Realizable sets of catenary degrees of numerical monoids

TL;DR

This paper characterizes which finite sets of nonnegative integers can be realized as the catenary degrees of numerical monoids, showing that almost all such sets are achievable by these monoids.

Contribution

It provides a classification of realizable catenary degree sets for numerical monoids, establishing that nearly all finite subsets are attainable.

Findings

Most finite subsets of nonnegative integers are realizable as catenary degree sets.

The paper identifies one exceptional subset that cannot be realized.

It advances understanding of factorization invariants in numerical monoids.

Abstract

The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a numerical monoid (i.e., a co-finite, additive submonoid of $\mathbb Z_{\ge 0}$). In particular, we show that, with one exception, every finite subset of $\mathbb Z_{\ge 0}$ that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

Figures

Equations

$${\mathsf{Z}}_S(n)=\{(a_1,\dots ,a_k):n=a_1{n}_1+\cdots +a_k{n}_k\}\subset {\mathbb{Z}}_{\ge 0}^k$$

$$\gcd (\mathbf{a},{\mathbf{a}}^{\prime })=(\min (a_1,b_1),\dots ,\min (a_k,b_k))\in {\mathbb{Z}}_{\ge 0}^k,$$

$$\mathsf{d}(\mathbf{a},{\mathbf{a}}^{\prime })=\max (|\mathbf{a}-\gcd (\mathbf{a},{\mathbf{a}}^{\prime })|,|{\mathbf{a}}^{\prime }-\gcd (\mathbf{a},{\mathbf{a}}^{\prime })|).$$

$$\mathrm{Betti}(S)=\{b\in S:{\nabla }_b\text{ is disconnected}\}$$

$$\mathsf{c}(S)\le \max \{\mathsf{c}(S_1),\mathsf{c}(S_2),{\mathsf{c}}_S(d)\}$$

$$\mathsf{c}_{T}(n)=\left\{\begin{array}[]{ll}\mathsf{c}_{S}(n/c)&n-cb\notin T\\ c&n-cb\in T\end{array}\right.$$

$$\mathsf{c}(T)=\max \{{\mathsf{c}}_T(m):m\in \mathrm{Betti}(T)\}={\mathsf{c}}_T(cb)=c,$$

$$a_{k+1}b+c(a_1{n}_1+\cdots +a_k{n}_k)=c(a_1^{\prime }{n}_1+\cdots +a_k^{\prime }{n}_k)$$

$$\mathsf{d}(\mathbf{a},{\mathbf{a}}^{\prime })\ge a_{k+1}\ge c,$$

Full text

1. Introduction

Nonunique factorization theory aims to classify and quantify the failure of elements of a cancellative commutative monoid $M$ to factor uniquely into irreducibles [9]. This is often achieved using arithmetic quantities called factoriation invariants. We consider here the catenary degree invariant (Definition 2.3), which is a non-negative integer $\mathsf{c}(m)$ measuring the distance between the irreducible factorizations of an element $m\in M$.

Many problems in factorization theory involve characterizing which possible values an invariant can take, given some minimal assumptions on the underlying monoid $M$. Solutions to such problems often take the form of a “realization” result in which a family of monoids achieving all possible values is identified. One such problem of recent interest concerns the possible values of the delta set invariant $\Delta(M)$, comprised of successive differences in lengths of factorizations of elements. It can be easily shown that $\min\Delta(M)=\gcd\Delta(M)$ for any atomic monoid [9], but no further restrictions were known. The so-called “delta set realization problem” [3, 6, 7] was recently solved by Geroldinger and Schmidt [10] by identifying a family of finitely generated Krull monoids whose delta sets achieve every finite set $D\subset\mathbb{Z}_{\geq 1}$ satisfying $\min(D)=\gcd(D)$.

