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\begin{slide}[7.3in,6in]
%\Huge
\begin{center} \large The conductor of submonoids of $\mathbb{N}^n$. \end{center}

\smallskip
\normalsize
\begin{center} Les Reid and Leslie G. Roberts \end{center}

Let $\mathbb{N}=\{0,1,2,3,\ldots\}$. The integers that can be written
as the sum of multiples of 3 and 5 are

3

5

6=3+3

8=3+5

9=3+3+3

10=5+5

\noindent and every larger integer (since we now have three consecutive
integers). However 7 cannot be so expressed).

%\vfill
%\break
%\newslide
\end{slide}
We can imagine that we have an unlimited supply of postage stamps of
denominations 3 cents and 5 cents. Then we can make up any postal rate
greater than or equal to 8 cents, but not 7 cents. (Once we get three
consecutive integers, here 8,9,10,   in the monoid we get all larger
integers by adding multiples of 3.

\begin{slide}[7.3in,6in]
More generally, given relatively prime $a,b\in \mathbb{N}$, then every
integer greater than or equal to $c:=(a-1)(b-1)=ab-a-b+1$ can be expressed as a sum of multiples of
$a$ and $b$, but $ab-a-b$ cannot.

Let $\mathcal{S}$ be the set of all integers of the form $Aa+Bb\ (A,B\in\mathbb{N})$.
Then $\mathcal{C}$ is closed under addition and contains 0, i.e. is a submonoid of
$\mathbb{N}$. Let $\mathcal{C}=\{x\in \mathbb{N}|x\geq c\}$. Then
$\mathcal{C}=\{x\in\mathbb{N}|x+\mathbb{N}\subseteq\mathcal{S}\}$,
and $\mathcal{C}$ (or abusively $c$) is called the conductor of $\mathcal{S}$
in $\mathbb{N}$.
%\newslide
\end{slide}
That should be positive integer multiples!

In the example of the first slide $c=2\cdot 4=8$.

\begin{slide}[7.3in,6in]
\noindent {\bf Problem:} Generalize the above to submonoids of $\mathbb{N}^n$.

\medskip
\noindent {\bf Definition:} If $\mathcal{S}\subseteq\overline{\mathcal{S}}$
are submonoids of $\mathbb{N}^n$ then the conductor $\mathcal{C}$ of
$\mathcal{S}$ in $\overline{\mathcal{S}}$ is defined by
\[\mathcal{C}=\{x\in\overline{\mathcal{S}}|x+\overline{\mathcal{S}}\subseteq\mathcal{S}\}.\]

\medskip
Note that $0\in\overline{\mathcal{S}}$ so if $x\in\mathcal{C}$ then
$x+0\in{\mathcal{S}}$ i.e. $\mathcal{C}\subseteq\mathcal{S}$.

\medskip
{\bf Hypotheses:} ${\mathcal S}$ is generated as a submonoid of
$\mathbb{N}^n$ by $n+1$ elements $\{\bf{v}_1,\bf{v}_2,\ldots,\bf{v}_{n+1}\}$, any
$n$ of which are linearly independent.
Furthermore we assume that $\{\bf{v}_1,\bf{v}_2,\ldots,\bf{v}_{n+1}\}$ generates
$\mathbb{Z}^n$ as an abelian group and
that $\overline{\mathcal{S}}=\mathbb{Z}^n\cap\sigma$ where $\sigma$ is the
cone spanned by $\{\bf{v}_1,\bf{v}_2,\ldots,\bf{v}_{n+1}\}$ (so that
$\overline{\mathcal{S}}$ is the normalization of $\mathcal{S}$ in $\mathbb{N}^n$).
Note that
\[\overline{\mathcal{S}}=\{x\in\mathbb{N}^n|dx\in\mathcal{S} {\rm\ for\ some\ }
d\in\mathbb{N}\backslash 0\}.\]
\end{slide}
How to generalize to $n>1$ is not totally obvious. 
The hypotheses correspond to $a,b>0$ and relatively prime, and seem appropriate.
We can assume without loss of generality  that the ${\bf v}_i$ generate
$\mathbb{Z}^n$, by change of basis.
\begin{slide}[7.3in,6in]
%\newslide
{\bf Theorem:} Under the above hypotheses the conductor $\mathcal{C}$ of
$\mathcal{S}$ in $\overline{\mathcal{S}}$ is non-empty and 
\[\mathcal{C} = \lambda+(\overset{\circ}{\sigma}\cap\overline{\mathcal{S}}).\]
where $\lambda=(\sum_{i=1}^{n+1}d_i{\bf v}_i)/2-\sum_{i=1}^{n+1}{\bf v}_i$,
$\overset{\circ}{\sigma}$ is the interior of $\sigma$ and $d_i$ is the order
of the subgroup of $\mathbb{Z}^n$ generated by
$\{{\bf v}_1,\ldots,{\bf v}_{i-1},{\bf v}_{i+1},\ldots,{\bf v}_n\}$.

