimal system of generators of S.

  * {m(S) + PF(S)}∪{m(S)} is a minimal system of generators of S.




	Let S_1 and S_2 be the PI-monoids of Example <ref>. For S_1 we have m(S_1)={(2,2)} and  Ap(S_1,(2,2))={(0,0),(3,3)} obtaining that {(2,2),(3,3)} is a system of generators of S_1 and that PF(S_1)={(1,1)}.
	For S_2, m(S_2)=(1,1) and
	Ap(S_2,(1,1))={(0,0)}∪{(x,1)| x∈𝐍∖{0,1}}∪{ (1,y)| y∈𝐍∖{0,1}}. So {(1,1)}∪{(x,1)| x∈𝐍∖{0,1}}∪{ (1,y)| y∈𝐍∖{0,1}} is a non-finite system of generators of S_2 and PF(S_2)={(x,0)| x∈𝐍∖{0}}∪{(0,y)| y∈𝐍∖{0}}.


Finally, our last results state the relationship between PI-monoids and MPD-semigroups.


Let S be a PI-monoid. Then S is an affine semigroup if and only if S is an MPD-semigroup. In this case, Ap(S,m(S)) is finite.



Let S be a finitely generated PI-monoid. By Corollary <ref>, PF(S) ≠∅. So, by Theorem <ref>, S is a MPD-semigroup. The converse is trivial as MPD-semigroups are affine semigroups by definition. The last part is a direct consequence of Theorem <ref> and Corollary <ref>.


The following result was inspired by <cit.>:


Let S be a PI-monoid. Then there exist a direct system (S_λ, i_λμ) of MPD-semigroups contained in S such that S = _λ∈Λ S_λ, where i_λμ : S_λ→ S_μ is the inclusion map.



Let Λ = {λ⊂{m(S) + PF(S)} | λ is finite}, partially ordered by inclusion, and define S_λ to be the affine semigroup generated by λ∪{m(S)}. Clearly, we have that S_λ⊆ S_μ if λ⊆μ; in this case, let i_λμ : S_λ→ S_μ is the inclusion map. Now, since S_λ⊆ S for every λ∈Λ, we conclude that S = _λ∈Λ S_λ by Theorem <ref>, because
{m(S) + PF(S)}∪{m(S)} is a minimal system of generators of S.

Finally, let us see that S_λ is a MPD-semigroup for every λ∈Λ. To do that, we first note that m(S_λ) = m(S) ∈ S_λ, for every λ∈Λ. Let 𝒜 = {𝐚_1, …, 𝐚_n}⊆PF(S) and let λ = {m(S) + 𝒜}∈Λ. Then, Ap(S_λ,m(S)) = {0, 𝐚_1, …, 𝐚_n} is finite, in particular, maximals_≼_S(Ap(S,m(S)) - m(S) is a non-empty finite set. Therefore, by Proposition <ref>, PF(S) ≠∅, that is to say, S_λ is a MPD-semigroup.


Acknowledgement.
This paper was originally motivated by a question made of Antonio Campillo and Félix Delgado about 𝒞-semigroups during a talk of the fourth author at the GAS seminar of the SINGACOM research group. The question is answered in a wider context by Corollary <ref>. Part of this paper was written during a visit of the second author to the Universidad de Cádiz (Spain) and to the IEMath-GR (Universidad de Granada, Spain), he thanks these institutions for their warm hospitality. The authors would like to thank Antonio Campillo, Félix Delgado and Pedro A. García-Sánchez for usseful suggestions and comments.



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<http://www.gap-system.org/Packages/numericalsgps.html>
 GAP
  The GAP Group, GAP – Groups, Algorithms, and Programming,
  Version 4.8.6;
  2016,
  .

OjVi
Ojeda, I.; Vigneron-Tenorio, A. Simplicial complexes and minimal free resolution of monomial algebras. J. Pure Appl. Algebra 214(6), 2010, 850–861.

ns-book
Rosales, J. C.; García-Sánchez, P. A. Numerical Semigroups. Developments in Mathematics. 20. Springer, New York, 2009.



