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| {{{#!latex2 \noindent {\bf Definition (Linear Set)} Let $N$ be the set of nonnegative integers and $k$ be a positive integer. A set $S\subseteq N^{k}$ is a \emph{linear set} if $\exists v_{0},v_{1},...,v_{t}$ in $N^{k}$ such that \[ S=\left\{ v~|~v=v_{0}+a_{1}v_{1}+...+a_{t}v_{t},a_{i}\in N\right\} \] 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 $S$. \bigskip |
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| \noindent {\bf Definition (Semilinear Set)} A set $S\subseteq N^{k}$ 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 }$N^{k}$\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. }}} |
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| The previous definition is from: [http://www.eecs.wsu.edu/~zdang/papers/catalytic.pdf Catalytic P Systems, Semilinear Sets, and Vector Addition Systems] | The definition are from: [http://www.eecs.wsu.edu/~zdang/papers/catalytic.pdf Catalytic P Systems, Semilinear Sets, and Vector Addition Systems] |
SemiLinearSets
See the definition for linear and semilinear sets in the following: attachment:SemiLinearSets.pdf
The old tex file is attached here: attachment:SemiLinearSets.tex
\noindent {\bf Definition (Linear Set)} Let $N$ be the set of nonnegative integers and $k$ be a positive
integer. A set $S\subseteq N^{k}$ is a \emph{linear set} if
$\exists v_{0},v_{1},...,v_{t}$ in $N^{k}$ such that
\[
S=\left\{ v~|~v=v_{0}+a_{1}v_{1}+...+a_{t}v_{t},a_{i}\in N\right\}
\]
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 $S$.
\bigskip
\noindent {\bf Definition (Semilinear Set)} A set $S\subseteq N^{k}$ 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 }$N^{k}$\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.The definition are from: [http://www.eecs.wsu.edu/~zdang/papers/catalytic.pdf Catalytic P Systems, Semilinear Sets, and Vector Addition Systems]