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\begin{document}
\vspace*{-60pt} 
\centerline{\smalltt INTEGERS: 
 \smallrm ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 5(1) 
(2005), \#A27} 
\vskip 50pt

\begin{center}
{\bf A CHARACTERIZATION OF MINIMAL ZERO-SEQUENCES OF INDEX ONE IN
FINITE CYCLIC GROUPS} \vskip 20pt {\bf Scott T.
Chapman\footnote{Part of this work was completed while the first
author was on an Academic Leave granted by the Trinity University
Faculty Development
Committee.}}\\
{\smallit Trinity University, Department of Mathematics, One Trinity Place, San Antonio, TX 78212-7200, USA}\\
{\tt schapman@trinity.edu}\\  \vskip 10pt
{\bf William W. Smith}\\
{\smallit The University of North Carolina at Chapel Hill, Department of Mathematics, Phillips Hall,
Chapel Hill, NC  27599-3250, USA}\\
{\tt wwsmith@email.unc.edu}\\
\end{center}
\vskip 30pt \centerline{\smallit Received: 4/27/05,
Revised: 10/26/05, Accepted: 11/16/05,
 Published: 11/29/05
} \vskip 30pt

\centerline{\bf Abstract}


\noindent Let $G\cong \mathbb{Z}_n$ where $n$ is a positive integer.
A finite sequence $S=\{g_1, \ldots ,g_k\}$ of not necessarily
distinct elements from $G$ for which $\sum_{i=1}^k g_i = 0$ is
called a zero-sequence.  If a zero-sequence $S$ contains no proper
subzero-sequence, then it is called a \textit{minimal
zero-sequence}.  The notion of the \textit{index} of a minimal
zero-sequence (see Definition \ref{basic}) in $\Z_n$ has been
recently addressed in the mathematical literature.  In this note, we
offer a characterization of minimal zero-sequences in $\Z_n$ with
index 1.

\pagestyle{myheadings} \markright{\smalltt INTEGERS: \smallrm
ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY \smalltt 5(1) (2005),
\#A27\hfill}

\thispagestyle{empty} \baselineskip=15pt \vskip 30pt


%\maketitle


%%%%%%%%%%%%%%%%%%%%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $G$ be an additive abelian group and $S=\{g_1, \ldots ,g_k\}$ a
finite sequence of not necessarily distinct elements from $G$.
Denote by $\mid S\mid=k$ the number of elements in $S$ (or the
\textit{length} of $S$) and let $\mathrm{supp}(S)=\{g\,\mid\, g\in
G\mbox{ with }g=g_i\mbox{ for some }i\}$ be the \textit{support} of
$S$. Various properties of the sequence $S$ have been considered
over the last several years in the mathematical literature.  Some of
these properties are among the following.
\begin{enumerate}
\item $S$ is \textit{zero-free} if $\sum_{i\in \mathscr{I}} g_i \neq 0$ for
any nonempty subset $\mathscr{I}\subseteq \{1,2,\ldots ,k\}$.
\item $S$ is a \textit{zero-sequence} if $\sum_{i=1}^k g_i =0$.
\item A zero-sequence $S$ is a \textit{minimal zero-sequence (or MZS)} if for every
nonempty $\mathscr{I}\subsetneq \{1,2,\ldots ,k\}$, the sequence
$\{g_i\}_{i\in \mathscr{I}}$ is zero-free.
\item A zero-sequence $S$ which is not an MZS is an \textit{almost minimal zero-sequence (or AMZS)} if for every
nonempty $\mathscr{I}\subsetneq \{1,2,\ldots ,k\}$ where the
sequence $\{g_i\}_{i\in \mathscr{I}}$ is a zero-sequence, then
$\{g_i\}_{i\in \mathscr{I}}$ is a minimal zero-sequence.
%\item $S$ is a \textit{pre-almost minimal zero-sequence (or PAMZS)} if $S$ is
%not a zero-sequence, there is a proper subset $\mathscr{I}\subset
%\{1,2,\ldots ,k\}$ such that $\{g_i\}_{i\in \mathscr{I}}$ is a
%zero-sequence, and for every $\mathscr{I}$ such that $\{g_i\}_{i\in
%\mathscr{I}}$ is a zero-sequence, then $\{g_i\}_{i\in \mathscr{I}}$
%is a minimal zero-sequence.
\end{enumerate}

