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\begin{document}


\section{Linear and Semilinear Set Definitions:}

\begin{definition}
Let $\mathbb{N}$ be the set of nonnegative integers and $k$ be a positive
integer. A set $S\subseteq \mathbb{N}^{k}$ is a \emph{linear set} if $%
\exists v_{0},v_{1},...,v_{t}$ in $\mathbb{N}^{k}$ such that 
\[
S=\left\{ v|v=v_{0}+a_{1}v_{1}+...+a_{t}v_{t},a_{i}\in \mathbb{N}\right\} 
\]%
The vector $v_{0}$ (referred to as the \emph{constant vector}) and $%
v_{1},v_{2},...,v_{t}$ (referred to as the \emph{periods}) are called the 
\emph{generators} of the linear set $S$.
\end{definition}

\begin{definition}
A set $S\subseteq \mathbb{N}^{k}$ is \emph{semilinear} if it is a finite
union of linear sets. $\emptyset $ is a trivial semilinear set where the set
of generators is empty. \emph{Every finite subset of }$\mathbb{N}^{k}$\emph{%
\ is semilinear} - it is a finite union of linear sets whose generators are
constant vectors. Clearly, \emph{semilinear} sets are closed under union and
projection. It is also know that semilinear sets are closed under
intersection and complementation.
\end{definition}

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