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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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| {{{#!latex2 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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| 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 $$v_{0}$$ (referred to as the ''constant vector'') and $$v_{1},v_{2},...,v_{t}$$ (referred to as the ''periods'') are called the ''generators'' of the linear set $$S$$. '''Definition (Semilinear Set)''' A set $$S\subseteq N^{k}$$ is ''semilinear'' if it is a finite union of linear sets. $$\emptyset$$ is a trivial semilinear set where the set of generators is empty. ''Every finite subset of '' $$N^{k}$$ ''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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| 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: SemiLinearSets.pdf
The old tex file is attached here: SemiLinearSets.tex
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 $$v_{0}$$ (referred to as the constant vector) and $$v_{1},v_{2},...,v_{t}$$ (referred to as the periods) are called the generators of the linear set $$S$$.
Definition (Semilinear Set) A set $$S\subseteq N^{k}$$ is ''semilinear'' if it is a finite union of linear sets. $$\emptyset$$ is a trivial semilinear set where the set of generators is empty. ''Every finite subset of '' $$N^{k}$$ 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]