\noindent In this article, we will consider a property of minimal
zero-sequences in finite cyclic groups which was introduced in the
literature in \cite{CFS} and consequently considered in greater
detail in \cite{Gao} and \cite{vadim}.  Some notation will be
necessary before giving a formal statement describing this property.
Since the ordering of the elements in a sequence $S$ does not
matter, we will view sequences as elements of $\mathcal{F}(G)$, the
free abelian monoid on $G$.  Hence, we write
\[
S=\prod_{g\in G} g^{n_g}
\]
where only finitely many of the $n_g$ are not zero.

Our goal is to offer a characterization of index 1 minimal
zero-sequences in $\Z_n$.  This will be done in terms of almost
minimal zero-sequences (see \cite[Chapter 5]{MF} for more
information on AMZSs). We will find the language of block monoids
useful for expressing and applying some of our arguments. For a
finite abelian group $G$, let $\mathcal{B}(G)$ represent the set of
elements in $\mathcal{F}(G)$ which are zero-sequences. Further, let
$\mathcal{U}(G)$ be the subset of $\mathcal{B}(G)$ consisting of the
minimal zero-sequences of $G$. If $S_1=\prod_{g\in G} g^{m_g}$ and
$S_2=\prod_{g\in G} g^{s_g}$ are in $\mathcal{B}(G)$, then
$\mathcal{B}(G)$ can be considered as a commutative cancellative
monoid under the operation
\[
S_1\ast S_2=\prod_{g\in G} g^{m_g+s_g}
\]
and is commonly called a \textit{block monoid} (more information on
block monoids can be found in \cite{GHK}). The irreducible elements
of $\mathcal{B}(G)$ are merely the elements of $\mathcal{U}(G)$ and
the \textit{empty block} (i.e., $S=\emptyset$) acts as the identity
of $\mathcal{B}(G)$.  An interpretation of an almost minimal
zero-sequence in terms of block monoids can be stated as follows:
$B\in \mathcal{B}(G)$ is an almost minimal zero-sequence if and only
if $B=B_1\cdots B_t$ with each $B_i$ in $\mathcal{U}(G)$ implies
that $t=2$.


\begin{definition}\label{basic}
Let $G$ be an abelian group. \begin{enumerate} \item[(1)] Let $g \in
G$ be a non-zero element with $\mathrm{ord}(g) = n > 1$. For a
sequence $S = (n_1g)\cdots (n_lg)$, where $l \in \mathbb{N}_0$ and
$n_1, \ldots , n_l \in [1, n]$, we define \[ \|S\|_g = \frac{n_1 +
\ldots + n_l}{n}\] to be the $g$-norm of $S$.  If $S=\emptyset$,
then set $\S=0$.
\item[(2)] Let $S$ be
a zero-sum sequence for which $\langle \mathrm{supp}(S)\rangle
\subset G$ is a nontrivial finite cyclic group. Then we call \[
\mathrm{index}(S) = \min\{\;\|S\|_g\; |\; g \in G \mbox{ with }
\langle \mathrm{supp}(S)\rangle = \langle g\rangle \} \in
\mathbb{N}_0\] the \textit{index} of S.
\end{enumerate}
\end{definition}


%\begin{definition}\label{basic}
%Let $\Z_n$ and $S=\{g_1, \ldots ,g_k\}$ be as above.  The integer
%\[
%\mathrm{index}(S)=\min \{\tau (\varphi(S))\,\mid\,
%\varphi\in\mathrm{Aut}(\Z_n)\},
%\]
%is known as the \textit{index} of $S$ in $\Z_n$.
%\end{definition}

\noindent Notice that the index of a sequence $S$ depends only on
$S$ and not the choice of the cyclic group $G$ which contains
$\mathrm{supp}(S)$.  Theorem 2 of \cite{CFS} indicates that as $n$
increases, there exist minimal zero-sequences of $\mathbb{Z}_n$ of
arbitrarily high index. The papers \cite{vadim} and \cite{Gao} have
both shown that for a fixed value of $n$, ``long'' minimal
zero-sequences must have index 1.  In particular, \cite[Section
2]{Gao} shows for $n \geq 10$ that a minimal zero-sequence $S$ in
$\Z_n$ with $\mid S\mid\,
> \frac{2n}{3}$ must have index 1.