In a similar vein, the catenary degree realization problem [11, Problem 4.1] asks which finite sets can occur as the set $\mathsf{C}(M)$ of catenary degrees achieved by elements of some atomic monoid $M$. A recent paper of Fan and Geroldinger [5] proves that, with minimal assumptions, the product $M=M_{1}\times M_{2}$ of two monoids $M_{1}$ and $M_{2}$ has set of catenary degrees $\mathsf{C}(M)=\mathsf{C}(M_{1})\cup\mathsf{C}(M_{2})$. While this does provide a complete solution to the catenary degree realization problem, the brevity of the proof raises the question: is such a result is possible without appealing to Cartesian products?

In this paper, we prove that with only a single exception, any finite set that can occur as the set of catenary degrees of some atomic monoid occurs as the set of catenary degrees of a numerical monoid (Theorem 4.2). Since numerical monoids are submonoids of a reduced rank one free monoid (namely, $\mathbb{Z}_{\geq 0}$), they never contain a Cartesian product of two nontrivial monoids.

Our method of constructing numerical monoids with prescribed sets of catenary degrees involves carefully gluing smaller numerical monoids (Definition 2.5) in such a way as to retain complete control of the catenary degree of every element (Theorem 3.3). The relationship between the catenary degree and gluings has been studied before [1], though noteable subtleties arise; Remark 3.2 clarifies some ambiguity in [1], and is another primary contribution of this manuscript.

2. Background

In what follows, $\mathbb{N}=\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.

Definition 2.1**.**

A numerical monoid $S$ is an additive submonoid of $\mathbb{N}$ with finite complement. When we write $S=\langle n_{1},\ldots,n_{k}\rangle$, we assume $n_{1}<\cdots<n_{k}$, and the chosen generators $n_{1},\ldots,n_{k}$ are minimal with respect to set-theoretic inclusion. These minimal generators are called irreducible elements or atoms.

Definition 2.2**.**

Fix $n\in S=\langle n_{1},\ldots,n_{k}\rangle$. A factorization of $n$ is an expression $n=u_{1}+\cdots+u_{r}$ of $n$ as a sum of atoms $u_{1},\ldots,u_{r}$ of $S$. Write

[TABLE]

for the set of factorizations of $n\in S$. Given ${\mathbf{a}}\in\mathsf{Z}_{S}(n)$, we denote by $|{\mathbf{a}}|$ the number of irreducibles in the factorization ${\mathbf{a}}$, that is, $|{\mathbf{a}}|=a_{1}+\cdots+a_{k}$.

We are now ready to define the catenary degree.

Definition 2.3**.**

Fix an element $n\in S=\langle n_{1},\ldots,n_{k}\rangle$ and factorizations ${\mathbf{a}},{\mathbf{a}}^{\prime}\in\mathsf{Z}_{S}(n)$. The greatest common divisor of ${\mathbf{a}}$ and ${\mathbf{a}}^{\prime}$ is given by

[TABLE]

and the distance between ${\mathbf{a}}$ and ${\mathbf{a}}^{\prime}$ (or the weight of $({\mathbf{a}},{\mathbf{a}}^{\prime})$) is given by

[TABLE]

Given $N\geq 0$, an $N$-chain from ${\mathbf{a}}$ to ${\mathbf{a}}^{\prime}$ is a sequence ${\mathbf{a}}={\mathbf{a}}_{1},{\mathbf{a}}_{2},\ldots,{\mathbf{a}}_{k}={\mathbf{a}}^{\prime}\in\mathsf{Z}_{S}(n)$ of factorizations such that $\mathsf{d}({\mathbf{a}}_{i-1},{\mathbf{a}}_{i})\leq N$ for all $i\leq k$. The catenary degree of $n$, denoted $\mathsf{c}_{S}(n)$, is the smallest $N\geq 0$ such that there exists an $N$-chain between any two factorizations of $n$. The set of catenary degrees of $S$ is the set $\mathsf{C}(S)=\{\mathsf{c}(m):m\in S\}$, and the catenary degree of $S$ is the supremum $\mathsf{c}(S)=\sup\mathsf{C}(S)$.

