A semigroup defining the Gröbner degeneration of a toric ideal

Abstract

We give an explicit set of generators for the semigroup of the Gröbner degeneration of a toric ideal. This set of generators is used to study algebraic properties of the semigroup it generates: approximation of semigroups, non-preservation of saturation, Betti elements, uniqueness of presentations, and Möbius functions.

1 Introduction

The Gröbner degeneration of an ideal is a flat family that deforms the ideal into a simpler one, for instance, a monomial ideal. Together with the theory of Gröbner bases, Gröbner degenerations are a powerful tool in the study of algebraic and geometric properties of an ideal or the corresponding algebraic variety.

In this paper we study Gröbner degenerations of toric ideals. The starting point for the whole discussion is the fact that the Gröbner degeneration of a toric ideal is also a toric ideal. Hence, it is defined by some semigroup. Several natural questions can be deduced concerning the algebraic or combinatorial properties of this semigroup. Yet, somehow surprisingly, they have not been systematically explored as far as we know. This paper aims at giving a first step towards that study.

Let $S\subset {\mathbb {Z}}^d$ be a semigroup defining a toric ideal I. Let $S_w\subset {\mathbb {Z}}^{d+1}$ be the semigroup of the Gröbner degeneration of I with respect to some weight vector w. Our first result provides an explicit set of generators of $S_w$ (Theorem 1.2). The resulting set of generators is our main tool to explore to what extent properties of the semigroup S are preserved for $S_w$.

We first study approximations of semigroups. It is known that the semigroup S can be approximated by its saturation ${\overline{S}}$. This means that there exists $a\in S$ such that $a+{\overline{S}}\subset S$. We prove that the corresponding result for the semigroup $S_w$ also involves the element $a\in S$ (Theorem 2.3). We stress that approximation of semigroups has a great deal of applications, including the study of valuations on graded algebras, the computation of dimension and degree of projective varieties, and intersection theory [10].

Next, we show that saturation is not preserved under Gröbner degenerations. We exhibit an infinite family of saturated semigroups for which a degeneration is saturated only for finitely many of them (Sect. 3). This suggests that saturation is actually rarely preserved under degenerations. Even though this phenomenon was somehow predictable, there are no explicit examples illustrating this fact in the literature, as far as we are aware.

We then study Betti elements of semigroups, a concept that is closely related to syzygies of toric ideals [7]. As before, the goal is to study the behaviour of Betti elements under Gröbner degenerations. We first show that each Betti element of the semigroup S gives rise to a Betti element of $S_w$ (Theorem 4.4). However, in general not all Betti elements of $S_w$ can be described like this (Example 4.7).

A particular case that we study is that of semigroups having a unique minimal generating set. These semigroups appear, for instance, in Algebraic Statistics [15]. We show that this property is preserved in the degeneration for certain families (Sects. 4.1 and 4.2). In our opinion, this gives enough evidence to conjecture that the uniqueness of a minimal generating set is always preserved under Gröbner degenerations (Conjecture 4.18).

The last section concerns Möbius functions of semigroups. This notion was introduced by G.-C. Rota and has a great deal of applications [13]. For a long time, Möbius functions were investigated only for numerical semigroups, although more general results were obtained in recent years [1]. In particular, an explicit formula for the Möbius function of semigroups with unique Betti element was given in [1, Theorem 4.1]. As an application of this formula, we conclude this paper by showing that the Möbius function of $S_w$ can be computed using only data of S and w, whenever both S and $S_w$ have a unique Betti element (Theorem 5.4).

2 An explicit semigroup defining the Gröbner degeneration of a toric ideal

Let us start by recalling the construction of a Gröbner degeneration of an ideal with respect to a weight vector.

Let $J\subset {\mathbb {K}}[x_1,\ldots ,x_n]$ be an ideal and $w=(w_1,\ldots ,w_n)\in {\mathbb {N}}^n$. For $f=\sum c_ux^u\in {\mathbb {K}}[x_1,\ldots ,x_n]$, let $d(f):=\max \{w\cdot u|c_u\ne 0\}$. Denote

$$\begin{aligned} f_t:=t^{d(f)}f(t^{-w_1}x_1,\ldots ,t^{-w_n}x_n)\in {\mathbb {K}}[x_1,\ldots ,x_n][t]. \end{aligned}$$

The ideal $J(t,w):=\langle f_t|f\in J\rangle$ is called the Gröbner degeneration of J. It is a classical fact that J(t, w) gives rise to a flat family deforming the affine algebraic variety defined by J to the variety of the initial ideal of J with respect to w [3, Theorem 15.17].

The following remark, which will be constantly used in this paper, provides a method to compute generators for the ideal J(t, w).

Remark 1.1

Let $G=\{g_1,\ldots ,g_s\}\subset J$ be a Gröbner basis of J with respect to any monomial order refining w. Then $G(t,w)=\{(g_1)_t,\ldots ,(g_s)_t\}\subset J(t,w)$ is a generating set of J(t, w) [3, Exercise 15.25].

Our first goal is to give an explicit generating set of the semigroup defining the Gröbner degeneration of a toric ideal. This generating set is described in terms of generators of the original semigroup and the weight vector.

The following notation will be constantly used throughout this paper.

Let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}\subset {\mathbb {Z}}^d$ be a finite subset. Consider the following map of semigroups,

$$\begin{aligned} \pi _{{\mathscr {A}}}:{\mathbb {N}}^n&\rightarrow {\mathbb {Z}}^d\\ (u_1,\ldots ,u_n)&\mapsto u_1a_1+\cdots +u_na_n. \end{aligned}$$

The image of $\pi _{{\mathscr {A}}}$ is denoted as ${\mathbb {N}}{\mathscr {A}}$. The previous map induces a map of ${\mathbb {K}}$-algebras,

$$\begin{aligned} {\hat{\pi }}_{{\mathscr {A}}}:{\mathbb {K}}[x_1,\ldots ,x_n]&\rightarrow {\mathbb {K}}\big [z_1,z_1^{-1},\ldots ,z_d,z_d^{-1}\big ]\\ x_i&\mapsto z^{a_i}. \end{aligned}$$

Denote $I_{{\mathscr {A}}}:=\ker {\hat{\pi }}_{{\mathscr {A}}}$. It is well known that this ideal is prime and generated by binomials. More precisely [14, Chapter 4],

$$\begin{aligned} I_{{\mathscr {A}}}=\langle x^{u}-x^{v}|\pi _{{\mathscr {A}}}(u)=\pi _{{\mathscr {A}}}(v)\rangle . \end{aligned}$$

The ideal $I_{{\mathscr {A}}}$ is called the toric ideal defined by ${\mathscr {A}}$.

The following theorem describes an explicit subset of ${\mathbb {Z}}^{d+1}$ that determines the Gröbner degeneration of $I_{{\mathscr {A}}}$.

Theorem 1.2

Let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}\subset {\mathbb {Z}}^d$ and $I_{{\mathscr {A}}}$ the corresponding toric ideal. Let $w=(w_1,\ldots ,w_n)\in {\mathbb {N}}^n$. Let

$$\begin{aligned} {\mathscr {A}}_w:=\{(a_1,w_1),\ldots ,(a_n,w_n),(0,\ldots ,0,1)\}\subset {\mathbb {Z}}^{d+1}. \end{aligned}$$

Then $I_{{\mathscr {A}}}(t,w)=I_{{{\mathscr {A}}}_w}$.

Proof

Let $u=(u_1,\ldots ,u_n)\in {\mathbb {N}}^n$ and $u_{n+1}\in {\mathbb {N}}$. We have the following relation:

$$\begin{aligned} \pi _{{\mathscr {A}}_w}(u,u_{n+1})&=u_1(a_1,w_1)+\cdots +u_n(a_n,w_n)+u_{n+1}(0,\ldots ,0,1)\nonumber \\&=(u_1a_1+\cdots +u_na_n,(u,u_{n+1})\cdot (w,1))\nonumber \\&=(\pi _{{\mathscr {A}}}(u),(u,u_{n+1})\cdot (w,1)). \end{aligned}$$

(1)

Let us first show that $I_{{\mathscr {A}}}(t,w)\subset I_{{{\mathscr {A}}}_w}$. Let $G\subset I_{{\mathscr {A}}}$ be a Gröbner basis consisting of binomials with respect to a refined monomial order $>_w$ and such that $G(t,w)=\{g_t|g\in G\}$ generates $I_{{\mathscr {A}}}(t,w)$ (see Remark 1.1). We show that $G(t,w)\subset I_{{{\mathscr {A}}}_w}$. Let $g=x^u-x^v\in G$, where $\pi (u)=\pi (v)$ and $x^u >_w x^v$. Then $g_t=x^u-x^vt^{w\cdot u-w\cdot v}$. By (1) it follows that

$$\begin{aligned} \pi _{{\mathscr {A}}_w}(v,w\cdot u-w\cdot v)&=(\pi _{{\mathscr {A}}}(v),(v,w\cdot u-w\cdot v)\cdot (w,1))\\&=(\pi _{{\mathscr {A}}}(v),w\cdot u)\\&=(\pi _{{\mathscr {A}}}(u),w\cdot u)\\&=\pi _{{\mathscr {A}}_w}(u,0). \end{aligned}$$

Thus, $g_t\in I_{{{\mathscr {A}}}_w}$ and so $I_{{\mathscr {A}}}(t,w)\subset I_{{{\mathscr {A}}}_w}$.