When restricting our attention to cyclic groups, the $g$-norm of an
zero-sequence can be used to draw some helpful conclusions.  We
determine some basic properties of the $g$-norm in the next
proposition. %\pagebreak
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{type}
Let $G$ be an abelian group, $g\in G$ a nonzero element and $S$,
$T\in \mathcal{B}(\langle g\rangle)$.
\begin{enumerate}
\item[(1)] $\| \cdot \|_g: \mathcal{B}(\langle g\rangle)\rightarrow
\mathbb{N}_0$ is a monoid homomorphism (i.e., $\|S\ast T\|_g = \|
S\|_g +\|T\|_g$).
\item[(2)] $\S = 0$ if and only if $S=\emptyset$.
\item[(3)] $\| 0\|_g=1$.
\item[(4)] If $\| S\|_g=1$, then $S$ is a MZS.
\item[(5)] If $\S =2$, then $S$ is an AMZS.
\end{enumerate}
\end{proposition}

\begin{proof}
The proofs of (1)-(3) are clear.  For (4), if $S=S_1\ast S_2$ with
$S_1$ and $S_2$ in $\mathcal{B}(\langle g\rangle)$, then $1=\S =
\|S_1\|_g+\|S_2\|_g \geq 2$, a contradiction. For (5), if $S$ is
neither an MZS or an AMZS, then $S=S_1\ast S_2\ast S_3$ for $S_1$,
$S_2$ and $S_3$ in $\mathcal{B}(\langle g\rangle)$.  The argument
now follows as in (4).
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We note that index one MZSs satisfy several interesting properties.
Two of these properties follow. Recall that if $S=\prod_{g\in G}
g^{n_g}$ is an MZS in $\Z_n$, then the \textit{cross number of} $S$
is defined as $\Bbbk (S)=\sum_{g\in G}
\frac{n_{g}}{\mathrm{ord}(g)}$ where $\mathrm{ord}(g)$ represents
the order of $g$ in $G$ (more information on the cross number can be
found in \cite{C}).  For $S\in \mathcal{B}(G)$ consider these
properties.


\textbf{(P1)} $S\ast S$ is an AMZS in $\Z_n$.


\textbf{(P2)} $\Bbbk (S)\leq 1$.


\noindent It follows directly from Proposition \ref{type} that
$S=\prod_{g\in G} g^{n_g}$ an MZS in $\Z_n$ with $\|S\|_g=1$
satisfies \textbf{(P1)}. That $\S=1$ implies $\Bbbk(S)\leq 1$ can be
seen as follows.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Suppose $S = (n_1g)\cdots (n_lg)$ is written as in Definition
\ref{basic} with $n=\mathrm{ord}(g)$.  Then
\[ \Bbbk(S)=
\sum_{i=1}^l \frac{1}{\mathrm{ord}(n_ig)}= \sum_{i=1}^l
\frac{1}{\frac{n}{\gcd {(n_i, n)}}} \leq \sum_{i=1}^k
\frac{n_i}{n}=\S = 1.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Hence we have the following.
\begin{proposition}
If $S$ is a MZS of $\Z_n$ with $\mathrm{index}(S)=1$, then $S$
satisfies properties $\mathrm{\mathbf{(P1)}}$ and
$\mathrm{\mathbf{(P2)}}$.
\end{proposition}

