On the structure of length sets with maximal elasticity

TL;DR

This paper investigates the structure of length sets in Krull monoids with finite class groups, showing that elements with maximal elasticity typically have length sets forming intervals, revealing insights into their factorization properties.

Contribution

It characterizes the structure of length sets with maximal elasticity in Krull monoids, demonstrating they are generally intervals, which advances understanding of factorization theory.

Findings

Length sets with maximal elasticity are intervals.

The structure of these length sets is characterized.

Insights into factorization properties of Krull monoids.

Abstract

Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor. Then every non-unit $a \in H$ has a factorization into atoms, say $a=u_1 \cdot\ldots \cdot u_k$ where $k$ is the factorization length and $u_1, \ldots, u_k$ are atoms of $H$. The set $\mathsf L (a)$ of all possible factorizaton lengths is the length set of $a$, and $\rho (H) = \sup \{ \max \mathsf L (a)/\min \mathsf L (a) \colon a \in H \}$ is the elasticity of $H$. We study the structure of length sets of elements with maximal elasticity and show that, in general, these length sets are intervals.

Equations

$$|U_1|=|U_2|=\mathsf{D}(G),g_1{g}_2|U_1,\text{and }(g_1+g_2)|U_2.$$

$$a=\prod _{p\in P}{p}^{{\mathsf{v}}_p(a)}$$

$$2+||z|-|z^{\prime }||\le \mathsf{d}(z,z^{\prime }).$$

$$2+\sup \Delta (\mathsf{L}(a))\le \mathsf{c}(a),$$

$$\mathsf{c}(H)=\text{sup}\{\mathsf{c}(a):a\in H\}\in {\mathbb{N}}_0\cup \{\infty \}.$$

$$\mathcal{C}(H)=\mathsf{q}(F)/\mathsf{q}(H_{\text{red}}),$$

$$\mathcal{L}(G_0):=\mathcal{L}(\mathcal{B}(G_0)),\Delta (G_0):=\Delta (\mathcal{B}((G_0)),\rho (G_0):=\rho (\mathcal{B}(G_0))\text{ and }{\rho }_k(G_0):={\rho }_k(\mathcal{B}(G_0)).$$

$$\mathsf{D}(G)=\max \{|U|:U\in \mathcal{A}(G)\}\in \mathbb{N}.$$

$$\mathcal{L}(H)=\mathcal{L}(B),\text{whence}\rho (H)=\rho (B)\text{and}\Delta (H)=\Delta (B).$$

$$L=y+(L^{\prime }\cup L^{\ast }\cup L^{\prime \prime })\subset y+\mathcal{D}+d\mathbb{Z},\text{where y∈Z},$$

$$\min \mathsf{L}(ab)\le \min \mathsf{L}(a)+\min \mathsf{L}(b)\le \max \mathsf{L}(a)+\max \mathsf{L}(b)\le \max \mathsf{L}(ab)$$

$$\rho (H)\ge \rho (\mathsf{L}(ab))\ge \frac{\max \mathsf{L}(a)+\max \mathsf{L}(b)}{\min \mathsf{L}(a)+\min \mathsf{L}(b)}\ge \text{min}\{\frac{\max \mathsf{L}(a)}{\min \mathsf{L}(a)},\frac{\max \mathsf{L}(b)}{\min \mathsf{L}(b)}\}=\rho (H).$$

$$\frac{\mathsf{D}(G)}{2}=\frac{\max \mathsf{L}(A)}{\min \mathsf{L}(A)}=\frac{m+\max \mathsf{L}(C)}{m+\min \mathsf{L}(C)}\le \frac{\max \mathsf{L}(C)}{\min \mathsf{L}(C)}\le \frac{\mathsf{D}(G)}{2}$$

$$2l\le \sum _{i=1}^l|V_i|=\sum _{j=1}^k|U_j|\le k\mathsf{D}(G)=2l,$$

$$\{U\in \mathcal{A}(G):|U|=\mathsf{D}(G)\}=\{U_1,\dots ,U_{\lambda }\}.$$

$$H=\{\overline{0}\}\cup \{(k_1,\dots ,k_{\lambda })\in {\mathbb{N}}_0^{\lambda }:\rho (\prod _{i=1}^{\lambda }{U}_i^{k_i})=\mathsf{D}(G)/2)\}\subset ({\mathbb{N}}_0^{\lambda },+).$$

$$\prod _{i=1}^{\lambda }{U}_i^{m_i}=\prod _{i=1}^s{V}_i\text{ and }\prod _{i=1}^{\lambda }{U}_i^{l_i}=\prod _{i=1}^t{V}_i.$$

$$\phi :H⟶{\mathcal{B}}_{\rho }(G),(k_1,\dots ,k_{\lambda })⟼\prod _{i=1}^{\lambda }{U}_i^{k_i}.$$

$$|U_1|=|U_2|=\mathsf{D}(G),g_1{g}_2|U_1,\text{and }(g_1+g_2)|U_2.$$

$$U_1{U}_2=((g_1+g_2)g_3{g}_4\cdot \dots \cdot g_l)g_1{g}_2{h}_2\cdot \dots \cdot h_l.$$

$$((-g_1)g_1)((-g_2)g_2)((-g_1-g_2)(g_1+g_2))=((g_1+g_2)(-g_1)(-g_2))((-g_1-g_2)g_1{g}_2)$$

$$\psi :\mathcal{F}(G)⟶{\mathbb{N}}_0^{|G|},S⟼({\mathsf{v}}_g(S){)}_{g\in G}$$

$$\underset{n\to \infty }{\lim }\frac{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):n-k_0\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}=1.$$

$$\underset{n\to \infty }{\lim }\frac{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):\mathsf{L}(A)\text{is an interval with}n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}=1.$$

$$\{A\in {\mathcal{B}}_{\rho }(G):n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\}=\underset{n-\text{fold product}}{\underbrace{\mathcal{A}({\mathcal{B}}_{\rho }(G))\times \dots \times \mathcal{A}({\mathcal{B}}_{\rho }(G))}}$$

$$\underset{n\to \infty }{\lim }\frac{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):n-k_0\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}{\left|\right\{A\in {\mathcal{B}}_{\rho }(G):n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\left\}\right|}=\underset{n\to \infty }{\lim }\frac{|(n-k_0)\Omega |}{|n\Omega |}=\underset{n\to \infty }{\lim }\frac{P(n-k_0)}{P(n)}=1.$$

$$\bigl\{A^{*}A:A\in\mathcal{B}_{\rho}(G)\bigr\}\subset\bigl\{A\in\mathcal{B}_{\rho}(G):\mathsf{L}(A)\ \text{is an interval}\bigr\},\$$

$$|\{A\in {\mathcal{B}}_{\rho }(G):n-k_0\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\}|=|\{A^{\ast }A:A\in {\mathcal{B}}_{\rho }(G),n-k_0\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\}|$$

$$\le |\{A\in {\mathcal{B}}_{\rho }(G):\mathsf{L}(A)\text{is an interval with}n\in {\mathsf{L}}_{{\mathcal{B}}_{\rho }(G)}(A)\}|,$$

$$\underset{n\to \infty }{\lim }\frac{\left|\right\{B\in {\mathcal{B}}_{\rho }(G):|B|\le n-\alpha \left\}\right|}{\left|\right\{B\in {\mathcal{B}}_{\rho }(G):|B|\le n\left\}\right|}=1.$$

Full text

This work was supported by FWF, Project Number DOC 183-N

1. Introduction

Let $H$ be a Krull monoid or a Krull domain with class group $G$ and suppose that each class contains a prime divisor. Then each (non-zero and non-invertible) element $a\in H$ has a factorization into atoms, say $a=u_{1}\cdot\ldots\cdot u_{k}$, where $k$ is the factorization length and $u_{1},\ldots,u_{k}$ are atoms (irreducible elements) of $H$. The set $\mathsf{L}(a)\subset\mathbb{N}$ of all possible factorization lengths $k$ is called the length set of $a$. Length sets and all parameters controlling their structure (such as elasticities and sets of distances) are the best investigated invariants in factorization theory (for a survey, we refer to [20]). To begin with the extreme cases, we have $|\mathsf{L}(a)|=1$ for all relevant $a\in H$ if and only if $|G|\leq 2$. If $G$ is infinite, then every finite nonempty subset of $\mathbb{N}_{\geq 2}$ occurs as a length set of some $a\in H$. Now suppose that $G$ is finite with $|G|\geq 3$. Then the length sets can be arbitrarily large, but they are all finite and well-structured (Proposition 2.2). Roughly speaking, they are sorts of generalized arithmetic progressions. In addition, it is known that almost all length sets are intervals (Proposition 3.1).