Now we show $I_{{{\mathscr {A}}}_w}\subset I_{{\mathscr {A}}}(t,w)$. Let $x^ut^{u_{n+1}}-x^vt^{v_{n+1}}\in I_{{{\mathscr {A}}}_w}$, i.e., $\pi _{{\mathscr {A}}_w}(u,u_{n+1})=\pi _{{\mathscr {A}}_w}(v,v_{n+1})$. By (1),

$$\begin{aligned} (\pi _{{\mathscr {A}}}(u),(u,u_{n+1})\cdot (w,1))=(\pi _{{\mathscr {A}}}(v),(v,v_{n+1})\cdot (w,1)). \end{aligned}$$

In particular, $\pi _{{\mathscr {A}}}(u)=\pi _{{\mathscr {A}}}(v)$ and so $x^u-x^v\in I_{{\mathscr {A}}}$. Now assume that $w\cdot u\ge w\cdot v$. Hence, $v_{n+1}=u_{n+1}+u\cdot w-v\cdot w$, and we conclude that

$$\begin{aligned} x^ut^{u_{n+1}}-x^vt^{v_{n+1}}&=x^ut^{u_{n+1}}-x^vt^{u_{n+1}+u\cdot w-v\cdot w}\\&=t^{u_{n+1}}(x^u-x^vt^{u\cdot w-v\cdot w})\\&=t^{u_{n+1}}(x^u-x^v)_t\in I_{{\mathscr {A}}}(t,w). \end{aligned}$$

$\square$

The previous theorem is our main tool to explore several combinatorial and algebraic properties of the Gröbner degeneration of toric ideals.

Remark 1.3

García-Puente, Sottile and Zhu introduced a set similar to ${\mathscr {A}}_w$ to define regular subdivisions of ${\mathscr {A}}$ [4, 16].

We conclude this section by defining the class of semigroups considered in this paper.

Definition 1.4

An affine semigroup S is a finitely generated submonoid of ${\mathbb {Z}}^d$ such that the group generated by S is ${\mathbb {Z}}^d$.

Remark 1.5

Given an affine semigroup generated by ${\mathscr {A}}=\{a_1,\ldots ,a_n\}\subset {\mathbb {Z}}^d$ and $w\in {\mathbb {N}}^n$, we have that ${\mathbb {N}}{\mathscr {A}}_w\subset {\mathbb {Z}}^{d+1}$ is a finitely generated semigroup. In addition, since ${\mathbb {N}}{\mathscr {A}}$ generates ${\mathbb {Z}}^d$ as a group and $(0,\ldots ,0,1)\in {\mathscr {A}}_w$, we obtain that ${\mathbb {N}}{\mathscr {A}}_w$ generates ${\mathbb {Z}}^{d+1}$ as a group. Hence, ${\mathbb {N}}{\mathscr {A}}_w$ is an affine semigroup.

3 Approximations of semigroups

Our first application of Theorem 1.2 concerns approximations of affine semigroups.

A well-known result in the theory of affine semigroups states that such semigroups can be approximated by their saturation. Recall that the saturation of an affine semigroup $S\subset {\mathbb {Z}}^d$ is defined as ${\overline{S}}\,{:=}\,{\mathbb {R}}_{\ge 0}S\cap {\mathbb {Z}}^d$. Equivalently, ${\overline{S}}=\{a\in {\mathbb {Z}}^d \mid ka\in S, \text { for } \text { some } k\in {\mathbb {N}}\}$.

Theorem 2.1

([10, Theorem 1.4]) Let $S\subset {\mathbb {Z}}^d$ be an affine semigroup. There exists $a\in S$ such that $a+{\overline{S}}\subset S$.

Our next goal is to study this theorem for the affine semigroup of the Gröbner degeneration of $I_{{\mathscr {A}}}$.

Lemma 2.2

Let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}\subset {\mathbb {Z}}^d$. Assume that there exists an element $a\in {\mathbb {N}}{\mathscr {A}}$ such that $a+\overline{{\mathbb {N}}{\mathscr {A}}}\subset {\mathbb {N}}{\mathscr {A}}$. Let $w\in {\mathbb {N}}^n$. Consider the following sets:

$$\begin{aligned} C&:=\left\{ \sum _{i=1}^n\alpha _ia_i \mid \alpha _i\in {\mathbb {R}}, 0\le \alpha _i\le 1\right\} ,\\ C_w&:=\left\{ \sum _{i=1}^n\alpha _i(a_i,w_i)+\alpha _{n+1}(0,\ldots ,0,1) \mid \alpha _i\in {\mathbb {R}}, 0\le \alpha _i\le 1\right\} . \end{aligned}$$

For each element $(c,c_{n+1})\in C_w\cap {\mathbb {Z}}^{d+1}$, where $c\in {\mathbb {Z}}^{d}$, there exists $\delta \in {\mathbb {N}}$ such that $(a,\delta )\in {\mathbb {N}}{\mathscr {A}}_w$ and $(a,\delta )+(c,c_{n+1})\in {\mathbb {N}}{\mathscr {A}}_w$.

Proof

Let $(c,c_{n+1})\in C_w\cap {\mathbb {Z}}^{d+1}$. Since $a\in {\mathbb {N}}{\mathscr {A}}$, we can write $a=\sum _{i=1}^nl_ia_i$. Now consider the following cases.

• (1) Suppose that $(c,c_{n+1})\in {\mathbb {N}}{\mathscr {A}}_w$. Let $\delta :=\sum _{i=1}^nl_iw_i$. Then $(a,\delta )=\sum _{i=1}^nl_i(a_i,w_i)\in {\mathbb {N}}{\mathscr {A}}_w$. In particular, $(a,\delta )+(c,c_{n+1})\in {\mathbb {N}}{\mathscr {A}}_w$.

• (2) Suppose that $(c,c_{n+1})\notin {\mathbb {N}}{\mathscr {A}}_w$. Notice that $(c,c_{n+1})\in C_w\cap {\mathbb {Z}}^{d+1}$ implies that $c\in C\cap {\mathbb {Z}}^d\subset \overline{{\mathbb {N}}{\mathscr {A}}}$. In particular, $a+c\in {\mathbb {N}}{\mathscr {A}}$. Let $a+c=\sum _{i=1}^n\beta _ia_i$ for some $\beta _i\in {\mathbb {N}}$. Now consider the following cases.

• (2.1) Suppose that $(a,0)+(c,c_{n+1})\in {\mathbb {N}}{\mathscr {A}}_w$. Let $\delta :=\sum _{i=1}^nl_iw_i$. Thus $(a,\delta )=\sum _{i=1}^nl_i(a_i,w_i)\in {\mathbb {N}}{\mathscr {A}}_w$. We obtain

  $$\begin{aligned} (a,\delta )+(c,c_{n+1})=(a,0)+(c,c_{n+1})+\delta (0,\ldots ,0,1)\in {\mathbb {N}}{\mathscr {A}}_w. \end{aligned}$$

• (2.2) Suppose that $(a,0)+(c,c_{n+1})\notin {\mathbb {N}}{\mathscr {A}}_w$. In particular, there is no $\beta _{n+1}\in {\mathbb {N}}$ such that

  $$\begin{aligned} (a,0)+(c,c_{n+1})=\sum _{i=1}^n\beta _i(a_i,w_i)+\beta _{n+1}(0,\ldots ,0,1). \end{aligned}$$

  So there is no $\beta _{n+1}\in {\mathbb {N}}$ such that $c_{n+1}=\sum _{i=1}^n\beta _iw_i+\beta _{n+1}$. Thus, $c_{n+1}<\sum _{i=1}^n\beta _iw_i$. Let $\delta :=\max \{\sum _{i=1}^n\beta _iw_i-c_{n+1},\sum _{i=1}^nl_iw_i\}$. Now notice that

  $$\begin{aligned} (a,\delta )&=\left( a,\sum _{i=1}^nl_iw_i\right) +\left( \delta -\sum _{i=1}^nl_iw_i\right) (0,\ldots ,0,1)\\&=\sum _{i=1}^nl_i(a_i,w_i)+\left( \delta -\sum _{i=1}^nl_iw_i\right) (0,\ldots ,0,1)\in {\mathbb {N}}{\mathscr {A}}_w. \end{aligned}$$

  Similarly,

  $$\begin{aligned} (a,\delta )+(c,c_{n+1})&=(a+c,\delta +c_{n+1})\\&=\left( a+c,\sum _{i=1}^n\beta _iw_i\right) +\left( \delta -\left( \sum _{i=1}^n\beta _iw_i-c_{n+1}\right) \right) (0,\ldots ,0,1)\\&=\sum _{i=1}^n\beta _i(a_i,w_i)+\left( \delta -\left( \sum _{i=1}^n\beta _iw_i-c_{n+1}\right) \right) (0,\ldots ,0,1)\in {\mathbb {N}}{\mathscr {A}}_w. \end{aligned}$$

$\square$

Theorem 2.3

Let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}\subset {\mathbb {Z}}^d$. Assume that there exists an element $a\in {\mathbb {N}}{\mathscr {A}}$ such that $a+\overline{{\mathbb {N}}{\mathscr {A}}}\subset {\mathbb {N}}{\mathscr {A}}$. Then there exists $\delta \in {\mathbb {N}}$ such that $(a,\delta )\in {\mathbb {N}}{\mathscr {A}}_w$ and $(a,\delta )+\overline{{\mathbb {N}}{\mathscr {A}}_w}\subset {\mathbb {N}}{\mathscr {A}}_w$.