{\bf Example:} If $n=1$ and $\mathcal{S}$ is the submonoid of $\mathbb{N}$
generated by relatively prime integers ${\bf v}_1=a$ and ${\bf v}_2=b$
then $d_1=b, d_2=a$, $\lambda=(ba+ba)/2-(a+b)=ab-a-b$, \newline
$\overline{\mathcal{S}}=
\mathbb{N}$ and $\overset{\circ}{\sigma}\cap\mathbb{N}=\mathbb{Z}_+$, yielding the
same conductor as before.
%\newslide
\end{slide}
Part of the theorem is that $\lambda$ is an integer.
The interior of $\sigma$ is in the obvious sense. In the example,
$\sigma=\{x\in\mathbb{R}|x\geq 0\}$ and $\overset{\circ}{\sigma}=
\{x\in\mathbb{R}|x> 0\}$. $\mathbb{Z}_+ = \{1,2,3,\ldots \}$. 
\begin{slide}[7.3in,6in]
{\bf Example:} Let $n=2$,
${\bf v}_1=\left [\begin{array}{c}2\\1\end{array}\right ]$,
${\bf v}_2=\left [\begin{array}{c}0\\2\end{array}\right ]$,
${\bf v}_3=\left [\begin{array}{c}3\\6\end{array}\right ]$.
Then $d_1=\left |\det\left (\begin{array}{cc}0&3\\2&6\end{array}\right )\right |=6$,
$d_2=9,\ d_3=4$. Thus
 \[\lambda = (d_1 {\bf v}_1+d_2{\bf v}_2+d_3{\bf v}_3)/2-{\bf v}_1-{\bf v}_2-{\bf v}_3 =\left [\begin{array}{c}7\\15\end{array}\right ]\]
If
${\bf v}_1=\left [\begin{array}{c}3\\1\end{array}\right ]$,
${\bf v}_2=\left [\begin{array}{c}1\\3\end{array}\right ]$,
${\bf v}_3=\left [\begin{array}{c}1\\2\end{array}\right ]$ then
$\lambda = \left [\begin{array}{c}3\\10\end{array}\right ]$.
\end{slide}
The first example is the one that is in our paper, but is perhaps misleading
because the conductor is principal. The second example was added for this
talk to show that the conductor need not be principal.
 
\begin{slide}[7.3in,6in]
 {\bf Interesting Observations:}
\begin{enumerate}
\item The translation to higher dimensions is not what one first might think
of. The closed interval $x\geq c$ becomes the {\em interior} of a cone. Thus
the closed interval $x\geq c$ should thus be thought of as the {\em interior} of
the interval $x \geq c-1$.
\item The conductor in dimension greater than
one need not be principal (as a $\overline{\mathcal{C}}$ ideal).
\item $\lambda$ is symmetric in the ${\bf v}_i$ even though
the ${\bf v}_i$ play a different geometric role.
\item In dimensions greater than two the cone $\sigma$ need not be
``simplicial''.        
\end{enumerate}
%\newslide
\end{slide}
Observation 3 should read ``play different geometric roles'', e.g. in
$\mathbb{N}^2$ one of the three is in the cone spanned by the other two.
If $n\geq 3$ the geometry is more complicated by the next observation:

\medskip
Simplicial means that $\sigma$ is spanned (over $\mathbb{R}_+$) by
$n$ vectors.
\begin{slide}[7.3in,6in]
{\bf Motivation:} Subrings of $\mathbb{Z}[t_1,\ldots,t_n]$ generated by
``terms'' $x_{\bf M} t^{\bf M},\ x_{\bf M}\in\mathbb{Z},\ {\bf M}=(m_1,\ldots,m_n)$
(where $t^{\bf M}=t^{m_1}t^{m_2}\cdots t^{m_N}$) arose naturally in our study
of weakly subintegral ring extensions. The set of all ${\bf M}$ such that
$x_{\bf M}\neq 0$ is a submonoid of $\mathbb{N}^n$ and an attempt to
describe geometrically all submonoids $\mathcal{T}$ of $\mathbb{N}^n$
with the property that if $x\in\mathbb{N}^n$ and $dn\in \mathcal{T}$ for
$d>>0$ then $n\in \mathcal{T}$ led to a discussion of the conductor of
monoids.

{\bf Reference:} Les Reid and L.Roberts, Monomial Subrings in Arbitrary
Dimension, To appear in the Journal of Algebra. (downloadable from my web page)

\smallskip Our Theorem on conductors 
and the first example for $n=2$ are respectively Theorems 5.2 and Example 5.3 of this paper.

\end{slide}
The referee seemed to think that our conductor theorem was a good one,
but then proceeded to show that it was not needed in our proofs. But it
was allowed to stay in the paper.
\end{document}