For a finite nonempty set $L\subset\mathbb{N}$, we denote by $\rho(L)=\max L/\min L$ the elasticity of $L$, and by $\rho(H)$, defined as the supremum of $\rho\big(\mathsf{L}(a)\big)$ over all $a$ of $H$, the elasticity of $H$. It is well-known that $\rho(H)=\mathsf{D}(G)/2$, where $\mathsf{D}(G)$ denotes the Davenport constant of $G$ (see Subsection 2.2). Recent research has seen much interest in the structure of length sets with maximal elasticity. It has turned out that these length sets have a much simpler structure than length sets in general (compare Propositions 2.2 and 2.4).

We continue the investigation of length sets with maximal elasticity and proceed in the more general setting of transfer Krull monoids (see Subsection 2.3). In analogy to the results on general length sets, we show that almost all length sets with maximal elasticity are intervals. We formulate the first of our main results.

Theorem 1.1**.**

Let $H$ be a transfer Krull monoid over a finite abelian group $G$ with $|G|\geq 3$. Suppose that $G$ has the following Property P.

• P.* There are $g_{1},g_{2}\in G$ and minimal zero-sum sequences $U_{1},U_{2}$ over $G$ such that*

[TABLE]

Then there exists some $a^{*}\in H$ such that for all elements $a\in H$ with $\rho(\mathsf{L}(a))=\rho(H)$, the length set $\mathsf{L}(a^{*}a)$ is an interval with elasticity $\rho(H)$.

In general, length sets $\mathsf{L}(a)$ with elasticity $\rho(H)$ need not be intervals (in other words, the above statement does not hold for $a^{*}=1$; see Proposition 2.4). The proof of Theorem 1.1 will be given in Section 3. We provide two corollaries stating that almost all (in a combinatorial sense) length sets with maximal elasticity are intervals (Corollaries 3.8 and 3.9). Then, in Section 4, we study Property P. Theorem 4.2 shows that Property P is satisfied by cyclic groups of odd order, by groups of rank two, by $p$-groups, and others. The only known groups, for which Property P does not hold, are cyclic groups of even order. Our second main result shows that also the statement of Theorem 1.1 does not hold for them. But, their length sets still satisfy a quite strong structural property.

Theorem 1.2**.**

Let $H$ be a transfer Krull monoid over a cyclic group $G$ of even order $|G|\geq 4$.

There is no element $a\in H$ with maximal elasticity such that $\max\mathsf{L}(a)-1\in\mathsf{L}(a)$. 2. 2.

If $|G|+1\notin\mathbb{P}$, then there exist $a^{*}\in H$ and $M\in\mathbb{N}_{0}$ such that for all $a\in H$ with $\rho(\mathsf{L}(a))=\rho(H)$, $\mathsf{L}(a^{*}a)\cap[\min\mathsf{L}(a^{*}a),\max\mathsf{L}(a^{*}a)-M]$ is an interval and $\rho(\mathsf{L}(a^{*}a))$ has elasticity $\rho(H)$.

The statement of Theorem 1.2.2 does not hold if $|G|+1$ is a prime smaller than or equal to $11$, see Remark 5.3.

2. Preliminaries

We denote by $\mathbb{N},\mathbb{N}_{0},$ and $\mathbb{P}$ the sets of positive integers, non-negative integers, and prime numbers, respectively. For real numbers $a,b\in\mathbb{R}$, $[a,b]=\{x\in\mathbb{Z}:a\leq x\leq b\}$ means the discrete interval between $a$ and $b$. Let $L,L^{\prime}\subset\mathbb{Z}$ be subsets. Then $L+L^{\prime}=\{a+b:a\in L,\hskip 2.84544ptb\in L^{\prime}\}$ denotes their sumset. The set of distances $\Delta(L)\subset\mathbb{N}$ is the set of all $d\in\mathbb{N}$ for which there is $a\in L$ such that $L\cap[a,a+d]=\{a,a+d\}$. If $L^{\prime}\subset L$ is finite, then $L^{\prime}$ is said to be an interval of $L$ if $L^{\prime}\neq\emptyset$ and $L^{\prime}=L\cap[\min L^{\prime},\max L^{\prime}]$. By an interval, we mean an interval of $\mathbb{Z}$, whence an interval is an arithmetic progression with difference $1$.

Let $G$ be an additively written finite abelian group. Let $r\in\mathbb{N}$ and $(e_{1},e_{2},\ldots,e_{r})$ be an $r$-tuple of non-zero elements of $G$. Then $(e_{1},e_{2},\ldots,e_{r})$ is said to be independent if for all $(m_{1},m_{2},\ldots,m_{r})\in\mathbb{Z}^{r}$ an equation $m_{1}e_{1}+m_{2}e_{2}+\ldots+e_{r}m_{r}=0$ implies $m_{i}e_{i}=0$ for all $i\in[1,r]$. An independent $r$-tuple of non-zero elements $(e_{1},e_{2},\ldots,e_{r})$ is said to be a basis if $G=\langle e_{1}\rangle\oplus\langle e_{2}\rangle\oplus\ldots\oplus\langle e_{r}\rangle$. For $n\in\mathbb{N}$, we let $C_{n}$ denote a cyclic group of order $n$.

2.1. Monoids and their Arithmetic.

Throughout this paper, by a monoid, we mean a commutative cancellative semigroup with an identity. Let $H$ be a multiplicatively written monoid. We denote by $H^{\times}$ the group of invertible elements of $H$, and we denote $H_{\textup{red}}=\{aH^{\times}:a\in H\}$ the associated reduced monoid of $H$. For two elements $a,b\in H$, we say $a$ divides $b$, written $a|b$, if $b\in aH$. The elements $a$ and $b$ are called associated, written $a\simeq b$, if $aH=bH$. The element $a$ is called an atom if $a=bc$ with $b,c\in H$ implies $b\in H^{\times}$ or $c\in H^{\times}$. The set of atoms of $H$ is denoted by $\mathcal{A}(H)$, and $H$ is said to be atomic if every non-unit element of $H$ is a finite product of atoms. For a set $P$, we let $\mathcal{F}(P)$ denote the free abelian monoid with basis $P$. Every $a\in\mathcal{F}(P)$ has a unique representation in the form

[TABLE]

where $\mathsf{v}_{p}:H\longrightarrow\mathbb{N}_{0}$ is the $p$-adic valuation of $a$. We call $|a|=\sum_{p\in P}\mathsf{v}_{p}(a)$ the length of $a$ and $\textup{supp}(a)=\{p\in P:\mathsf{v}_{p}(a)>0\}$ the support of $a$.

The free abelian monoid $\mathsf{Z}(H)=\mathcal{F}(\mathcal{A}(H_{\textup{red}}))$ is called the factorization monoid of $H$ and the map $\pi:\mathsf{Z}(H)\longrightarrow H_{\textup{red}}$, defined by $\pi(u)=u$ for all $u\in\mathcal{A}(H_{\textup{red}})$, is called the factorization homomorphism of $H$. Let $a\in H$. Then $\mathsf{Z}(a)=\pi^{-1}(aH^{\times})\subset\mathsf{Z}(H)$ is the set of factorizations of $a$, $\mathsf{L}(a)=\{|a|:z\in\mathsf{Z}(a)\}\subset\mathbb{N}_{0}$ is the length set of $a$, and $\mathcal{L}(H)=\{\mathsf{L}(a):a\in H\}$ is the system of length sets of $H$. By definition, the monoid $H$ is atomic if $\mathsf{Z}(a)\neq\emptyset$ for all $a\in H$. The monoid $H$ is said to be a BF*-monoid*, if $\mathsf{L}(a)$ is finite and nonempty for all $a\in H$, and we call $H$ half-factorial if $|\mathsf{L}(a)|=1$ for all $a\in H$.