Definition 2.4**.**

Fix a finitely generated monoid $S$. For each nonzero $n\in S$, consider the graph $\nabla_{n}$ with vertex set $\mathsf{Z}(n)$ in which two vertices ${\mathbf{a}},{\mathbf{a}}^{\prime}\in\mathsf{Z}(n)$ share an edge if $\gcd({\mathbf{a}},{\mathbf{a}}^{\prime})\neq 0$. If $\nabla_{n}$ is not connected, then $n$ is called a Betti element of $S$. We write

[TABLE]

for the set of Betti elements of $S$.

We conclude this section by defining the gluing operation for numerical monoids. For a more thorough introduction to gluing, see [12, Chapter 8].

Definition 2.5**.**

Fix numerical monoids $S_{1}$ and $S_{2}$, and positive integers $d_{1}$ and $d_{2}$. The monoid $S=d_{1}S_{1}+d_{2}S_{2}$ is a gluing of $S_{1}$ and $S_{2}$ by $d$ if $d=\operatorname{lcm}(d_{1},d_{2})\in S_{1}\cap S_{2}$.

3. Catenary degree of numerical monoid gluings

Theorem 3.3 provides the primary technical result used in Theorem 4.2 to construct a numerical monoid with a prescribed set of catenary degrees, and is closely related to Theorem 3.1, an ambiguously worded (Remark 3.2) result appearing in [1].

Theorem 3.1** ([1, Corollary 4]).**

If $S$ is a gluing of $S_{1}$ and $S_{2}$ by $d$, then

[TABLE]

and equality holds if $\mathsf{c}(S_{1})$ and $\mathsf{c}(S_{2})$ are each computed by taking the maximum of $\mathsf{c}_{S}(n)$ over elements of $S_{1}$ and $S_{2}$, respectively.

Remark 3.2**.**

The wording of Theorem 3.1 differs significantly from the original statement appearing in [1]. In particular, the original wording was ambiguous, and it was unclear that $\mathsf{c}(S_{1})$ and $\mathsf{c}(S_{2})$ should be computed in the (nonstandard) way specified above. If these values are computed in the usual way, then the inequality in Theorem 3.1 can indeed be strict (see, for instance, Examples 3.4 and 3.5); the unambiguous statement given in Theorem 3.1 is the result of discussions with the second author of [1] that took place after such a monoid $S$ was found.

Remark 3.2 is an example of one of the many subtleties one encounters when working with the catenary degree. It remains an interesting question to determine for which $S$ equality is achieved in Theorem 3.1; Theorem 3.3 gives one such setting.

Theorem 3.3**.**

Suppose $S=\langle n_{1},\ldots,n_{k}\rangle$ is a numerical monoid. Fix $c>\mathsf{c}(S)$, $b\in S$ nonzero with $\gcd(b,c)=1$, and let $T=\langle cn_{1},\ldots,cn_{k},b\rangle$. Then

[TABLE]

for any $n\in T$. In particular, $\mathsf{c}(T)=c$.

Proof.

Since $b\in S$ and $\gcd(b,c)=1$, $T$ is a gluing of $S$ and $\mathbb{N}$ by $cb$ by [12, Theorem 8.2], meaning $\operatorname{Betti}(T)=c\operatorname{Betti}(S)\cup\{cb\}$ by [1, Corollary 2]. As such, if $n-cb\notin T$, then $a_{k+1}$ is constant among all factorizations ${\mathbf{a}}\in\mathsf{Z}_{T}(n)$, and thus $\mathsf{c}_{T}(n)=\mathsf{c}_{S}(n/c)$.

Now, suppose $n-cb\in T$. By [2, Theorem 3.1],

[TABLE]

so it suffices to prove that $\mathsf{c}_{T}(n)\geq c$. To this end, we will show that if ${\mathbf{a}},{\mathbf{a}}^{\prime}\in\mathsf{Z}_{T}(n)$ with $a_{k+1}\neq a^{\prime}_{k+1}$, then $\mathsf{d}({\mathbf{a}},{\mathbf{a}}^{\prime})\geq c$. Without loss of generatity, it suffices to assume $\gcd({\mathbf{a}},{\mathbf{a}}^{\prime})=0$ and $a_{k+1}>0$. Since

[TABLE]

and $\gcd(b,c)=1$, we must have $c\mid a_{k+1}$. This implies

[TABLE]

which completes the proof. ∎

We conclude this section with several examples demonstrating that no hypotheses in Theorem 3.3 can be safely omitted.