Proof

Consider the notation of Lemma 2.2. Let $\{c_1,\ldots ,c_m\}=C_w\cap {\mathbb {Z}}^{d+1}$. By the lemma, there exists $\delta _i\in {\mathbb {N}}$ such that $(a,\delta _i)\in {\mathbb {N}}{\mathscr {A}}_w$ and $(a,\delta _i)+c_i\in {\mathbb {N}}{\mathscr {A}}_w$. Let $\delta :=\max _{i}\{\delta _i\}$. Let $v\in \overline{{\mathbb {N}}{\mathscr {A}}_w}$. Since $\overline{{\mathbb {N}}{\mathscr {A}}_w}={\mathbb {R}}_{\ge 0}{\mathscr {A}}_w\cap {\mathbb {Z}}^{d+1}$, we can write $v=\sum _{i=1}^nr_i(a_i,w_i)+r_{n+1}(0,\ldots ,0,1)$, where $r_i\in {\mathbb {R}}_{\ge 0}$. Thus,

$$\begin{aligned} (a,\delta )+v&=(a,\delta )+\sum _{i=1}^nr_i(a_i,w_i)+r_{n+1}(0,\ldots ,0,1)\\&=(a,\delta )+\sum _{i=1}^n\lfloor r_i \rfloor (a_i,w_i)+\lfloor r_{n+1} \rfloor (0,\ldots ,0,1)\\&\quad +\sum _{i=1}^n(r_i-\lfloor r_i \rfloor )(a_i,w_i)+(r_{n+1}-\lfloor r_{n+1} \rfloor )(0,\ldots ,0,1). \end{aligned}$$

Notice that $\sum _{i=1}^n\lfloor r_i \rfloor (a_i,w_i)+\lfloor r_{n+1} \rfloor (0,\ldots ,0,1)\in {\mathbb {N}}{\mathscr {A}}_w$ and $\sum _{i=1}^n(r_i-\lfloor r_i \rfloor )(a_i,w_i)+(r_{n+1}-\lfloor r_{n+1} \rfloor )(0,\ldots ,0,1)\in C_w\cap {\mathbb {Z}}^{d+1}$. Assume that this last element is $c_1$. Then

$$\begin{aligned} (a,\delta )+c_1=(\delta -\delta _1)(0,\ldots ,0,1)+(a,\delta _1)+c_1\in {\mathbb {N}}{\mathscr {A}}_w. \end{aligned}$$

We conclude that

$$\begin{aligned} (a,\delta )+v=\sum _{i=1}^n\lfloor r_i \rfloor (a_i,w_i)+\lfloor r_{n+1} \rfloor (0,\ldots ,0,1)+(a,\delta )+c_1\in {\mathbb {N}}{\mathscr {A}}_w. \end{aligned}$$

$\square$

4 Saturation

In this section we show that saturation of affine semigroups is not preserved by Gröbner degeneration. Recall that an affine semigroup S is saturated if $S={\overline{S}}$.

Let $m\in {\mathbb {N}}$. Consider the set

$$\begin{aligned} {\mathscr {A}}(m):=\{(1,0),(1,1),(m,m+1)\}\subset {\mathbb {Z}}^2. \end{aligned}$$

It is well known that ${\mathbb {N}}{\mathscr {A}}(m)$ is a saturated semigroup. Let $w=(1,1,1)$. We show that ${\mathbb {N}}{\mathscr {A}}(m)_w$ is saturated if and only if $m\le 2$. We use this notation throughout this section.

Proposition 3.1

The semigroup ${\mathbb {N}}{\mathscr {A}}(1)_w$ is saturated.

Proof

Let $(a,b,c)\in {\mathbb {N}}^3$ be such that $\lambda (a,b,c)\in {\mathbb {N}}{\mathscr {A}}(1)_w$, for some $\lambda \ge 1$. We want to show that $(a,b,c)\in {\mathbb {N}}{\mathscr {A}}(1)_w$. There exist $\alpha _i\in {\mathbb {N}}$ such that

$$\begin{aligned} \lambda a&=\alpha _1+\alpha _2+\alpha _3,\\ \lambda b&=\alpha _2+2\alpha _3,\\ \lambda c&=\alpha _1+\alpha _2+\alpha _3+\alpha _4=\lambda a+\alpha _4. \end{aligned}$$

By the third equation, it follows that $c\ge a$. Let $c=a+d$, for some $d\ge 0$. Then $(a,b,c)=(a,b,a)+d(0,0,1)$. Thus, we need to show that $(a,b,a)\in {\mathbb {N}}{\mathscr {A}}(1)_w$. Since $\lambda (a,b,c)\in {\mathbb {N}}{\mathscr {A}}(1)_w$, it follows that $\lambda (a,b)\in {\mathbb {N}}{\mathscr {A}}(1)$. Since this semigroup is saturated, $(a,b)\in {\mathbb {N}}{\mathscr {A}}(1)$, i.e.,

$$\begin{aligned} (a,b)=\beta _1(1,0)+\beta _2(1,1)+\beta _3(1,2). \end{aligned}$$

We conclude that $(a,b,a)=\beta _1(1,0,1)+\beta _2(1,1,1)+\beta _3(1,2,1)\in {\mathbb {N}}{\mathscr {A}}(1)_w$. $\square$

Proposition 3.2

The semigroup ${\mathbb {N}}{\mathscr {A}}(2)_w$ is saturated.

Proof

Let $b_1=(1,0,1)$, $b_2=(1,1,1)$, $b_3=(2,3,1)$, and $b_4=(0,0,1)$. By definition, ${\mathscr {A}}(2)_w=\{b_1,b_2,b_3,b_4\}$. To show that ${\mathbb {N}}{\mathscr {A}}(2)_w$ is saturated we prove that ${\mathbb {N}}{\mathscr {A}}(2)_w={\mathbb {R}}_{\ge 0}({\mathscr {A}}(2)_w)\cap {\mathbb {Z}}^3$. Clearly, ${\mathbb {N}}{\mathscr {A}}(2)_w\subset {\mathbb {R}}_{\ge 0}({\mathscr {A}}(2)_w)\cap {\mathbb {Z}}^3$. For the other inclusion it is enough to show that $A_1\cap {\mathbb {Z}}^3\subset {\mathbb {N}}{\mathscr {A}}(2)_w$, where $A_1=\{\sum _i\lambda _ib_i|\lambda _i\in {\mathbb {R}},0\le \lambda _i\le 1\}$.

Let $(x,y,z)\in A_1\cap {\mathbb {Z}}^3$. Then there exist $\lambda _i\in [0,1]$ such that

$$\begin{aligned} x&=\lambda _1+\lambda _2+2\lambda _3,\nonumber \\ y&=\lambda _2+3\lambda _3,\nonumber \\ z&=\lambda _1+\lambda _2+\lambda _3+\lambda _4. \end{aligned}$$

(2)

By the third equation it follows that $z\le 4$.

• $z=4\Rightarrow \lambda _1=\lambda _2=\lambda _3=\lambda _4=1\Rightarrow (x,y,z)=(4,4,4)\in {\mathbb {N}}{\mathscr {A}}(2)_w$.

• $z=0\Rightarrow \lambda _1=\lambda _2=\lambda _3=\lambda _4=0\Rightarrow (x,y,z)=(0,0,0)\in {\mathbb {N}}{\mathscr {A}}(2)_w$.

By doing elementary operations on the equations in (2) we obtain:

$$\begin{aligned} \lambda _3&=x-z+\lambda _4, \end{aligned}$$

(3)

$$\begin{aligned} \lambda _2+3\lambda _4&=-3x+y+3z, \end{aligned}$$

(4)

$$\begin{aligned} \lambda _1+\lambda _2+2\lambda _4&=2z-x. \end{aligned}$$

(5)

On the other hand, since $(x,y,z)\in A_1\cap {\mathbb {Z}}^3$ it follows that

$$\begin{aligned} (x,y)\in&\{\lambda _1(1,0)+\lambda _2(1,1)+\lambda _3(2,3)|\lambda _i\in [0,1]\}\cap {\mathbb {Z}}^2\\&=\{(0,0),(1,0),(1,1),(2,1),(2,2),(2,3),(3,3),(3,4),(4,4)\}. \end{aligned}$$

Now we are ready to study the remaining values of z.

• $z=1$. Of the nine options for (x, y, z) only (0, 0, 1), (1, 0, 1), (1, 1, 1), (2, 3, 1) are in $A_1$ (for the other cases $-3x+y+3z<0$, contradicting (4)). These four elements belong to ${\mathbb {N}}{\mathscr {A}}(2)_w$.

• $z=2$. If $(0,0,2)\in A_1$ then, by Eq. (3), $\lambda _4=\lambda _3+2\ge 2$, which is a contradiction. Thus $(0,0,2)\notin A_1$. Similarly, $(4,4,2)\notin A_1$. On the other hand, $(2,2,2)=\frac{1}{3}(1,0,1)+(1,1,1)+\frac{1}{3}(2,3,1)+\frac{1}{3}(0,0,1)\in A_1$ and $(2,2,2)=2(1,1,1)\in {\mathbb {N}}{\mathscr {A}}(2)_w$. The other six options for (x, y, z) are the sum of two different elements of ${\mathscr {A}}(2)_w$. Therefore they belong to $A_1$ and also to ${\mathbb {N}}{\mathscr {A}}(2)_w$.

• $z=3$. If $(0,0,3)\in A_1$ then, by Eq. (3), $\lambda _4=\lambda _3+3\ge 3$, which is a contradiction. Thus, $(0,0,3)\notin A_1$. Similarly, $(1,0,3),(1,1,3)\notin A_1$. If $(2,2,3)\in A_1$, again by (3), $0\le \lambda _4=\lambda _3+1\le 1$ implying $\lambda _3=0$ and $\lambda _4=1$. Thus, by (4), $\lambda _2=5-3=2$, which is a contradiction. Thus, $(2,2,3)\notin A_1$. Similarly, $(2,3,3)\notin A_1$. The remaining four options for (x, y, z) are the sum of three different elements of ${\mathscr {A}}(2)_w$. Therefore they belong to $A_1$ and also to ${\mathbb {N}}{\mathscr {A}}(2)_w$.

We conclude that $A_1\cap {\mathbb {Z}}^3\subset {\mathbb {N}}{\mathscr {A}}(2)_w$ and so ${\mathbb {R}}_{\ge 0}({\mathscr {A}}(2)_w)\cap {\mathbb {Z}}^3\subset {\mathbb {N}}{\mathscr {A}}(2)_w$, implying that ${\mathbb {N}}{\mathscr {A}}(2)_w$ is saturated. $\square$

Proposition 3.3

The semigroup ${\mathbb {N}}{\mathscr {A}}(m)_w$ is not saturated for all $m\ge 3$.