Suppose that $H$ is a BF-monoid. To describe the structure of $\mathcal{L}(H)$, we define the following invariants. We denote by $\Delta(H)=\bigcup_{L\in\mathcal{L}(H)}\Delta(L)$ the set of distances of $H$, and by $\rho(H)=\textup{sup }\{\rho(L):L\in\mathcal{L}(H)\}$ the elasticity of $H$, and lastly, we call $\rho_{k}(H)=\textup{sup }\bigcup_{k\in L\in\mathcal{L}(H)}L$ the $k$-th elasticity of $H$. We say that $H$ has accepted elasticity if $\rho(H)=\rho(\mathsf{L}(a))$ for some $a\in H$.

Let $H$ be an atomic monoid and suppose that $x,y\in\mathsf{Z}(H)$ and $z=\textup{gcd}(x,y)$. Then we call $\mathsf{d}(x,y)=\textup{max}\{|z^{-1}x|,|z^{-1}y|\}$ the distance between $x$ and $y$. Explicitly, if $x=u_{1}\cdot\ldots\cdot u_{n}w_{1}\cdot\ldots\cdot w_{r}$ and $y=v_{1}\cdot\ldots\cdot v_{m}w_{1}\cdot\ldots\cdot w_{r}$ with $n,m,r\in\mathbb{N}_{0},$

$u_{1},\ldots,u_{n},v_{1},\ldots,v_{m},w_{1},\ldots,w_{r}\in\mathcal{A}(H_{\textup{red}})$ and $\{u_{1},\ldots,u_{n}\}\cap\{v_{1},\ldots,v_{m}\}=\emptyset$, then $\mathsf{d}(x,y)=\textup{max}\{n,m\}$. The distance function $\mathsf{d}:\mathsf{Z}(H)\times\mathsf{Z}(H)\longrightarrow\mathbb{N}_{0}$ is a metric. We define the catenary degree $\mathsf{c}(a)$ for $a\in H$ to be the smallest $N\in\mathbb{N}_{0}\cup\{\infty\}$ such that, for any two factorizations $z,z^{\prime}\in\mathsf{Z}(a)$, there exists a finite sequence $z=z_{0},z_{1},\ldots,z_{k}=z^{\prime}\in\mathsf{Z}(a)$ satisfying that $\mathsf{d}(z_{i-1},z_{i})\leq N$ for all $i\in[1,k]$. If $z,\,z^{\prime}\in\mathsf{Z}(a)$ and $z\neq z^{\prime}$, then

[TABLE]

If $|\mathsf{Z}(a)|\geq 2$, then

[TABLE]

whence $\mathsf{c}(a)\leq 3$ implies that $\mathsf{L}(a)$ is an interval. Furthermore, we define the *catenary degree * $\mathsf{c}(H)$ of $H$ as

[TABLE]

2.2. Krull Monoids

A monoid homomorphism $\varphi\colon H\to D$ is called a divisor homomorphism if, for all $a,b\in H$, $\varphi(a)\,|\,\varphi(b)$ (in $D$) implies that $a\,|\,b$ (in $H$). A submonoid $S\subset H$ is called saturated if the inclusion $S\hookrightarrow H$ is a divisor homomorphism. A monoid $H$ is a Krull monoid if one of the following equivalent conditions is satisfied (see [6, Chapter 2]).

• (a)

$H$ is completely integrally closed and satisfies the ascending chain condition on divisorial ideals.

• (b)

There is a divisor homomorphism $\varphi\colon H\to D$, where $D$ is a free abelian monoid, such that for every $\alpha\in D$ there exists a nonempty finite set $A\subset H$ with $\alpha=\gcd\varphi(A)$.

• (c)

There is a divisor homomorphism from $H$ into a free abelian monoid.

If $D$ is a domain, then $D$ is a Krull domain if and only if its monoid of nonzero elements is a Krull monoid. Condition (a) shows that every integrally closed noetherian domain is Krull. Let $H$ be a Krull monoid. Then there exists a free abelian monoid $F=\mathcal{F}(P)$ such that $H_{\textup{red}}\subset F$ is a submonoid, and the inclusion map $H_{\textup{red}}\hookrightarrow F$ is a divisor homomorphism that satisfies Condition (b). In this case, $F$ and $P$ are uniquely determined by $H$ (up to isomorphism). Consequently, the group

[TABLE]

where $\mathsf{q}(F)$ and $\mathsf{q}(H_{\textup{red}})$ are the quotient groups of $F$ and $H_{\textup{red}}$, respectively, depends only on $H$ and is called the class group of $H$. For background and further examples of Krull monoids, we refer to [11, 6].

Next, we introduce a Krull monoid having a combinatorial flavor, which plays a central role when studying the arithmetic of general Krull monoids.

Let $G$ be a finite abelian group, and let $G_{0}\subset G$ be a subset. Then the elements of the free abelian monoid $\mathcal{F}(G_{0})$ with basis $G_{0}$ are called sequences over $G_{0}$. For a sequence $S=g_{1}\cdot\ldots\cdot g_{\ell}\in\mathcal{F}(G_{0})$, we call $\sigma(S)=\sum_{i=1}^{\ell}g_{i}\in G$ the sum of $S$, $|S|=\ell$ the length of $S$, and $\operatorname{supp}(S)=\{g_{1},\ldots,g_{\ell}\}\subset G$ the support of $S$. If $\varphi\colon G\to G^{\prime}$ is a map from $G$ to a group $G^{\prime}$, we set $\varphi(S)=\varphi(g_{1})\cdot\ldots\cdot\varphi(g_{l})$, and we define $-S=(-g_{1})\cdot\ldots\cdot(-g_{l})$. The sequence $S$ is called

• •

a zero-sum sequence if $\sigma(S)=0$,

• •

zero-sumfree if $\sum_{i\in I}g_{i}\neq 0$ for all $\emptyset\neq I\subset[1,\ell]$, and

• •

squarefree if $|S|=|\operatorname{supp}(S)|$.

Furthermore, $\mathcal{B}(G_{0})=\{S\in\mathcal{F}(G_{0}):\sigma(S)=0\}$ is the monoid of zero-sum sequences over $G_{0}$. The monoid $\mathcal{B}(G_{0})$ is viewed as a submonoid of $\mathcal{F}(G_{0})$. $\mathcal{B}(G_{0})$ is a finitely generated monoid, and since the inclusion $\mathcal{B}(G_{0})\hookrightarrow\mathcal{F}(G_{0})$ is a divisor homomorphism, Condition (c) implies that $\mathcal{B}(G_{0})$ is a Krull monoid. We denote $\mathcal{A}(G_{0})$ the set of atoms of $\mathcal{B}(G_{0})$, and we call the elements of $\mathcal{A}(G_{0})$ the minimal zero-sum sequences over $G_{0}$. We set

[TABLE]

Lastly, the Davenport constant of $G$ is defined by

[TABLE]

It is known that $\rho(\mathcal{B}(G))=\mathsf{D}(G)/2$ for all finite abelian groups $G$.

2.3. Transfer Homomorphisms and Transfer Krull Monoids.

We start with the definition of transfer homomorphisms.

Definition 2.1**.**

Let $H$ and $B$ be atomic monoids. Then a monoid homomorphism $\theta:H\longrightarrow B$ is called a transfer homomorphism if the following properties are satisfied.

1. $B=\theta(H)B^{\times}$ and $\theta^{-1}(B^{\times})=H^{\times}$.

2. If $u\in H,\hskip 2.84544ptb,c\in B$ and $\theta(u)=bc$, then there exist $v,w\in H$ such that $u=vw,\theta(v)\simeq b$ and $\theta(w)\simeq c$.