\begin{example}
Properties \textbf{(P1)} and \textbf{(P2)} do not characterize MZSs
of index 1. Notice that all of the index 2 MZSs in \cite{CFS} do not
satisfy \textbf{(P1)} (see in particular the proof of \cite[Theorem
2]{CFS}).  A slight modification of the construction used in
\cite{CFS} yields the following example.  Let $G=\Z_{23}$ and set
$S=2\cdot 7\cdot 9\cdot 11\cdot 17$. It is a routine calculation to
check the 22 possible values of $\S$ and determine that
$\mathrm{index}(S) = 2$. Since $\Bbbk(S)\leq 1$, $S$ satisfies
\textbf{(P2)}.  For considering property \textbf{(P1)}, note that
$\|S\|_1 = 2$ and so $\| S\ast S\|_1 = 4$. To establish that $S\ast
S$ is an AMZS, one needs only observe that if it were not, then
$S\ast S = A\ast B\ast C$ for some zero sequences $A$, $B$, and $C$.
It follows that this has to be done (with the proper choice of $g$)
so that $\|A\|_g=\|B\|_g = 1$ and $\|C\|_g=2$. The key then to
observing such a decomposition is impossible is to note that
$7^2\cdot 9$ is the only subsequence of $S\ast S$ that sums to $23$.
\end{example}

While \textbf{(P1)} and \textbf{(P2)} do not offer the
characterization of index 1 MZSs we desire, a relatively simple
condition involving the AMZS's which contain an MZS $S$ does provide
a characterization.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% New Version of Main Theorem with Referee's Proof %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{theorem}
Let $G$ be an abelian group and $S$ a minimal zero-sequence over $G$
such that $\mathrm{supp}(S)$ generates a cyclic group $H$ of order
$n \geq 2$. Then the following statements are equivalent:
\begin{enumerate}
\item[(a)] There exists some AMZS $A \in \mathcal{F}(H)$ of length $|A| = |S| + n$
where $S$ divides $A$ in $\mathcal{B}(G)$. \item[(b)] There exists
some $g \in H$ such that $g^nS$ is an AMZS.
\item[(c)] $\mathrm{index}(S) = 1$.
\end{enumerate}
\end{theorem}

\begin{proof}
(a) $\Rightarrow$ (b) Let $A = ST$ be an AMZS of length $|S| + n$
for some $T \in \mathcal{F}(H)$. Then $T$ is a minimal zero-sum
sequence of length $n$. Thus, for example by \cite[Lemma 13]{G},
there exists some $g \in H$ such that $T = g^n$.

(b) $\Rightarrow$ (c) Let $g \in H$ and $A = g^nS$ an AMZS. Then
there are $m_1, \ldots ,m_l \in [1, n - 1]$ with $m_1 \leq \ldots
\leq m_l$ such that $S = \prod_{i=1}^l (m_ig)$. We assert that $\|
S\|_g= 1$.  Assume to the contrary that
\[
\| S\|_g = \frac{m_1 + \ldots + m_l}{n} = k \mbox{ with } k \geq 2.
\]
Since $S$ is a minimal zero-sum sequence, there exist $u$, $v \in
[1, l - 1]$ such that
\[
(k - 2)n < m_1 + \ldots + m_u < (k - 1)n < m_1 + \ldots + m_u +
m_{u+1}
\]
and
\[
m_{u+1} + \ldots + m_v < n < m_{u+1} + \ldots + m_v + m_{v+1}.
\]
We set \[ r = (k - 1)n - (m_1 + \ldots + m_u),\] \[s = n - (m_{u+1}
+ \ldots + m_v)
\]
and we define \[ N_1 = g^r \prod_{i=1}^u (m_ig),\; N_2 = g^s
\prod_{i=u+1}^v (m_ig) \mbox{ and } N_3 = g^{n-(r+s)}
\prod_{i=v+1}^l (m_ig). \]
 By construction, $N_1$, $N_2$ and $N_3$ are
zero-sum sequences with $A = N_1N_2N_3$, a contradiction to the fact
that $A$ is an AMZS.

(c) $\Rightarrow$ (a) Let $g \in H$ such that $\| S\|_g= 1$. We set
$A = g^nS$, and since $\| A\|_g = 2$, it follows that $A$ is an
AMZS.
\end{proof}

\noindent
\section*{\normalsize{Acknowledgement}}


The authors would like to thank the referee for many helpful
suggestions.




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\bibitem{CFS} S. T. Chapman, M. Freeze, and W. W. Smith, Minimal
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\bibitem{MF} M. Freeze, Lengths of Factorizations in Dedekind
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\end{thebibliography}


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