attachment:SemiLinearSets.tex of SemiLinearSets

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   8 %TCIDATA{Created=Monday, February 07, 2005 13:43:30}
   9 %TCIDATA{LastRevised=Monday, February 07, 2005 13:58:44}
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  38 \begin{document}
  39 
  40 
  41 \section{Linear and Semilinear Set Definitions:}
  42 
  43 \begin{definition}
  44 Let $\mathbb{N}$ be the set of nonnegative integers and $k$ be a positive
  45 integer. A set $S\subseteq \mathbb{N}^{k}$ is a \emph{linear set} if $%
  46 \exists v_{0},v_{1},...,v_{t}$ in $\mathbb{N}^{k}$ such that 
  47 \[
  48 S=\left\{ v|v=v_{0}+a_{1}v_{1}+...+a_{t}v_{t},a_{i}\in \mathbb{N}\right\} 
  49 \]%
  50 The vector $v_{0}$ (referred to as the \emph{constant vector}) and $%
  51 v_{1},v_{2},...,v_{t}$ (referred to as the \emph{periods}) are called the 
  52 \emph{generators} of the linear set $S$.
  53 \end{definition}
  54 
  55 \begin{definition}
  56 A set $S\subseteq \mathbb{N}^{k}$ is \emph{semilinear} if it is a finite
  57 union of linear sets. $\emptyset $ is a trivial semilinear set where the set
  58 of generators is empty. \emph{Every finite subset of }$\mathbb{N}^{k}$\emph{%
  59 \ is semilinear} - it is a finite union of linear sets whose generators are
  60 constant vectors. Clearly, \emph{semilinear} sets are closed under union and
  61 projection. It is also know that semilinear sets are closed under
  62 intersection and complementation.
  63 \end{definition}
  64 
  65 \end{document}