A transfer homomorphism $\theta:H\longrightarrow B$ allows us to pull back arithmetic properties from $B$ to $H$ ([6, Proposition 3.2.3]). In particular, we have

[TABLE]

A monoid $H$ (respectively, a domain $D$) is said to be transfer Krull monoid over $G$ (respectively, a transfer Krull domain) if there exists a transfer homomorphism $\theta:H\longrightarrow\mathcal{B}(G)$ (respectively, $\theta:D\setminus\{0\}\longrightarrow\mathcal{B}(G)$). It is a classic result that a Krull monoid with class group $G$, which has a prime divisor in each class, is a transfer Krull monoid over $G$ ([6, Theorem 3.4.10]). Deep results of the last decade found various classes of rings and monoids that are transfer Krull but fail to be Krull. They include non-commutative Dedekind domains, non-principal orders in Dedekind domains, and others (e.g., [21, 16, 4, 3], and the survey [9]).

2.4. Length Sets with Maximal Elasticity

Our goal is to understand the structure of length sets with maximal elasticity. We begin by recalling the Structure Theorem for Length Sets (this result will be needed later on in the proof of Theorem 1.2). To do so, we need the concepts of AAMPs (almost arithmetic multiprogressions) and of AAPs (almost arithmetic progressions).

Let $\ell,d\in\mathbb{N}$, let $M\in\mathbb{N}_{0}$, and let $\mathcal{D}\subset\mathbb{Z}$ be a subset with $\{0,d\}\subset\mathcal{D}\subset[0,d]$. Let $L\subset\mathbb{Z}$ be a finite, nonempty subset. Then $L$ is an almost arithmetic multiprogression (AAMP) with difference $d$, period $\mathcal{D}$, length $\ell$ and bound $M$ if

[TABLE]

• •

$\min L^{*}=0$, $L^{*}$ is an interval of $\min L^{*}+\mathcal{D}+d\mathbb{Z}$, $\ell$ is maximal for the property that $\min L^{*}+(\ell-1)d\in L^{*}$,

• •

$L^{\prime}\subset[-M,-1]$, and $L^{\prime\prime}\subset\max L^{*}+[1,M]$.

We say $L$ is an almost arithmetic progression (AAP) with difference $d$, length $\ell$, and bound $M$ if it is an AAMP with difference $d$, period $\{0,d\}$, length $\ell$, and bound $M$.

Proposition 2.2** (The Structure Theorem for Length Sets).**

Let $H$ be a transfer Krull monoid over a finite abelian group $G$. There exists some $M\in\mathbb{N}_{0}$ such that every $L\in\mathcal{L}(H)$ is an AAMP with some difference $d\in\Delta(H)$ and bound $M$.

Proof.

Since $\mathcal{L}(H)=\mathcal{L}(G)$ and $\mathcal{B}(G)$ is finitely generated, this follows from [6, Theorem 4.4.11]. ∎

A realization theorem by Schmid shows that the above structural description is best possible ([18]). In order to describe the structure of length sets with maximal elasticity, we need one more invariant. Then we state some results as well as a standing conjecture (Conjecture 2.3) about length sets with maximal elasticity. Let $\Delta_{\rho}(G)$ denote the set of all $d\in\mathbb{N}$ with the following property:

• For every $k\in\mathbb{N}$, there is $L_{k}\in\mathcal{L}(G)$ which is an AAP with difference $d$, length $\ell\geq k$, and $\rho(L_{k})=\rho(G)$.

Conjecture 2.3**.**

Let $G$ be a finite abelian group with $|G|>4$. Then $\Delta_{\rho}(G)=\{1\}$ if and only if $G$ is neither cyclic nor an elementary $2$-group.

It is easy to see that $\Delta_{\rho}(G)\neq\{1\}$ if $G$ is cyclic or an elementary $2$-group. The reverse implication has been confirmed, among others, for groups of rank two, and for groups isomorphic either to $C_{2}\oplus C_{2}\oplus C_{2n}$ or to $C_{p^{k}}^{r}$, where $n,r\geq 2,k\geq 1$ and $p$ is a prime number with $p^{k}\geq 3$ (see [2, 8]).

Proposition 2.4** ([2], Corollary 2.3).**

Let $H$ be a transfer Krull monoid over a finite abelian group $G$ and suppose that $\Delta_{\rho}(G)=\{1\}$. Then there exists some $M\in\mathbb{N}_{0}$ such that every $L\in\mathcal{L}(H)$ with $\rho(L)=\mathsf{D}(G)/2$ is an AAP with difference $1$ and bound $M$.

If the above statement holds for $M$, then it holds for every $M^{\prime}$ with $M^{\prime}\geq M$. However, apart from some exceptional cases, the statement does not hold for $M=0$. Comparing Proposition 2.4 with Proposition 2.2, we observe that the structure of length sets with maximal elasticity is much more rigid than those of general length sets. Moreover, if $G$ is cyclic or an elementary $2$-group with $|G|>4$, then $\Delta_{\rho}(G)\neq\{1\}$ and the statement of Proposition 2.4 does not hold.

3. Proof of Theorem 1.1

The goal of this section is to prove Theorem 1.1. In order to highlight its significance, we first recall a result on general length sets. If $H=\mathcal{B}(G)$, for some finite abelian group $G$, then Proposition 3.1 implies that the length sets of almost all elements (in a combinatorial sense) are intervals. If $H$ is an analytic Krull monoid (in the sense of [12]), then Proposition 3.1 implies that the length sets of almost all (in the sense of Dirichlet density) elements of $H$ are intervals ([6, Theorem 9.4.11]).

Proposition 3.1**.**

Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor. Then there exists some $a^{*}\in H$ such that, for all $a\in H$, the catenary degree $\mathsf{c}(a^{*}a)\leq 3$, whence the length set $\mathsf{L}(a^{*}a)$ is an interval.

Proof.

This easily follows from [6, Theorem 7.6.9]. ∎

For the proof of Theorem 1.1, we proceed via a series of lemmas.

Lemma 3.2**.**

Let $H$ be a BF-monoid with accepted elasticity and let $a,b\in H$ with $\rho(\mathsf{L}(a))=\rho(\mathsf{L}(b))=\rho(H)$. Then $\rho(\mathsf{L}(ab))=\rho(H)$, $\max\mathsf{L}(ab)=\max\mathsf{L}(a)+\max\mathsf{L}(b)$ and $\min\mathsf{L}(ab)=\min\mathsf{L}(a)+\min\mathsf{L}(b)$.

Proof.

Using the inclusion $\mathsf{L}(a)+\mathsf{L}(b)\subset\mathsf{L}(ab)$ for all $a,b\in H$, we get

[TABLE]

and hence

[TABLE]

Thus we obtain $\rho(\mathsf{L}(ab))=\rho(H)$, and also $\max\mathsf{L}(ab)=\max\mathsf{L}(a)+\max\mathsf{L}(b)$ and $\min\mathsf{L}(ab)=\min\mathsf{L}(a)+\min\mathsf{L}(b)$. ∎

Lemma 3.3** ([8], Lemma 3.2 (i)).**

Let $G$ be a finite abelian group with $|G|\geq 3$. A sequence $A\in\mathcal{B}(G)$ satisfies the equality $\rho(\mathsf{L}(A))=\mathsf{D}(G)/2$ if and only if there exist $k,l\in\mathbb{N}$ and $U_{1},.\hskip 1.42271pt.\hskip 1.42271pt.\hskip 1.42271pt,U_{k},V_{1},.\hskip 1.42271pt.\hskip 1.42271pt.\hskip 1.42271pt,V_{l}\in\mathcal{A}(G)$ with $|U_{1}|=\ldots=|U_{k}|=\mathsf{D}(G),\hskip 2.84544pt|V_{1}|=\ldots=|V_{l}|=2$ such that $A=U_{1}\cdot\ldots\cdot U_{k}=V_{1}\cdot\ldots\cdot V_{l}$.

Proof.