Proof

By definition, ${\mathscr {A}}(m)_w=\{(1,0,1),(1,1,1),(m,m+1,1),(0,0,1)\}$. Assume first that $m=2r+1$ for some $r\ge 1$. Let $q=r-1\ge 0$ and notice:

$$\begin{aligned} (m+1,m+1,r+1)=(1,0,1)+(m,m+1,1)+q(0,0,1)\in {\mathbb {N}}{\mathscr {A}}(m)_w. \end{aligned}$$

On the other hand, $(2,2,1)\notin {\mathbb {N}}{\mathscr {A}}(m)_w$. Since $(m+1,m+1,r+1)=(r+1)(2,2,1)$ we conclude that ${\mathbb {N}}{\mathscr {A}}(m)_w$ is not saturated in this case.

Now assume that $m=2r$ for some $r\ge 2$. Let $q=r-2\ge 0$. Then,

$$\begin{aligned} (m+2,m+2,r+1)=(1,0,1)+(1,1,1)+(m,m+1,1)+q(0,0,1)\in {\mathbb {N}}{\mathscr {A}}(m)_w. \end{aligned}$$

As before, $(m+2,m+2,r+1)=(r+1)(2,2,1)$ implying that ${\mathbb {N}}{\mathscr {A}}(m)_w$ is not saturated in this case. $\square$

5 Betti elements

As in previous sections, let $S\subset {\mathbb {Z}}^d$ be an affine semigroup generated by ${\mathscr {A}}=\{a_1,\ldots ,a_n\}$. Given $w\in {\mathbb {N}}^n$, denote $S_w:={\mathbb {N}}{\mathscr {A}}_w$. In this section we study the behaviour of Betti elements of affine semigroups under Gröbner degenerations. We assume that S is pointed, that is, $S\cap (-S)=\{0\}$. We first recall the basic definitions we need.

Let $\sim _{{\mathscr {A}}}$ denote the kernel congruence of $\pi _{{\mathscr {A}}}$, i.e., $\alpha \sim _{{\mathscr {A}}}\beta$ if $\pi _{{\mathscr {A}}}(\alpha )=\pi _{{\mathscr {A}}}(\beta )$. It is well-known that $\sim _{{\mathscr {A}}}$ is finitely generated. A presentation $\rho \subset {\mathbb {N}}^n\times {\mathbb {N}}^n$ for S is a system of generators of $\sim _{{\mathscr {A}}}$. A minimal presentation for S is a minimal system of generators of $\sim _{{\mathscr {A}}}$. Notice that this is equivalent to ask for a minimal set of binomial generators of the corresponding toric ideal.

Definition 4.1

Let ${\text{ Betti }}(S):=\{\pi _{{\mathscr {A}}}(\alpha )\mid (\alpha ,\beta )\in \rho \}$, where $\rho \subset {\mathbb {N}}^n\times {\mathbb {N}}^n$ is any minimal presentation of S. The set ${\text {Betti}}(S)$ does not depend on $\rho$ [7, Chapter 9]. It is called the set of Betti elements of S.

The terminology in the previous definition comes from the Betti numbers of $I_{{\mathscr {A}}}$. Recall that $I_{{\mathscr {A}}}$ is S-graded, where $\textrm{deg}_{S}(x_i)=\pi _{{\mathscr {A}}}(e_i)$, and $e_i$ is the i-th element of the canonical basis of ${\mathbb {N}}^n$. The first Betti number of degree $a\in S$ of ${\mathbb {K}}[S]:={\mathbb {K}}[x]/I_{{\mathscr {A}}}$, denoted by $\beta _{1,a}({\mathbb {K}}[S])$ is the number of minimal generators of degree a of $I_{{\mathscr {A}}}$. It is well known that it does not depend on the minimal set of generators of $I_{{\mathscr {A}}}$. The first Betti number of ${\mathbb {K}}[S]$ is the cardinality of a minimal set of generators of $I_{{\mathscr {A}}}$, denoted by $\beta _{1}({\mathbb {K}}[S])$, so $\beta _{1}({\mathbb {K}}[S])=\sum _{a\in S}\beta _{1,a}({\mathbb {K}}[S])$.

Remark 4.2

In view of the previous paragraph, for $a\in {\mathbb {N}}^{d}$, $a\in {\text {Betti}}(S)$ if and only if $\beta _{1,a}({\mathbb {K}}[S])\ne 0$.

Our first result relates the Betti elements of S with those of $S_w$. First we prove a simple lemma.

Lemma 4.3

Let $w\in {\mathbb {N}}^ n$. Let $\{g_1,\ldots ,g_s\}\subset I_{{\mathscr {A}}}$ be a generating set of $I_{{\mathscr {A}}}$. Assume that $\{(g_1)_t,\ldots ,(g_r)_t\}$ generates $I_{{{\mathscr {A}}}_w}$, for some $r\le s$. Then $\{g_1,\ldots ,g_r\}$ generates $I_{{\mathscr {A}}}$.

Proof

If $r=s$, there is nothing to prove. Suppose $r<s$ and let $j\in \{r+1,\ldots ,s\}$. By hypothesis, $(g_j)_t=\sum _{i=1}^r h_{i}(x,t)(g_i)_t.$ Making $t=1$ it follows $g_j=\sum _{i=1}^r h_{i}(x,1)g_i$. $\square$

Theorem 4.4

Let $w\in {\mathbb {N}}^n$. Then, for each $b\in {\text {Betti}}(S)$ there exists $\lambda \in {\mathbb {N}}$ such that $(b,\lambda )\in {\text {Betti}}(S_w)$.

Proof

Let $\{g_1,\ldots ,g_s\}\subset I_{{\mathscr {A}}}$ be a Gröbner basis with respect to a refined order $>_w$, where $g_i=x^{\alpha _i}-x^{\beta _i}$ and $\alpha _i\cdot w\ge \beta _i\cdot w$, for each i. Then, $I_{{{\mathscr {A}}}_w}=\langle (g_1)_t,\ldots ,(g_s)_t \rangle$ by Remark 1.1.

After reordering the $g_i$’s if necessary, we may assume that $\{(g_1)_t,\ldots ,(g_r)_t\}$ is a minimal generating set, for some $r\le s$. In particular, ${\text {Betti}}(S_w)=\{\pi _{{\mathscr {A}}_w}(\alpha _i,0)=(\pi _{{\mathscr {A}}}(\alpha _i),\alpha _i\cdot w)\}_{i=1}^r$. On the other hand, by Lemma 4.3, it follows that $\{g_1,\ldots ,g_r\}$ generates $I_{{\mathscr {A}}}$ and so this set contains a minimal generating set. This implies the theorem. $\square$

Remark 4.5

A similar result to Theorem 4.4 was proved in [11, Theorem 8.29].

In view of Theorem 4.4, several natural questions arise. Is the element $\lambda$ unique? Does every Betti element of $S_w$ have as first coordinate a Betti element of S? In the following examples we show that every scenario could actually happen. Example 4.6 shows that $\lambda$ may not be unique. Example 4.7 shows that there are Betti elements of $S_w$ whose first coordinate is not a Betti element of S. Finally, Example 4.8 exhibits an infinite family of numerical semigroups where the map ${\text {Betti}}(S)\rightarrow {\text {Betti}}(S_w)$, $b\mapsto (b,\lambda )$ is well-defined and bijective. These examples show that Theorem 4.4 is the best result we can expect regarding Betti elements of S and $S_w$.

Example 4.6

Let $a_1=6, a_2=10, a_3=15$ and ${\mathscr {A}}=\{a_1,a_2,a_3\}$. Then $I_{{\mathscr {A}}}$ is minimally generated by $\{x_1^5-x_2^3,x_2^3-x_3^2\}$. Thus, ${\text {Betti}}(S)=\{30\}$. Let $w=(1,1,1)$. The previous set of generators is also a Gröbner basis for $I_{{\mathscr {A}}}$ with respect to $>_w$, where > is the lexicographical order. Hence, $I_{{{\mathscr {A}}}_w}$ is generated by $\{x_1^5-x_2^3t^2,x_2^3-x_3^2t\}$. Actually, it is a minimal generating set. In particular, ${\text {Betti}}(S_w)=\{(30,5),(30,3)\}$.

Example 4.7

Let ${\mathscr {A}}=\{(1,0),(1,1),(1,2),(1,3)\}\subset {\mathbb {N}}^2$. Then $I_{{\mathscr {A}}}$ is minimally generated by $\{ac-b^2,ad-bc,bd-c^2\}$. Thus, ${\text {Betti}}(S)$ is the set $\{(2,2),(2,3),(2,4)\}$. Let $w=(3,7,2,5)$. A Gröbner basis for $I_{{\mathscr {A}}}$ with respect to $>_{w}$, where > is the lexicographical order, is $\{b^2-ac,bc-ad,bd-c^2,ad^2-c^3\}$. Hence, $I_{{{\mathscr {A}}}_w}$ is generated by $\{b^2-act^9,bc-adt,bd-c^2t^8,ad^2-c^3t^7\}.$ Actually, it is a minimal generating set. In particular, ${\text {Betti}}(S_w)=\{(2,2,14),(2,3,9),(2,4,12),(3,6,13)\}$.

Example 4.8

Let $b_1,\ldots ,b_n\ge 2$ be pairwise relatively prime integers and $a_i:=\prod _{j\ne i}b_j$. Let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}$ and $S={\mathbb {N}}{\mathscr {A}}\subset {\mathbb {N}}$. In this case, ${\text {Betti}}(S)=\{b\}$, where $b=\prod _{j=1}^n b_j$ [6, Example 12]. Let $w=(w_1,\ldots ,w_n)\in {\mathbb {N}}^n$. Consider a permutation $(i_1,\ldots ,i_n)$ of $(1,\ldots ,n)$ such that $b_{i_1}w_{i_1}\ge b_{i_2}w_{i_2}\ge \cdots \ge b_{i_n}w_{i_n}$.