Example 3.4**.**

The numerical monoid $T=\langle 6,9,10,14\rangle=2S+9\mathbb{N}$ is a gluing, where $S=\langle 3,5,7\rangle$. In this case, $\mathsf{c}(T)=3$, even though $\mathsf{c}(S)=4$.

Example 3.5**.**

The numerical monoid $T=\langle 15,25,35,18,27\rangle=5S_{1}+9S_{2}$ is a gluing, where $S_{1}=\langle 3,5,7\rangle$ and $S_{2}=\langle 2,3\rangle$. In this case, $\mathsf{c}(T)=3$, even though $\mathsf{c}(S_{1})=4$ and both scaling factors $5$ and $9$ are strictly larger than both $\mathsf{c}(S_{1})$ and $\mathsf{c}(S_{2})$.

4. Realizable sets of catenary degrees

In this section, we apply Theorem 3.3 to characterize which finite subsets of $\mathbb{Z}_{\geq 0}$ are realized as the set of catenary degrees of a numerical monoid, thus providing an alternative answer to [11, Problem 4.1] from that appearing in [5].

Example 4.1**.**

Theorem 4.2 implies $S=\langle 90,91,96,120,150\rangle$ has set of catenary degrees $\{0,2,3,5,6\}$. Figure 1 depicts the catenary degrees of the elements of $S$. The Betti element $480\in\operatorname{Betti}(S)$ has catenary degree $\mathsf{c}(480)=5$, and the elements $n\in S$ with catenary degree $\mathsf{c}(n)\geq 5$ are precisely those divisible by $480$ (in the monoid-theoretic sense, i.e. $n-480\in S$).

Theorem 4.2**.**

Fix a finite set $C\subset\mathbb{Z}_{\geq 0}$. Then there exists a numerical monoid with set of catenary degrees $C$ if and only if (i) $0\in C$, (ii) $1\notin C$, and (iii) $c=\max C\geq 3$.

Proof.

For the backward direction, (i) and (ii) both clearly follow from Definition 2.3, and (iii) follows from the fact that no numerical monoid is half-factorial [8].

For the converse direction, fix a finite set $C$ satisfying conditions (i), (ii) and (iii). First, if $C=\{0,c\}$ or $C=\{0,2,c\}$, then $C$ is the set of catenary degrees of a monoid by [11, Remark 4.2] and [11, Theorem 4.3], respectively. In all other cases, $C^{\prime}=C\cap[0,c)$ satisfies conditions (i), (ii) and (iii) above, so we can inductively assume that $C^{\prime}$ is the set of catenary degrees of some numerical monoid $S=\langle n_{1},\ldots,n_{k}\rangle$. Choosing $b\in S$ such that $b>n_{k}$ and $\gcd(b,c)=1$ and applying Theorem 3.3 yields a numerical monoid $T$ with set of catenary degrees $C$, as desired. ∎

Remark 4.3**.**

In the last sentence of the proof of Theorem 4.2, the choice $b>n_{k}$ simply ensures the monoid $T$ is minimally generated.

Example 4.4**.**

Let $C=\{0,2,7,20,26,57\}$. Following the proof of Theorem 4.2, we begin with the monoid $S=\langle 3,8,13\rangle$, which has set of catenary degrees $\{0,2,7\}$. Subsequent choices of $b$ outlined in the following table yields a numerical monoid with set of catenary degrees $C$.

[TABLE]

Note that each choice of $b$ lies in the monoid $S$ from the previous row, as required by Theorem 4.2 (for instance, $1301=11\cdot 51+7\cdot 60+2\cdot 160$). This can be readily checked with the GAP package numericalsgps [4].