Suppose that $\rho(\mathsf{L}(A))=\mathsf{D}(G)/2$ for some $A\in\mathcal{B}(G)$. If $A=0^{m}C$ for $m\in\mathbb{N}_{0}$ and $C\in\mathcal{B}(G\setminus\{0\})$, then from the inequality

[TABLE]

we obtain $m=0$. Now write $A=U_{1}\cdot\ldots\cdot U_{k}=V_{1}\cdot\ldots\cdot V_{l}$, where $k=\min\mathsf{L}(A),l=\max\mathsf{L}(A)$ and $U_{1},.\hskip 1.42271pt.\hskip 1.42271pt.\hskip 1.42271pt,U_{k},\hskip 1.42271ptV_{1},\hskip 1.42271pt.\hskip 1.42271pt.\hskip 1.42271pt.\hskip 1.42271pt,V_{k}\in\mathcal{A}(G)$. Then $l/k=\mathsf{D}(G)/2$ and

[TABLE]

which implies that $|U_{i}|=\mathsf{D}(G)$ for all $i\in[1,k]$ and $|V_{j}|=2$ for all $j\in[1,l]$. The reverse implication is straightforward. ∎

Lemma 3.4**.**

Let $H$ be a BF-monoid with accepted elasticity and let $a\in H$ be an element with maximal elasticity such that $\min\mathsf{L}(a)+1$ and $\max\mathsf{L}(a)-1$ are lengths of $a$. Then there is an $N\in\mathbb{N}$ such that $\mathsf{L}(a^{n})$ is an interval for all $n\geq N$.

Proof.

Let $k:=\textup{min }\mathsf{L}(a)$, $l:=\textup{max }\mathsf{L}(a)$, and suppose that $[k,k+1]\cup[l-1,l]\subset\mathsf{L}(a)$. Then the $(l-k)$-fold sumset $(l-k)\mathsf{L}(a)$ is an interval. Since $\rho(\mathsf{L}(a))=\rho(H)$, Lemma 3.2 implies that maximal and minimal elements of $(l-k)\mathsf{L}(a)$ coincide with the maximal and minimal elements of $\mathsf{L}(a^{l-k})$, respectively. Hence, $\mathsf{L}(a^{l-k})$ is an interval itself. Moreover, $\mathsf{L}(a^{n})$ is also an interval for all $n\geq l-k$, which again follows from Lemma 3.2.∎

Lemma 3.5** ([6], Theorem 3.1.4 and Proposition 2.7.5).**

Let $H$ be a monoid such that $H_{\textup{red}}$ is finitely generated. Then the set of distances $\Delta(H)$ is finite and the elasticity $\rho(H)$ is rational. Moreover, if $S\subset H$ is a saturated submonoid, then $S_{{\text{\rm red}}}$ is finitely generated.

Let $\mathcal{B}_{\rho}(G)$ be the set of all zero-sum sequences with maximal elasticity and the identity element of $\mathcal{B}(G)$. Then Lemma 3.2 provides that $\mathcal{B}_{\rho}(G)$ is a submonoid of $\mathcal{B}(G)$.

Proposition 3.6**.**

Let $G$ be a finite abelian group. Then the submonoid $\mathcal{B}_{\rho}(G)\subset\mathcal{B}(G)$ is finitely generated.

Proof.

Since $\mathcal{B}(G)$ is finitely generated, $\mathcal{A}(G)$ is finite, say

[TABLE]

Let

[TABLE]

Suppose that $(l_{1},\ldots,l_{\lambda})\in H$ divides $(m_{1},\ldots,m_{\lambda})\in H$ in $\mathbb{N}_{0}^{\lambda}$, which means that $(m_{1}-l_{1},\ldots,m_{\lambda}-l_{\lambda})\in\mathbb{N}_{0}^{\lambda}$. We set $s:=(\sum_{i=1}^{\lambda}m_{i})\mathsf{D}(G)/2$ and $t:=(\sum_{j=1}^{\lambda}l_{j})\mathsf{D}(G)/2$. Then there exist atoms $V_{1},\ldots,V_{s}$ with length two such that

[TABLE]

Hence, we have $\prod_{i=1}^{\lambda}U_{i}^{m_{i}-l_{i}}=\prod_{i=t+1}^{s}V_{i}$. By Lemma 3.3, we infer that $\rho(\prod_{i=1}^{\lambda}U_{i}^{m_{i}-l_{i}})=\mathsf{D}(G)/2$. Therefore, $(m_{1}-l_{1},\ldots,m_{\lambda}-l_{\lambda})\in H$, whence $H$ is a saturated submonoid of the additive monoid $(\mathbb{N}^{\lambda}_{0},+)$. Thus, Lemma 3.5 implies that $H$ is finitely generated. Now consider the following monoid homomorphism

[TABLE]

By Lemma 3.3 and by construction of $H$, the map $\varphi$ is an epimorphism, whence $\mathcal{B}_{\rho}(G)$ is finitely generated. ∎

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let $H$ be a transfer Krull monoid over a finite abelian group $G$ with $|G|\geq 3$ and which satisfies Property P. By (2.2), it is sufficient to prove the assertion for the monoid $H=\mathcal{B}(G)$. Let $g_{1},g_{2}\in G$ and let $U_{1},U_{2}\in\mathcal{A}(G)$ be such that

[TABLE]

We set $U_{1}=g_{1}g_{2}g_{3}\cdot\ldots\cdot g_{l}$ and $U_{2}=(g_{1}+g_{2})h_{2}h_{3}\cdot\ldots\cdot h_{l}$, where $l=\mathsf{D}(G)$. Now, consider the sequence $A^{\prime}=(-U_{1})U_{1}(-U_{2})U_{2}$. By Lemma 3.3, this is a sequence with maximal elasticity. We aim to show that $U_{1}U_{2}$ can be expressed as a product of three atoms. Observe that

[TABLE]

Since the sequence $g_{1}g_{2}h_{2}\cdot\ldots\cdot h_{l}$ has length $l+1>\mathsf{D}(G)$, and its subsequence $h_{2}h_{3}\cdot\ldots\cdot h_{l}$ is a zero-sumfree sequence, it follows that this sequence can be factored into two atoms only. Additionally, observe that the sequence $(g_{1}+g_{2})g_{3}g_{4}\cdot\ldots\cdot g_{l}$ is also an atom. Hence, we conclude that $\min\mathsf{L}(A^{\prime})+1\in\mathsf{L}(A^{\prime})$. Moreover, using the equality

[TABLE]

we deduce that $\max{\mathsf{L}}(A^{\prime})-1\in\mathsf{L}(A^{\prime})$. Thus, by Lemma 3.4 there is $n\in\mathbb{N}$ such that $\mathsf{L}((A^{\prime})^{n})$ is an interval. Since $\mathcal{B}(G)$ is a finitely generated reduced monoid, Lemma 3.5 implies that its set of distances $\Delta(G)$ is finite. Let $k=\max\{\max\Delta(G),n\}$, and define $A^{*}=(A^{\prime})^{k}$. We assert that for any sequence $A\in\mathcal{B}(G)$ with maximal elasticity, $\mathsf{L}(A^{*}A)$ forms an interval. Indeed, since $\max\Delta(\mathsf{L}(A))\leq k$ and $\mathsf{L}(A^{*})$ is an interval of length greater than $k$, we get that $\mathsf{L}(A)+\mathsf{L}(A^{*})$ is an interval. Lastly, applying Lemma 3.2, we conclude that $\mathsf{L}(A^{*}A)$ is also an interval and has elasticity $\mathsf{D}(G)/2$. ∎

As applications of Theorem 1.1, we present two corollaries, which demonstrate that almost all (in a combinatorial sense) length sets with maximal elasticity are intervals. To establish the first result, we require a lemma concerning the growth behavior of the $n$-fold sumset of a finite set. Khvanskii first proved that the $n$-fold sumset of a finite set $A\subset\mathbb{Z}^{d}$ grows like a polynomial. His result was generalized and refined, and the bounds were made effective by many authors ([15, 10, 5]). We cite a version due to Granville, Shakan, and Walker ([10]).

Lemma 3.7**.**

Let $A\subset\mathbb{Z}^{d}$, with $d\geq 1$, be a finite subset. Then there exist a polynomial $P_{A}\in\mathbb{Q}[x]$ and a constant $N_{A}$ such that $|nA|=P_{A}(n)$ for all $n\geq N_{A}$.