It is known that $I_{{\mathscr {A}}}=\langle f_2,\ldots ,f_n\rangle$, where $f_i=x_1^{b_1}-x_i^{b_i}$, for each $i\in \{2,\ldots ,n\}$ [6, Example 12]. Define

$$\begin{aligned}&g_{i_2}=x_{i_1}^{b_{i_1}}-x_{i_2}^{b_{i_2}},{} & {} h_{i_2}=x_{i_1}^{b_{i_1}}-x_{i_2}^{b_{i_2}},\\&g_{i_3}=x_{i_2}^{b_{i_2}}-x_{i_3}^{b_{i_3}},{} & {} h_{i_3}=x_{i_1}^{b_{i_1}}-x_{i_3}^{b_{i_3}},\\&\vdots{} & {} \vdots \\&g_{i_n}=x_{i_{n-1}}^{b_{i_{n-1}}}-x_{i_n}^{b_{i_n}},{} & {} h_{i_n}=x_{i_1}^{b_{i_1}}-x_{i_n}^{b_{i_n}}. \end{aligned}$$

A direct computation shows that

$$\begin{aligned} \langle g_{i_2},\ldots ,g_{i_n} \rangle =\langle h_{i_2},\ldots ,h_{i_n} \rangle =\langle f_2,\ldots ,f_n \rangle . \end{aligned}$$

Notice that the binomials $f_i$, $h_{i_j}$, and $g_{i_j}$ are S-homogeneous. In addition, $\{f_2,\ldots ,f_n\}$ is a minimal generating set of $I_{{\mathscr {A}}}$. Indeed, assume, for instance, that $f_2$ can be written in terms of $f_3,\ldots ,f_n$. Then, evaluating at $(0,1,0,\ldots ,0)$ we obtain a contradiction. By the cardinality of $\{g_{i_2},\ldots ,g_{i_n}\}$, it follows that this set is also a minimal generating set of $I_{{\mathscr {A}}}$.

Let > denote the lexicographical order on ${\mathbb {K}}[x_1,\ldots ,x_n]$ with the variables ordered as $x_{i_1}>\cdots >x_{i_n}$. Then, for each $j\in \{2,\ldots ,n\}$, we have ${\text {LT}}_{>_{w}}(g_{i_j})=x_{i_{j-1}}^{b_{i_{j-1}}}$. In particular, for any $1<j<k\le n$, the leading terms ${\text {LT}}_{>_{w}}(g_{i_j})$ and ${\text {LT}}_{>_{w}}(g_{i_k})$ are relatively prime. Hence, $\{g_{i_2},\ldots ,g_{i_n}\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$ with respect to $>_{w}$ [9, Corollary 2.3.4]. It follows that

$$\begin{aligned} I_{{{\mathscr {A}}}_w}=\langle (g_{i_2})_t,\ldots ,(g_{i_n})_t \rangle . \end{aligned}$$

Notice that each $(g_{i_j})_t=x_{i_{j-1}}^{b_{i_{j-1}}}-x_{i_{j}}^{b_{i_{j}}}t^{w_{i_{j-1}}b_{i_{j-1}}-w_{i_{j}}b_{i_{j}}}$ is $S_w$-homogeneous of degree $(b,b_{i_{j-1}}w_{i_{j-1}})$.

We already proved that $\{g_{i_2},\ldots ,g_{i_n}\}$ is a minimal generating set of $I_{{\mathscr {A}}}$. Therefore, $\{(g_{i_2})_t,\ldots ,(g_{i_n})_t\}$ is also a minimal generating set of $I_{{{\mathscr {A}}}_w}$ (see Remark 1.1 and Lemma 4.3). We conclude that ${\text {Betti}}(S_w)$ equals the set $\{(b,b_{i_{1}}w_{i_{1}}),\ldots ,(b,b_{i_{n-1}}w_{i_{n-1}})\}.$ In particular, $|{\text {Betti}}(S_w)|=1$ if and only if $b_{i_1}w_{i_1}=\cdots =b_{i_{n-1}}w_{i_{n-1}}$. In this case, the map ${\text {Betti}}(S)\rightarrow {\text {Betti}}(S_w)$, $b\mapsto (b,b_{i_1}w_{i_1})$ is well-defined and bijective.

5.1 Semigroups with unique minimal generating set

In this section we explore further consequences of Theorem 4.4 in the context of affine semigroups having a unique minimal generating set. Such semigroups have been studied, for instance, in [2, 5, 12].

We say that an affine semigroup S is uniquely presented if it has a unique minimal presentation. Notice that this is equivalent to the corresponding toric ideal having a unique minimal generating set of binomials, up to scalar multiplication.

Remark 4.9

The notion of uniquely presented is not to be confused with the previous notion of minimally presented. The former asks for a unique minimal generating set of a toric ideal whereas the latter just asks for a minimal generating set, up to scalar multiplication.

We define a partial order $>_S$ on S as follows (recall that we assume $S\cap (-S)=\{0\}$): $\alpha >_S\beta$ if $\alpha -\beta \in S$. We say that $\alpha \in {\text {Betti}}(S)$ is Betti minimal if it is minimal with respect to the order $>_S$. The set of such elements is denoted as ${\text {Betti-min}}(S)$.

Remark 4.10

It is known that $I_{{\mathscr {A}}}$ is uniquely presented if and only if ${\text {Betti}}(S)={\text {Betti-min}}(S)$ and the cardinality of ${\text {Betti}}(S)$ is equal to the cardinality of a minimal generating set of binomials of $I_{{\mathscr {A}}}$ [5, Corollary 6]. By Remark 4.2, this is also equivalent to ${\text {Betti}}(S)={\text {Betti-min}}(S)$ and $\beta _{1,a}({\mathbb {K}}[x]/I_{{\mathscr {A}}})=1$ for all $a\in {\text {Betti}}(S)$.

In the following proposition we show that, for uniquely presented semigroups, the Betti elements of $S_w$ coming from Betti elements of S are Betti-minimal.

Let S be an affine semigroup with Betti elements ${\text {Betti}}(S)=\{b_1,\ldots ,b_r\}$. Let $w\in {\mathbb {N}}^n$. By Theorem 4.4, there exist some $\lambda _i\in {\mathbb {N}}$ such that $(b_i,\lambda _i)\in {\text {Betti}}(S_w)$, for each $i\in \{1,\ldots ,r\}$.

Theorem 4.11

With the previous notation, assume in addition that S is uniquely presented. Then each $(b_i,\lambda _i)$ is Betti-minimal.

Proof

Let $\{g_1,\ldots ,g_r\}\subset I_{{\mathscr {A}}}$ be the only minimal binomial generating set of $I_{{\mathscr {A}}}$. Using Buchberger’s algorithm, we extend this set to a Gröbner basis $\{g_1,\ldots ,g_r$, $g_{r+1},\ldots ,g_s\}$ with respect to $>_w$. In particular, $g_{r+1},\ldots ,g_s$ are binomials as well. We can assume that for each $j\in \{r+1,\ldots ,s\}$, the binomial $g_j$ is not an scalar multiple of $g_i$, for all $i\in \{1,\ldots ,r\}$.

We have that $\{(g_1)_t,\ldots ,(g_s)_t\}$ generates $I_{{{\mathscr {A}}}_w}$. We claim that, for $i\in \{1,\ldots ,r\}$, the binomial $(g_i)_t$ is not generated by $\{(g_j)_t\}_{j\ne i}$. Indeed, suppose this is the case for some i. Lemma 4.3 implies that $\{g_j\}_{j\ne i}$ is a generating set of $I_{{\mathscr {A}}}$. By removing redundant elements of this set, we obtain a minimal binomial generating set of $I_{{\mathscr {A}}}$ not containing $g_i$. This contradicts the uniqueness of the minimal binomial generating set of $I_{{\mathscr {A}}}$.

Consider any subset of $\{(g_i)_t\}_{i=1}^s$ that minimally generates $I_{{{\mathscr {A}}}_w}$. By the previous paragraph, such a subset must contain $\{(g_1)_t,\ldots ,(g_r)_t\}$. For each $i\in \{1,\ldots ,r\}$ we write $(g_i)_t=x^{\alpha _i}-x^{\beta _i}t^{\tau _i}, \text{ for } \text{ some } \tau _i\in {\mathbb {N}},$ where $\pi _{{\mathscr {A}}_w}(\alpha _i,0)=\pi _{{\mathscr {A}}_w}(\beta _i,\tau _i)=(b_i,\lambda _i)\in {\text {Betti}}(S_w)$.

By the uniqueness condition on S, each $b_i\in {\text {Betti}}(S)$ is Betti-minimal and $\pi _{{\mathscr {A}}}^{-1}(b_i)=\{\alpha _i,\beta _i\}$ for each $i\in \{1,\ldots ,r\}$ [5, Section 3]. This implies that

$$\begin{aligned} \pi _{{\mathscr {A}}_w}^{-1}(b_i,\lambda _i)=\{(\alpha _i,0),(\beta _i,\tau _i)\}. \end{aligned}$$

Indeed, if $\pi _{{\mathscr {A}}_w}(\gamma ,l)=(b_i,\lambda _i)$ then $\gamma =\alpha _i$ or $\gamma =\beta _i$ implying that $l=0$ or $l=\tau _i$, respectively.

