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See the definition for linear and semilinear sets in the following: attachment:SemiLinearSets.pdf See the definition for linear and semilinear sets in the following: [[attachment:SemiLinearSets.pdf]]
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The old tex file is attached here: attachment:SemiLinearSets.tex The old tex file is attached here: [[attachment:SemiLinearSets.tex]]
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{{{#!latex
\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		 '''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 ''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 (referred to as the ''constant vector'') and (referred to as the ''periods'') are called the ''generators'' of the linear set .

'''Definition (Semilinear Set)''' A set is ''semilinear'' if it is a finite union of linear sets. is a trivial semilinear set where the set of generators is empty. ''Every finite subset of '' ''is semilinear'' - it is a finite union of linear sets whose generators are constant vectors. Clearly, ''semilinear'' sets are closed under union and projection. It is also know that semilinear sets are closed under
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}}}

SemiLinearSets

See the definition for linear and semilinear sets in the following: SemiLinearSets.pdf

The old tex file is attached here: SemiLinearSets.tex

Definition (Linear Set) Let be the set of nonnegative integers and be a positive integer. A set is a ''linear set'' if in such that

$

The vector (referred to as the constant vector) and (referred to as the periods) are called the generators of the linear set .

Definition (Semilinear Set) A set is ''semilinear'' if it is a finite union of linear sets. is a trivial semilinear set where the set of generators is empty. ''Every finite subset of '' is semilinear - it is a finite union of linear sets whose generators are constant vectors. Clearly, 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]

SemiLinearSets (last edited 2020-01-26 21:07:54 by scot)