Note that there is a canonical isomorphism from the free abelian monoid $\mathcal{F}(G)$ to $(\mathbb{N}_{0}^{|G|},+)$ defined by

[TABLE]

We have $\mathcal{A}(\mathcal{B}_{\rho}(G))\subset\mathcal{B}_{\rho}(G)\subset\mathcal{F}(G)$. Set $\Omega:=\psi(\mathcal{A}(\mathcal{B}_{\rho}(G)))$. In particular, the monoid $\mathcal{B}_{\rho}(G)$ is isomorphic to the monoid $S:=[\Omega]$ generated by $\Omega$.

Corollary 3.8**.**

Let $G$ be a finite abelian group with $|G|\geq 3$.

For every $k_{0}\in\mathbb{N}$, we have

[TABLE] 2. 2.

If $G$ satisfies Property $\bf P$, then

[TABLE]

Proof.

1. Proposition 3.6 implies that the set $\Omega\subset\mathbb{N}_{0}^{|G|}$, defined above, is finite. Moreover, since

[TABLE]

we get that $|\{A\in\mathcal{B}_{\rho}(G):n\in\mathsf{L}_{\mathcal{B}_{\rho}(G)}(A)\}|=|n\Omega|$ for all $n\in\mathbb{N}$. By Lemma 3.7, there exist a polynomial $P\in\mathbb{Q}[x]$ and a constant $N$ such that $|n\Omega|=P(n)$ for all $n\geq N$. Thus

[TABLE]

2. Assume that $G$ has Property $\bf P$. Then Theorem 1.1 implies that there is some $A^{*}\in\mathcal{B}_{\rho}(G)$ such that $\mathsf{L}(A^{*}A)$ is an interval with elasticity $\mathsf{D}(G)/2$ for all $A\in\mathcal{B}_{\rho}(G)$. Let $k_{0}\in\mathsf{L}_{\mathcal{B}_{\rho}(G)}(A^{*})$. Thus

[TABLE]

which implies that

[TABLE]

[TABLE]

whence the assertion follows from Part 1. ∎

Corollary 3.9**.**

Let $G$ be a finite abelian group with $|G|\geq 3$.

For every $\alpha\in\mathbb{N}$, we have

[TABLE] 2. 2.

If $G$ satisfies Property $\bf P$, then

[TABLE]

Proof.

1. Let $|G|=m$. For every $a=(a_{1},\ldots,a_{m})\in\mathbb{N}_{0}^{m}$, we define $|a|=a_{1}+\ldots+a_{m}$ and $X^{a}=X_{1}^{a_{1}}\ldots X_{m}^{a_{m}}$, where $X=(X_{1},\ldots,X_{m})$. We consider the algebra

[TABLE]

where $\mathbb{C}[S]_{n}$ is the linear space generated by monomials of degree $n$ in $\mathbb{C}[S]$. Hence $\mathbb{C}[S]$ is a finitely generated graded commutative algebra. Let the Hilbert series be

[TABLE]

The Hilbert-Serre Theorem (see [1, Theorem 11.1] or [13, Theorem 6.1]) implies that there exists a polynomial $Q(t)\in\mathbb{Z}[t]$ such that $H(\mathbb{C}[S],t)=\frac{Q(t)}{\prod_{i=1}^{\beta}(1-t^{m_{i}})}$ for some $\beta,m_{1},\ldots,m_{\beta}\in\mathbb{N}$. Thus by [22, Proposition 4.4.1], there exist $N\in\mathbb{N}$ and polynomials $f_{0},f_{1},\ldots,f_{N-1}$ such that for all sufficiently large $n$, we have

[TABLE]

Let $\alpha_{N}\in\mathbb{N}$ be such that $\alpha_{N}N\geq\alpha$. We claim that

[TABLE]

In fact, for every $i\in[0,N-1]$, we have

[TABLE]

Hence the claim follows. Next, since

[TABLE]

it suffices to show that

[TABLE]

Let $b_{n}=|\{a\in S:|a|=n\}|,c_{n}=b_{n-\alpha_{N}N},$ and $\epsilon_{n}=1-\frac{c_{n}}{b_{n}}$. Then our claim implies $\lim_{n\to\infty}\epsilon_{n}=0$. Note that $\lim_{n\to\infty}\sum_{i=1}^{n}b_{i}=\infty$. Therefore

[TABLE]

[TABLE]

[TABLE]

For every $\epsilon>0$, there exists $N_{\epsilon}\in\mathbb{N}$ with $N_{\epsilon}>\alpha_{N}N$ such that for every $i\geq N_{\epsilon}$, we have $\epsilon_{i}<\epsilon$. It follows that

[TABLE]

whence $\lim_{n\to\infty}\frac{\sum_{i=\alpha_{N}N+1}^{n}b_{i}\epsilon_{i}}{\sum_{i=1}^{n}b_{i}}=0$ and the assertion follows.

2. Assume that $G$ has Property $\bf P$. Then again Theorem 1.1 implies that there is some $A^{*}\in\mathcal{B}_{\rho}(G)$ such that $\mathsf{L}(A^{*}B)$ is an interval with elasticity $\mathsf{D}(G)/2$ for all $B\in\mathcal{B}_{\rho}(G)$. Let $|A^{*}|=\alpha$. Thus

[TABLE]

which implies that

[TABLE]

[TABLE]

whence the assertion follows from Part 1. ∎

4. On property P occurring in Theorem 1.1

Let $G=C_{n_{1}}\oplus\ldots\oplus C_{n_{r}}$ with $r\in\mathbb{N}_{0}$ and $1<n_{1}\,|\,\ldots\,|\,n_{r}$, and set $\mathsf{D}^{*}(G)=1+\sum_{i=1}^{r}(n_{i}-1)$. It only takes a few lines to check that

[TABLE]

and it is well-known that equality holds for $p$-groups, groups of rank $r\leq 2$, and others (e.g., [6, Theorems 5.5.9 and 5.8.3], [7, page 56], [19]). But, there are groups with $\mathsf{D}^{*}(G)<\mathsf{D}(G)$ (see, for example, [14] for some recent progress). However, so far there is only one series of groups with $\mathsf{D}(G)>\mathsf{D}^{*}(G)$, for which the precise value of the Davenport constant is known. We summarize this result in the next lemma.

Lemma 4.1** ([17] Theorem 5.8 and Theorem 5.9).**

Let $G=C_{2}^{4}\oplus C_{2k}$ with $k\geq 70$.

We have

[TABLE] 2. 2.

Suppose that $k$ is odd. Then a sequence $U\in\mathcal{F}(G)$ is a minimal zero-sum sequence of length $\mathsf{D}(G)$ if and only if there exists a basis $(e_{1},e_{2},e_{3},e_{4},e)$ of $G$, with $\operatorname{ord}(e_{2})=2$ for $i\in[1,4]$ and $\operatorname{ord}(e)=2k$, such that $U=e^{2k-3}V_{1}V_{2}$, where

[TABLE]

Theorem 4.2**.**

Let $G$ be a finite abelian group with $|G|\geq 3$. Then Property

• P.* There are $g_{1},g_{2}\in G$ and minimal zero-sum sequences $U_{1},U_{2}$ over $G$ such that*

[TABLE]

is satisfied in each of the following cases:

1. (a)

$G$* is a cyclic group of odd order* 2. (b)

$G$* is not cyclic and $\mathsf{D}(G)=\mathsf{D}^{*}(G)$* 3. (c)

$G$* has odd order and there is some $U\in\mathcal{A}(G)$ of length $|U|=\mathsf{D}(G)$ that is not squarefree* 4. (d)

$G=C_{2}^{4}\oplus C_{2k}$* with $k\geq 70$.*

Remark. The only known groups, which do not satisfy Property P are cyclic groups of even order. They will be studied in Section 5. Moreover, there is no known group of odd order that does not satisfy Condition (c).

Proof.