Summarizing, for each $i\in \{1,\ldots ,r\}$ we have $(b_i,\lambda _i)\in {\text {Betti}}(S_w)$ and $|\pi _{{\mathscr {A}}_w}^{-1}(b_i,\lambda _i)|=2$. We conclude that $(b_i,\lambda _i)$ is Betti-minimal [5, Corollary 5]. $\square$

Corollary 4.12

Let S be a uniquely presented affine semigroup. Let $w\in {\mathbb {N}}^n$ be such that some minimal generating set of $I_{{\mathscr {A}}}$ is also a Gröbner basis with respect to some refined order $>_w$. Then all Betti elements of $S_w$ are Betti-minimal. In particular, $I_{{{\mathscr {A}}}_w}$ is also uniquely presented.

Proof

Let $\{g_1,\ldots ,g_r\}\subset I_{{\mathscr {A}}}$ be a minimal generating set that is also a Gröbner basis with respect to $>_w$. Then $\{(g_1)_t,\ldots ,(g_r)_t\}\subset I_{{{\mathscr {A}}}_w}$ is a generating set. In addition, by Lemma 4.3, it is minimal. Thus, every Betti element of $S_w$ is of the form $(b,\lambda )$, for some $\lambda \in {\mathbb {N}}$ and $b\in {\text {Betti}}(S)$. By Theorem 4.11, such Betti elements are Betti-minimal. The last statement of the corollary follows from Remark 4.10. $\square$

Let us look at an example where the conditions of the previous corollary are satisfied.

Example 4.13

Let $S=\langle a,a+1,a+2 \rangle \subset {\mathbb {N}}$, where $a=2q\ge 4$. The only minimal set of binomial generators of $I_{{\mathscr {A}}}$ is $\{y^2-xz,x^{q+1}-z^q\}$ [5, Theorem 15]. Let $w\in {\mathbb {N}}^3$ be such that $2w_2>w_1+w_3$. Then the leading terms of these two binomials with respect to any refined order $>_w$ are relatively prime. Hence, they form a Gröbner basis with respect to $>_w$. By Corollary 4.12, $I_{{{\mathscr {A}}}_w}$ is uniquely presented.

Now we show an example of a uniquely presented affine semigroup such that any of its Gröbner degenerations is also uniquely presented.

As usual, let ${\mathscr {A}}=\{a_1,\ldots ,a_n\}$. The Lawrence ideal of ${\mathscr {A}}$, denoted $I_{\Lambda ({\mathscr {A}})}$, is the ideal of ${\mathbb {K}}[x_1,\ldots ,x_n,y_1,\ldots ,y_n]$ generated by

$$\begin{aligned} \left\{ x^uy^v-x^vy^u|\sum _{i=1}^{n}u_ia_i=\sum _{i=1}^{n}v_ia_i\right\} . \end{aligned}$$

This ideal is studied, for instance, in [14, Chapter 7]. There, Lawrence ideals are used as an auxiliary tool to compute Graver bases. The relevant fact for us is that Lawrence ideals are uniquely presented [12, Corollary 16, Proposition 4].

Corollary 4.14

Any Gröbner degeneration of the Lawrence ideal is uniquely presented.

Proof

It is known that any minimal binomial generating set $\{g_1,\ldots ,g_r\}$ of $I_{\Lambda ({\mathscr {A}})}$ is a reduced Gröbner basis with respect to any order [14, Theorem 7.1]. By Corollary 4.12, any Gröbner degeneration of the Lawrence ideal is uniquely presented. $\square$

5.2 A further example of Gröbner degenerations preserving the uniqueness of a presentation

In this section we present a further example showing that Gröbner degenerations preserve the property of being uniquely presented. We stress that, as opposed to Example 4.13 or Corollary 4.14, the results of this section do not rely on Corollary 4.12.

Proposition 4.15

Let $S=\langle a,a+1,a+2 \rangle$, $a\in {\mathbb {N}}$, $a=2q+2$, $q\ge 1$, and $w=(w_1,w_2,w_3)\in {\mathbb {N}}^3{\setminus }\{(0,0,0)\}$. The following are Gröbner bases of $I_{{\mathscr {A}}}$ for the w-degrevlex order (the leading monomials are listed first):

• 1. Suppose $2w_2\ge w_1+w_3$. Then $G=\{y^2-xz,x^{q+2}-z^{q+1}\}.$

• 2. Suppose $2w_2<w_1+w_3$ and $(q+1)w_3\le (q+2)w_1$.

  • (a) Suppose $(q+2)w_3\le (q+1)w_1+2w_2$. Then $G=\{xz-y^2,x^{q+2}-z^{q+1},x^{q+1}y^2-z^{q+2}\}.$

  • (b) Suppose $(q+1)w_1+2w_2<(q+2)w_3$. Then $G=\{xz-y^2,x^{q+2}-z^{q+1},z^{q+2}-x^{q+1}y^2\}.$

• 3. Suppose $2w_2<w_1+w_3$, $(q+2)w_1<(q+1)w_3$.

  • (a) If $(q+i+3)w_1<2(i+1)w_2+(q-i)w_3$ for all $0\le i\le q$, then $G=\{xz-y^2,z^{q+1}-x^{q+2},y^2z^q-x^{q+3},y^4z^{q-1}-x^{q+4},\ldots ,y^{2(q+1)}-x^{2q+3}\}$.

  • (b) Suppose there exists $n\in {\mathbb {N}}$, $n\le q$ such that $2(n+1)w_2+(q-n)w_3\le (q+n+3)w_1$, and for all $0\le i <n$, $(q+i+3)w_1<2(i+1)w_2+(q-i)w_3$. Then $G=\{xz-y^2,z^{q+1}-x^{q+2},y^2z^q-x^{q+3},y^4z^{q-1}-x^{q+4},\ldots ,y^{2n}z^{q-(n-1)}-x^{q+(n-1)+3},x^{q+n+3}-y^{2(n+1)}z^{q-n}\}.$

Proof

Let $p_1=y^2-xz$ and $p_2=x^{q+2}-z^{q+1}$. A straightforward computation shows that $I_{{\mathscr {A}}}=\langle p_1,p_2 \rangle$. More generally, generators for toric ideals of semigroups generated by intervals can be found in [8, Theorem 8].

• 1. Suppose $2w_2\ge w_1+w_3$. Then the initial monomials of $p_1$ and $p_2$ are relatively prime. Hence, $G=\{p_1,p_2\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$.

• 2. Suppose $2w_2<w_1+w_3$ and $(q+1)w_3\le (q+2)w_1$. The corresponding S-polynomial of $p_1$ and $p_2$ is $S(p_1,p_2)=z^{q+2}-x^{q+1}y^2$. An application of Buchberger’s algorithm shows that $G=\{p_1,p_2,S(p_1,p_2)\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$, whether $z^{q+2}$ or $x^{q+1}y^2$ is the initial monomial of $S(p_1,p_2)$.

• 3. Suppose $2w_2<w_1+w_3$, $(q+2)w_1<(q+1)w_3$. Consider $p_3:=S(p_1,p_2)=x^{q+3}-y^2z^q$.

  If $(q+3)w_1\ge 2w_2+qw_3$ then, by Buchberger’s algorithm, $G=\{p_1,p_2,p_3\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$.

  Suppose $(q+3)w_1< 2w_2+qw_3$. Let $p_4:=S(p_1,p_3)=x^{q+4}-y^4z^{q-1}$. In addition, notice $S(p_2,p_3)=x^{q+3}z-x^{q+2}y^{2}=x^{q+2}p_1$.

  If $(q+4)w_1\ge 4w_2+(q-1)w_3$ then, by Buchberger’s algorithm, $G=\{p_1,p_2,p_3,p_4\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$.

  Suppose $(q+4)w_1< 4w_2+(q-1)w_3$. If $q=1$, then by Buchberger’s algorithm $G=\{p_1,p_2,p_3,p_4\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$. Otherwise $q\ge 2$. Let $p_5=S(p_1,p_4)=x^{q+5}-y^6z^{q-2}$. In addition, notice $S(p_2,p_4)=z^{2}x^{q+4}-y^4x^{q+2}=(x^{q+3}z+x^{q+2}y^{2})p_1$ and $S(p_3,p_4)=zx^{q+4}-y^{2}x^{q+3}=x^{q+3}p_1$.

  If $(q+5)w_1\ge 6w_2+(q-2)w_3$ then, by Buchberger’s algorithm, $G=\{p_1,p_2,p_3,p_4,p_5\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$.

  Suppose $(q+5)w_1< 6w_2+(q-2)w_3$. If $q=2$, then by Buchberger’s algorithm $G=\{p_1,p_2,p_3,p_4,p_5\}$ is a Gröbner basis of $I_{{\mathscr {A}}}$. Otherwise we continue as before and we have two possibilities:

  • i. For all $n\in {\mathbb {N}}$, $n\le q$, $(q+i+3)w_1<2(i+1)w_2+(q-i)w_3$. In this case we obtain the statement 3a of this proposition.

  • ii. There exists $n\in {\mathbb {N}}$, $n<q$ such that $(q+n+3)w_1\ge 2(n+1)w_2+(q-n)w_3$, and for all $0\le i <n$, $(q+i+3)w_1<2(i+1)w_2+(q-i)w_3$. In this case we obtain the statement 3b of this proposition. $\square$

Lemma 4.16

Let $I_{{\mathscr {A}}}$ be a toric ideal and $G_0=\left\{ x^{\alpha _i}-x^{\beta _i}:i\in \{1,\dots ,m\}\right\}$ be a minimal generating set of $I_{{\mathscr {A}}}$. Let $M_S=\{x^{\alpha _i},x^{\beta _i}:i\in \{1,\dots , m\}\}$. Suppose that the ideal $\langle M_S\rangle$ is minimally generated by $M_S$. In addition, suppose that $\pi _{{\mathscr {A}}}(\alpha _i)\ne \pi _{{\mathscr {A}}}(\alpha _j)$, for all $i,j\in \{1,\dots ,m\}$, $i\ne j$. Then $I_{{\mathscr {A}}}$ is uniquely presented and $G_0$ is the unique minimal generating set of binomials, up to scalar multiplication.