(a) Suppose $G$ is cyclic of odd order $n\geq 3$. Let $g\in G$ be any generator. Clearly, $2g\in G$ is also a generator of the group, since the order of $G$ is odd. Now, consider $U_{1}=g^{n}$ and $U_{2}=(2g)^{n}$. Since both $U_{1}$ and $U_{2}$ are atoms with length $\mathsf{D}(G)$, our condition holds.

(b) Suppose that $G\cong C_{n_{1}}\oplus\ldots\oplus C_{n_{r}}$ with $1<n_{1}\mid\ldots\mid n_{r}$ and that $\mathsf{D}^{*}(G)=\mathsf{D}(G)$. Let $(e_{1},\ldots,e_{r})$ be a basis of $G$ with $\operatorname{ord}(e_{i})=n_{i}$ for $i\in[1,r]$ and with $r\geq 2$. Then the sequence $U=(e_{1}+e_{2}+\ldots+e_{r})\prod_{i=1}^{r}e_{i}^{n_{i}-1}$ has the property that $|U|=\mathsf{D}(G)$. Moreover, the map $\varphi:G\longrightarrow G$, given by $e_{i}\mapsto e_{i}$ for all $i\in[1,r-1]$ and $e_{r}\mapsto e_{r}+e_{r-1}$, is an isomorphism. Since $U$ is a minimal zero-sum sequence, $\varphi(U)$ is as well, and therefore, $|\varphi(U)|=|U|=\mathsf{D}(G)$ and $e_{r-1}+e_{r}\in\textup{supp }(\varphi(U))$. If we choose $U_{1}=U$ and $U_{2}=\varphi(U)$, then the condition holds.

(c) Suppose $G$ has odd order and there is some $U\in\mathcal{A}(G)$ of length $|U|=\mathsf{D}(G)$ that is not squarefree. Then the map defined by the multiplication by two, $\varphi:G\longrightarrow G,\hskip 5.69046ptg\mapsto 2g$ for every $g\in G$, is an isomorphism. Since $U$ is not squarefree, there exists $g\in G$ such that $g^{2}\hskip 2.84544pt|\hskip 2.84544ptU$. Then $\varphi(U)$ is a minimal zero-sum sequence of length $|\varphi(U)|=|U|=\mathsf{D}(G)$ and $2g\in\textup{supp }(\varphi(U))$. Thus, the condition holds with $U_{1}=U$ and $U_{2}=\varphi(U)$.

(d) Suppose $G=C_{2}^{4}\oplus C_{2k}$ with $k\geq 70$. If $k$ is even, then $\mathsf{D}(G)=\mathsf{D}^{*}(G)$ by Lemma 4.1, whence Property P holds by Condition (b).

Now suppose that $k$ is odd. Let $(e_{1},e_{2},e_{3},e_{4},e)$ be a basis of $G$ as in Lemma 4.1. Then $U=e^{2k-3}V_{1}V_{2}$, where $V_{1}$ and $V_{2}$ are as of Lemma 4.1, is a minimal zero-sum sequence of length $\mathsf{D}(G)$. One can see that $(e_{1},e_{2},e_{3},e_{4},e_{1}+e_{2}+e_{3}+\frac{k+1}{2}e)$ is also a basis of $G$. Consider the map $\varphi:G\longrightarrow G$ defined by $e_{i}\mapsto e_{i}$ for $i\in[1,4]$ and $e\mapsto e_{1}+e_{2}+e_{3}+\frac{k+1}{2}e$. This is an isomorphism, which implies that $\varphi(U)\in\mathcal{A}(G)$ and $|\varphi(U)|=|U|=\mathsf{D}(G)$. Moreover, we have $e(e_{1}+e_{2}+e_{3}+\frac{k-1}{2}e)\hskip 2.84544pt|\hskip 2.84544ptU$ and $e_{1}+e_{2}+e_{3}+\frac{k+1}{2}e\hskip 2.84544pt|\hskip 2.84544pt\varphi(U)$ as desired. ∎

We end this section by considering a Property P∗ of finite abelian groups and its arithmetic consequences. Theorem 4.3 establishes a result on the catenary degree for elements with maximal elasticity. It is analogous to the result given by Proposition 3.1, but its validity is quite restricted. Let $H$ be an atomic monoid and let $a\in H$. Then the basic inequality (2.1) shows that $\mathsf{c}(a)\leq 3$ implies that $\mathsf{L}(a)$ is an interval. But the converse is far from being true. Thus, the conclusion of Theorem 4.3 is stronger than that of Theorem 1.1.

Define Property P∗ as

• P∗. For every nonzero element $g\in G$, there is some $A_{g}\in\mathcal{A}(G)$ with $|A_{g}|=\mathsf{D}(G)$

P∗. and $g\in\operatorname{supp}(A_{g})$.

Clearly, Property P∗ implies Property P. To give an example of a group satisfying P∗, suppose that $G$ is an elementary $p$ group of rank $r$ and let $g=e_{1}\in G\setminus\{0\}$. Then $e_{1}$ can be extended to a basis, say $(e_{1},e_{2},\ldots,e_{r})$. Then

[TABLE]

is a minimal zero-sum sequence of length $|A_{g}|=\mathsf{D}^{*}(G)=\mathsf{D}(G)$ and with $g\in\operatorname{supp}(A_{g})$. On the other hand, finite cyclic groups satisfy Property P∗ if and only if they are of prime order. Similarly, groups of rank two need not satisfy Property P∗ ([6, Theorem 5.8.4]).

Theorem 4.3**.**

Let $H$ be a transfer Krull monoid over a finite abelian group $G$, and suppose that $G$ satisfies Property P∗. Then there exists some $a^{*}\in H$ such that all elements $a\in H$ with $\rho(\mathsf{L}(a))=\rho(H)$ have catenary degree $\mathsf{c}(aa^{*})\leq 3$, whence the length sets $\mathsf{L}(a^{*}a)$ are intervals with elasticity $\rho(H)$.

Proof.

By [6, Theorem 3.4.10], it suffices to prove the assertion for the monoid $\mathcal{B}(G)$. Consider the following sequence

[TABLE]

where the sequences $A_{g}$ are as in the Definition of Property P∗. Lemma 3.3 follows that the length set of $A^{*}$ has maximal elasticity. Let $A\in\mathcal{B}(G)$ be such that $\rho(\mathsf{L}(A))=\mathsf{D}(G)/2$. Then $\mathsf{\rho}(\mathsf{L}(A^{*}A))=\mathsf{D}(G)/2$ by Lemma 3.2. Since $\textup{supp }(A^{*})\cup\{0\}=G$, we get $\textup{supp }(A^{*}A)\cup\{0\}=G$. Thus, [6, Theorem 7.6.8] implies that $\mathsf{c}(A^{*}A)\leq 3$, and consequently, the length set $\mathsf{L}(A^{*}A)$ forms an interval. It has elasticity $\rho(H)$ by Lemma 3.2. ∎

5. On cyclic groups of even order: Proof of Theorem 1.2

In this section, we prove Theorem 1.2.

Lemma 5.1** ([7], Theorem 5.3.1).**

Let $G$ be a cyclic group of order $n\geq 2$. Then for every $k\in\mathbb{N}$ we have $\rho_{2k}(\mathcal{B}(G))=kn$ and $\rho_{2k+1}(\mathcal{B}(G))=kn+1$.

Lemma 5.2**.**

Let $G$ be a cyclic group of order $n\geq 3$. Then any sequence $A\in\mathcal{B}(G)$ with maximal elasticity is of the form:

[TABLE]

where $G_{0}=\{g\in G:\operatorname{ord}(g)=n\}$ and $n_{g}\in\mathbb{N}_{0}$.

Proof.

Let $A\in\mathcal{B}(G)$ and suppose that $\rho(\mathsf{L}(A))=\mathsf{D}(G)/2$. Then, by Lemma 3.3, there exist atoms $U_{1},.\hskip 2.84544pt.\hskip 2.84544pt.\hskip 2.84544pt,U_{m},\hskip 2.84544ptV_{1},.\hskip 2.84544pt.\hskip 2.84544pt.\hskip 2.84544pt,V_{l}\in\mathcal{A}(G)$ such that $A=U_{1}\cdot\ldots\cdot U_{m}=V_{1}\cdot\ldots\cdot V_{l}$, where $|U_{i}|=\mathsf{D}(G)$ and $|V_{j}|=2$ for all $i\in[1,m]$ and $j\in[1,l]$. Lemma 5.1 ensures that $m$ is even, say $m=2k$ for some $k\in\mathbb{N}$. By applying [6, Lemma 1.4.9], we obtain that $U_{i}=g_{i}^{n}$, where $g_{i}\in G_{0}$ for all $i\in[1,k]$. Thus, we conclude that the sequence $A\in\mathcal{B}(G)$ with maximal elasticity is equal to $\prod_{i=1}^{k}U_{i}(-U_{i})=\prod_{i=1}^{k}g_{i}^{n}(-g_{i})^{n}$ (after reindexing if necessary). ∎

Proof of Theorem 1.2.

Let $H$ be a transfer Krull monoid over a cyclic group $G$ of even order $n\geq 4$. By (2.2), it is sufficient to prove the assertion for the monoid $H=\mathcal{B}(G)$.

1. Take a sequence $A\in\mathcal{B}(G)$ with maximal elasticity. Then by Lemma 5.2, the sequence $A$ is of the form:

$A=\prod_{g\in G_{0}}\big(g^{n}(-g)^{n}\big)^{n_{g}}$

where $G_{0}=\{g\in G:\text{ord}(g)=n\}$. Now, we claim that $\max\mathsf{L}(A)-1\notin\mathsf{L}(A)$. Let $N=\max\mathsf{L}(A)=\sum_{g\in G_{0}}nn_{g}$. Assume that $N-1\in\mathsf{L}(A)$, which means that $A=\prod_{i=1}^{N-1}V_{i}$ for some atoms $V_{1},\ldots,V_{N-1}\in\mathcal{A}(G_{0})$. Thus we get $2N=|A|=\sum_{i=1}^{N-1}|V_{i}|$. Note that $2\leq|V_{i}|\leq n$ for all $i\in[1,N-1]$. Therefore, we have two cases:

Case 1: Among the sets $|V_{i}|$, exactly one has cardinality four, while all others have cardinality two. Without loss of generality, assume $|V_{i}|=2$ for all $i\in[1,N-2]$ and $|V_{N-1}|=4$. This configuration is impossible, as having $|V_{N-1}|=4$ implies $V_{N-1}=g(-g)h(-h)$ for some $g,h\in G_{0}$, which does not constitute an atom.

Case 2: Exactly two of the $|V_{i}|$ values are equal to three, while all remaining values are equal to two. Without loss of generality, we assume $|V_{i}|=2$ for all $i\in[1,N-3]$, and $|V_{N-2}|=|V_{N-1}|=3$. Then for a generator $g\in G_{0}$, there exist $a,b,c\in[1,n]$ such that $ag+bg+cg=0$ and $ag,bg,cg\in G_{0}$, that is, $\textup{gcd}(abc,n)=1$ and $n\hskip 2.84544pt|\hskip 2.84544pta+b+c$. But since $n$ is even while $a+b+c$ is an odd number, this is not possible.

2. Assume that $n+1\notin\mathbb{P}$. Then there exist $a,b\geq 3$ such that $n+1=ab$. We consider the zero-sum sequence

[TABLE]

which has the property that $\min\mathsf{L}((-A^{\prime})A^{\prime})+1\in\mathsf{L}((-A^{\prime})A^{\prime})$. Observe that the zero-sum sequence $(-A^{\prime})A^{\prime}$ has maximal elasticity. Thus, for any $k\in\mathbb{N}$ we have

[TABLE]

On the other hand, Proposition 2.2 provides that there exists $M>0$ such that every $L\in\mathcal{L}(G)$ is an AAMP with some difference $d$ and bound $M$. Let $l=M+\max\Delta(G)$, and choose $A^{*}=((-A^{\prime})A^{\prime})^{l}$. Then we have that $\mathsf{L}(A^{*})$ is an AAMP with bound $M$ and $[\min\mathsf{L}(A^{*}),\textup{ min }\mathsf{L}(A^{*})+l]\subset\mathsf{L}(A^{*})$. Therefore $\mathsf{L}(A^{*})\cap[\min\mathsf{L}(A^{*}),\hskip 2.84544pt\max\mathsf{L}(A^{*})-M]$ forms an interval. Let $A\in\mathcal{B}(G)$ with maximal elasticity. Then $\mathsf{L}(A^{*}A)$ has elasticity $\mathsf{D}(G)/2$ by Lemma 3.2. Again, since $\mathsf{L}(A^{*}A)$ is an AAMP and since

[TABLE]

we infer that $\mathsf{L}(A^{*}A)\cap[\min\mathsf{L}(A^{*}A),\hskip 2.84544pt\max\mathsf{L}(A^{*}A)-M]$ forms an interval. ∎

Remark 5.3**.**

Let $G$ be a cyclic group of order $2n$ with $n\geq 2$ and suppose that $2n+1$ is a prime.

1. Let $2n\in\{4,6,10\}$. Then [8, Theorem 3.5] states that $1\notin\Delta_{\rho}(G)$. We claim that this implies that $\min\mathsf{L}(A)+1\notin\mathsf{L}(A)$ for every sequences $A\in\mathcal{B}(G)$ with maximal elasticity. Suppose, seeking a contradiction, that there exists a sequence $A\in\mathcal{B}(G)$ with maximal elasticity such that $\min\mathsf{L}(A)+1\in\mathsf{L}(A)$. By applying the same argument as in the proof of Theorem 1.2, we deduce that there exists $M\in\mathbb{N}$ such that $\mathsf{L}(A^{*}_{k})\cap[\min\mathsf{L}(A^{*}_{k}),\max\mathsf{L}(A^{*}_{k})-M]$ is an interval for all sufficiently large $k$, where $A^{*}_{k}=A^{k}$. This is equivalent to asserting that $1\in\Delta_{\rho}(G)$, which leads to a contradiction.

2. Let $2n\in\{12,16,18,22,30,36\}$ and let $A\in\mathcal{B}(G)$ be such that $\rho(\mathsf{L}(A))=\mathsf{D}(G)/2=n$. Then there exist atoms $U_{1},\ldots,U_{k}\in\mathcal{A}(G)$ with $|U_{i}|=\mathsf{D}(G)$ for all $i\in[1,k]$ such that $A=\prod_{i=1}^{k}U_{i}$. Observe that $\textup{min }\mathsf{L}(A)=k$. Assume that $k+1\in\mathsf{L}(A)$. Then there exist atoms $V_{1},\ldots,V_{k+1}\in\mathcal{A}(G)$ such that $\prod_{i=1}^{k+1}V_{i}=\prod_{i=1}^{k}U_{i}$. By performing suitable cancellations, we obtain $\prod_{i=1}^{k^{\prime}+1}V^{\prime}_{i}=\prod_{i=1}^{k^{\prime}}U^{\prime}_{i}$, where $\{V^{\prime}_{1},\ldots,V^{\prime}_{k^{\prime}+1}\}\subset\{V_{1},\ldots,V_{k+1}\}$ and $\{U^{\prime}_{1},\ldots,U^{\prime}_{k^{\prime}}\}\subset\{U_{1},\ldots,U_{k}\}$. A computer program has shown that the equation

[TABLE]

does not have a solution for $k^{\prime}\leq 3$. (We refer to https://zenodo.org/records/15676314 for further details).

We end with the following open question.

Open Problem 5.4**.**

Let $G$ be a cyclic group of even order and suppose that $|G|+1$ is a prime. Does there exist some $A\in\mathcal{B}(G)$ with $\rho\big(\mathsf{L}(A)\big)=\mathsf{D}(G)/2$ and $1+\min\mathsf{L}(A)\in\mathsf{L}(A)$?

Acknowledgment. I want to thank Alfred Geroldinger and Qinghai Zhong for their support and guidance, which were helpful in the completion of this work. I also thank Benjamin Hackl for his support in verifying the computations mentioned in Remark 5.3.

Conflict of interest statement. The author states that there is no conflict of interest.