Proof

By [2, Proposition 3.1], for any monomial $m\in M_S$ and for any minimal generating set of binomials G of $I_{{\mathscr {A}}}$, there exists $\gamma \in {\mathbb {N}}^n$, such that $m-x^{\gamma }\in G$ or $x^{\gamma }-m\in G$.

Suppose there exists $l\in \{1,\dots ,m\}$ such that $x^{\alpha _l}-x^{\beta _l}\notin G$ and $x^{\beta _l}-x^{\alpha _l}\notin G$, for some minimal generating set of binomials G of $I_{{\mathscr {A}}}$. As $\pi _{{\mathscr {A}}}(\alpha _l)=\pi _{A}(\beta _l)$, by Remark 4.2$\beta _{1,\pi _{{\mathscr {A}}}(\alpha _l)}({\mathbb {K}}[x]/I_{{\mathscr {A}}})\ge 2$.

By hypothesis, $\pi _{{\mathscr {A}}}(\alpha _i)\ne \pi _{{\mathscr {A}}}(\alpha _j)$ for all $i\ne j$, and $G_0$ is a minimal generating set of $I_{{\mathscr {A}}}$. By Remark 4.2$\beta _{1,\pi _{{\mathscr {A}}}(\alpha _i)}({\mathbb {K}}[x]/I_{{\mathscr {A}}})=1$, for each $i\in \{1,\ldots ,m\}$. This is a contradiction. Thus $G_0$ is the unique minimal generating set of binomials, up to scalar multiplication. $\square$

Proposition 4.17

Let $S=\langle a,a+1,a+2 \rangle$, $a\in {\mathbb {N}}$, $2<a$ and $a=2q+2$. Then, for any $w\in {\mathbb {N}}^3{\setminus }\{(0,0,0)\}$, the ideal $I_{{{\mathscr {A}}}_w}$ is uniquely presented.

Proof

We divide the proof based on the three cases of Proposition 4.15. The ideal $I_{{\mathscr {A}}}$ is uniquely presented by [5, Theorem 15]. In addition, its unique minimal generating set is $\{y^2-xz,x^{q+2}-z^{q+1}\}$ (see the proof of Proposition 4.15). Hence, case 1 follows from Corollary 4.12.

Case 2(a). From the Gröbner basis of $I_{{\mathscr {A}}}$ we can produce a generating set of $I_{{{\mathscr {A}}}_w}$. The monomials appearing in this generating set are:

$$\begin{aligned} \big \{&xz,y^2t^{w_1+w_3-2w_2},x^{q+2},z^{q+1}t^{(q+2)w_1-(q+1)w_3},\\&x^{q+1}y^2,z^{q+2}t^{(q+1)w_1+2w_2-(q+2)w_3}\big \}. \end{aligned}$$

By the inequalities satisfied by w in this case, it follows that this set is a minimal generating set of the ideal it generates. In addition, the ${\mathscr {A}}_w$-degrees of these monomials are:

$$\begin{aligned} \{(2a+2,w_1+w_3),((q+2)a,(q+2)w_1),((q+2)(a+2),(q+1)w_1+2w_2)\}. \end{aligned}$$

Since these degrees are different, we conclude that $I_{{{\mathscr {A}}}_w}$ is uniquely presented by Lemma 4.16.

Case 2(b). As in the previous case, we obtain the following set of monomials:

$$\begin{aligned} \big \{&xz,y^2t^{w_1+w_3-2w_2},x^{q+2},z^{q+1}t^{(q+2)w_1-(q+1)w_3},\\&z^{q+2},x^{q+1}y^2t^{(q+2)w_3-(q+1)w_1-2w_2}\big \}. \end{aligned}$$

If $(q+2)w_1>(q+1)w_3$, then proceed exactly as in case 2(a). Suppose that $(q+2)w_1=(q+1)w_3$. Then $(q+2)w_3-(q+1)w_1-2w_2=w_1+w_3-2w_2=:\delta .$ Thus, $z^{q+2}-x^{q+1}y^2t^{\delta }=x^{q+1}(xz-y^2t^{\delta })-z(x^{q+2}-z^{q+1}).$ This implies that $I_{{{\mathscr {A}}}_w}$ is generated by $\{xz-y^2t^{\delta },x^{q+2}-z^{q+1}\}$. Hence, $I_{{{\mathscr {A}}}_w}$ is uniquely presented by Lemma 4.16.

Case 3(a). As before, we obtain the following set of monomials:

$$\begin{aligned} \big \{&xz,y^2t^{w_1+w_3-2w_2},z^{q+1},x^{q+2}t^{(q+1)w_3-(q+2)w_1}\big \}\\&\cup \big \{y^{2(i+1)}z^{q-i},x^{q+i+3}t^{2(i+1)w_2+(q-i)w_3-(q+i+3)w_1}\big \}_{0 \le i \le q}. \end{aligned}$$

By the inequalities satisfied by w in this case, it follows that this set is a minimal generating set of the ideal it generates. It remains to prove that the ${\mathscr {A}}_w$-degrees are all different. It is enough to show this for the first entry of the ${\mathscr {A}}_w$-degrees. Indeed, the first entries are

$$\begin{aligned} \{2a+2,(q+2)a,\ldots ,(q+i+3)a,\ldots ,(2q+3)a\}. \end{aligned}$$

We conclude that $I_{{{\mathscr {A}}}_w}$ is uniquely presented by Lemma 4.16.

Case 3(b). Proceed exactly as in case 3(a). $\square$

Remark 4.18

By computing Gröbner bases and using Lemma 4.16, we verified that Proposition 4.17 also holds for other families of uniquely presented numerical semigroups generated by intervals. Moreover, we used the same method to study this property for other families of numerical semigroups. Our computations give enough evidence to conjecture that the uniqueness of a presentation of a toric ideal is preserved under Gröbner degenerations.

6 Möbius functions

In this final section we study Möbius functions of affine semigroups. Several authors have provided explicit formulas for Möbius functions of some families of semigroups. In particular, the case of semigroups with a unique Betti element was studied in [1]. As a final application of Theorem 1.2, we present some relations among the Möbius functions of S and $S_w$, in the case where both S and $S_w$ have a unique Betti element.

Let $S\subset {\mathbb {Z}}^d$ be a pointed affine semigroup. As in previous sections, consider the following partial order: for $x,y\in {\mathbb {Z}}^d$, $x <_S y$ if $y-x\in S$. An interval on ${\mathbb {Z}}^d$ with respect to $<_S$ is defined as $[x,y]_{{\mathbb {Z}}^d}:=\{z\in {\mathbb {Z}}^d\mid x\le _{S}z\le _{S}y\}.$ Denote as $c_l(x,y)$ the cardinality of

$$\begin{aligned} \{\{x<_{S}z_1<_{S}z_2<_{S}\cdots <_{S}z_l=y\}\subset [x,y]_{{\mathbb {Z}}^d}\}. \end{aligned}$$

The Möbius function of S, denoted $\mu _S$, is defined as

$$\begin{aligned} \mu _S:{\mathbb {Z}}^d\rightarrow {\mathbb {Z}}, y\mapsto \sum _{l\ge 0}(-1)^lc_l(0,y). \end{aligned}$$

This sum is always finite [1, Section 2]. Notice that if $y\notin S$ then $\mu _S(y)=0$ (since, in this case, $c_l(0,y)=0$ for all $l\ge 0$). Thus, we restrict the domain of $\mu _S$ to S.

The following formula is the starting point of our discussion.

Theorem 5.1

([1, Theorem 4.1]) Let $S=\langle a_1,\ldots ,a_n\rangle \subset {\mathbb {Z}}^{d}$ be a pointed affine semigroup. Suppose that ${\text {Betti}}(S)=\{b\}$. Then,

$$\begin{aligned} \mu _S(z)=\sum _{j=1}^t(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) , \end{aligned}$$

where $\{A_1,\ldots ,A_t\}=\{A\subset \{1,\ldots ,n\}\mid \exists k_{A}\in {\mathbb {N}} \text { such } \text { that } z=\sum _{i\in A}a_i+k_Ab\}$.

We introduce some notation that we use throughout this section.

Let $w\in {\mathbb {N}}^n$ and $S_w=\langle a'_1,\ldots ,a'_n,a'_{n+1}\rangle \subset {\mathbb {Z}}^{d+1}$, where $a'_i=(a_i,w_i)$ for $i\in \{1,\ldots ,n\}$ and $a'_{n+1}=({\underline{0}},1)$. Assume that ${\text {Betti}}(S)=\{b\}$ and ${\text {Betti}}(S_w)=\{(b,d_w)\}$. For $z\in S$ and $(z,\lambda )\in S_w$, denote:

• ${\mathfrak {a}}_z:=\{A\subset \{1,\ldots ,n\}\mid \exists k_{A}\in {\mathbb {N}} \text { such } \text { that } z=\sum _{i\in A}a_i+k_Ab\}$.

• ${\mathfrak {b}}_{(z,\lambda )}:=\{B\subset \{1,\ldots ,n+1\}\mid \exists k_{B}\in {\mathbb {N}} \text { such } \text { that } (z,\lambda )=\sum _{i\in B}a'_i+k_B(b,d_w)\}$.

Lemma 5.2

Let $z\in S$ and $(z,\lambda )\in S_w$. Then,

• (i) ${\mathfrak {b}}_{(z,\lambda )}':=\{B{\setminus }\{n+1\}\mid B\in {\mathfrak {b}}_{(z,\lambda )}\}\subset {\mathfrak {a}}_z$. In addition, if $B\setminus \{n+1\}=A$, for some $A\in {\mathfrak {a}}_z$, then $k_{B}=k_A$.

• (ii) If $B\ne C$ in ${\mathfrak {b}}_{(z,\lambda )}$ then $B{\setminus }\{n+1\}\ne C{\setminus }\{n+1\}$. In particular, $|{\mathfrak {b}}_{(z,\lambda )}|=|{\mathfrak {b}}_{(z,\lambda )}'|\le |{\mathfrak {a}}_z|$.

Proof

Let $B\in {\mathfrak {b}}_{(z,\lambda )}$. There exists $k_B\in {\mathbb {N}}$ such that $(z,\lambda )=\sum _{i\in B}a'_i+k_B(b,d_w)$. In particular,

$$\begin{aligned} z=\sum _{i\in B}a_i+k_Bb=\sum _{i\in B{\setminus }\{n+1\}}a_i+k_Bb. \end{aligned}$$

Thus, $B\setminus \{n+1\}\in {\mathfrak {a}}_z$. This shows the first part of (i). For the same B, let $A\in {\mathfrak {a}}_z$ be such that $A=B{\setminus }\{n+1\}$. Then $\sum _{i\in B{\setminus }\{n+1\}}a_i+k_Bb=z=\sum _{i\in A}a_i+k_Ab$. Hence, $k_A=k_B$.

Now we prove (ii). The result is clear if $n+1$ is contained in B and C or if it is not contained in either. Thus, we can assume that $n+1\in B$ and $n+1\notin C$. Suppose that $B\setminus \{n+1\}=C{\setminus }\{n+1\}$. In particular, there exist $k_B,k_C\in {\mathbb {N}}$ such that $\sum _{i\in B{\setminus }\{n+1\}}a_i+k_Bb=z=\sum _{i\in C{\setminus }\{n+1\}}a_i+k_Cb$. Hence $k_B=k_C$. This implies

$$\begin{aligned} \sum _{i\in B}a_i'=(z,\lambda )-k_B(b,d_w)=(z,\lambda )-k_C(b,d_w)=\sum _{i\in C}a_i'. \end{aligned}$$

In particular, $\sum _{i\in B}w_i=\sum _{i\in C}w_i$. This is a contradiction since, by the assumption on C,

$$\begin{aligned} \sum _{i\in C}w_i=\sum _{i\in C\setminus \{n+1\}}w_i<1+\sum _{i\in C\setminus \{n+1\}}w_i=1+\sum _{i\in B\setminus \{n+1\}}w_i=\sum _{i\in B}w_i. \square \end{aligned}$$

Lemma 5.3

Let $z\in S$. Suppose that ${\mathfrak {a}}_z=\{A_1,\ldots ,A_r\}$, $r\ge 1$. Let $l_j=\sum _{i\in A_j}w_i+k_{A_j}d_w$, for each $j\in \{1,\ldots ,r\}$. Let $\lambda \in {\mathbb {N}}$ be such that $(z,\lambda )\in S_w$. Then,

$$\begin{aligned} {\mathfrak {b}}_{(z,\lambda )}\ne \emptyset \Longleftrightarrow \lambda =l_j \text{ or } \lambda =l_j+1 \text{ for } \text{ some } j\in \{1,\ldots ,r\}. \end{aligned}$$

Proof

Let $B\in {\mathfrak {b}}_{(z,\lambda )}$. We have two cases:

• $n+1\notin B$. By (i) of Lemma 5.2, $B=A_j$ for some j. Hence, $(z,\lambda )=\sum _{i\in B}a_i'+k_B(b,d_w)=\sum _{i\in A_j}a_i'+k_B(b,d_w)$. We also know that $k_B=k_{A_j}$. We conclude that $\lambda =\sum _{i \in B}w_i+k_Bd_w=\sum _{i\in A_j}w_i+k_{A_j}d_w=l_j$.

• $n+1\in B$. Like in the previous item, $B{\setminus }\{n+1\}=A_j$ and $k_B=k_{A_j}$, for some j. It follows that $\lambda =\sum _{i\in B}w_i+k_Bd_w=(1+\sum _{i\in A_j}w_i)+k_{A_j}d_w=1+l_j$.

Now suppose that $\lambda =l_j$ (resp. $\lambda =l_j+1$) for some $j\in \{1,\ldots ,r\}$. Then, by definition, $A_j\in {\mathfrak {b}}_{(z,\lambda )}$ (resp. $A_j\cup \{n+1\}\in {\mathfrak {b}}_{(z,\lambda )}$). $\square$

We are now ready to prove the main result of this section.

Theorem 5.4

Let $S\subset {\mathbb {Z}}^{d}$ be a pointed affine semigroup. Let $w\in {\mathbb {N}}^n$. Assume that ${\text {Betti}}(S)=\{b\}$ and ${\text {Betti}}(S_w)=\{(b,d_w)\}$. Then the Möbius function of $S_w$ can be computed in terms of data of S. More precisely, for $(z,\lambda )\in S_w$, $\mu _{S_w}(z,\lambda )=0$ whenever ${\mathfrak {a}}_z=\emptyset$ or $\lambda \notin \{l_j,l_j+1\}_{j=1}^r$. If ${\mathfrak {a}}_z=\{A_1,\ldots ,A_r\}$ and $\lambda \in \{l_j,l_j+1\}_{j=1}^r$, then

$$\begin{aligned} \mu _{S_w}(z,\lambda )=\sum _{\lambda =l_j}(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) -\sum _{\lambda =l_j+1}(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) . \end{aligned}$$

Proof

The fact that S is pointed implies that $S_w$ is pointed. By Theorem 5.1:

$$\begin{aligned} \mu _{S_w}(z,\lambda )= \left\{ \begin{array}{ll} 0,&{}{}\quad {\mathfrak {b}}_{(z,\lambda )}=\emptyset , \\ \sum _{B\in {\mathfrak {b}}_{(z,\lambda )}}(-1)^{|B|}\left( {\begin{array}{c}k_B+n-d-1\\ k_B\end{array}}\right) , &{}{}\quad {\mathfrak {b}}_{(z,\lambda )}\ne \emptyset .\\ \end{array} \right. \end{aligned}$$

(*)

Lemmas 5.2 (ii) and 5.3, together with (*) imply that $\mu _{S_w}(z,\lambda )=0$ whenever ${\mathfrak {a}}_z=\emptyset$ or $\lambda \notin \{l_j,l_j+1\}_{j=1}^r$ (in both cases ${\mathfrak {b}}_{(z,\lambda )}=\emptyset$).

Let ${\mathfrak {a}}_z=\{A_1,\ldots ,A_r\}$, $r\ge 1$. Let $j_0\in \{1,\ldots ,r\}$ be such that $\lambda =l_{j_0}$ or $\lambda =l_{j_0}+1$. Let ${\mathfrak {a}}_z^0:=\{A_j|\lambda =l_j\}$ and ${\mathfrak {a}}_z^1:=\{A_j|\lambda =l_j+1\}$. The proof of Lemma 5.3 shows that ${\mathfrak {a}}_z^0=\{B\in {\mathfrak {b}}_{(z,\lambda )}\mid n+1\notin B\}$ and ${\mathfrak {a}}_z^1=\{B{\setminus }\{n+1\}\mid B\in {\mathfrak {b}}_{(z,\lambda )}, n+1\in B\}$. Using these facts and Lemma 5.2, (*) implies:

$$\begin{aligned} \mu _{S_w}(z,\lambda )&=\sum _{\begin{array}{c} B\in {\mathfrak {b}}_{(z,\lambda )}\\ n+1\notin B \end{array}}(-1)^{|B|}\left( {\begin{array}{c}k_B+n-d-1\\ k_B\end{array}}\right) +\sum _{\begin{array}{c} B\in {\mathfrak {b}}_{(z,\lambda )}\\ n+1\in B \end{array}}(-1)^{|B|}\left( {\begin{array}{c}k_B+n-d-1\\ k_B\end{array}}\right) \\&=\sum _{A_j\in {\mathfrak {a}}_z^0}(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) +\sum _{A_j\in {\mathfrak {a}}_z^1}(-1)^{|A_j|+1}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) \\&=\sum _{A_j\in {\mathfrak {a}}_z^0}(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) -\sum _{A_j\in {\mathfrak {a}}_z^1}(-1)^{|A_j|}\left( {\begin{array}{c}k_{A_j}+n-d-1\\ k_{A_j}\end{array}}\right) . \end{aligned}$$

$\square$

Example 5.5

Let S and w be a numerical semigroup and a weight vector, respectively, such that $|{\text {Betti}}(S)|=|{\text {Betti}}(S_w)|=1$ (see Example 4.8). It is known that $|{\mathfrak {a}}_z|\le 1$ for all $z\in S$ [1, Proof of Corollary 4.2]. By Theorem 5.1,

$$\begin{aligned}\mu _{S}(z)= \left\{ \begin{array}{ll} 0,&{}\quad {\mathfrak {a}}_z=\emptyset ,\\ (-1)^{|A|}\left( {\begin{array}{c}k_{A}+n-2\\ k_{A}\end{array}}\right) , &{}\quad {\mathfrak {a}}_z=\{A\}.\\ \end{array} \right. \end{aligned}$$

Comparing this formula with Theorem 5.4 we obtain

$$\begin{aligned}\mu _{S_w}(z,\lambda )= \left\{ \begin{array}{ll} 0,&{}\quad {\mathfrak {a}}_z=\emptyset ,\\ \mu _S(z), &{}\quad {\mathfrak {a}}_z=\{A\},\lambda =l,\\ -\mu _S(z), &{}\quad {\mathfrak {a}}_z=\{A\},\lambda =l+1,\\ 0, &{}\quad {\mathfrak {a}}_z=\{A\},\lambda \notin \{l,l+1\}. \end{array} \right. \end{aligned}$$

Remark 5.6

The formulas of the previous example are also valid for any semigroup S satisfying the conditions of Theorem 5.4 with the extra assumption $|{\mathfrak {a}}_z|\le 1$ for all $z\in S$.