61 lines
1.5 KiB
TeX
61 lines
1.5 KiB
TeX
\documentclass{article}
|
|
|
|
\usepackage{bbm}
|
|
\usepackage{amsmath}
|
|
|
|
\newcommand{\N}{\mathbbm{N}}
|
|
|
|
\newtheorem{Def}{Definition}
|
|
\newtheorem{The}{Theorem}[Def]
|
|
|
|
\DeclareMathOperator{\Dom}{Dom}
|
|
\newcommand{\Dim}[1]{\tilde{#1}}
|
|
|
|
\begin{document}
|
|
|
|
Note: we assume here that dimensions are cells instead of strings.
|
|
|
|
\begin{Def}[space]
|
|
A ZigZag space is the triplet $Z = (X, c, C)$ where $X$ is the set of
|
|
cells, $c : X \rightarrow \Sigma^*$ is the cell content mapping
|
|
and $C \subset X^3$ is the set of connections, if the following hold:
|
|
%
|
|
\begin{enumerate}
|
|
%
|
|
\item If $(d, e, f) \in C$ and $(d, e', f) \in C$ then $e = e'$.
|
|
%
|
|
\item If $(d, e, f) \in C$ and $(d, e, f') \in C$ then $f = f'$.
|
|
%
|
|
\end{enumerate}
|
|
\end{Def}
|
|
|
|
\begin{Def}[subspace]
|
|
A ZigZag space $Z' = (X', c', C')$ is a subspace of another ZigZag
|
|
space $Z = (X, c, C)$ if the following hold:
|
|
%
|
|
\begin{enumerate}
|
|
%
|
|
\item $X' \subset X$
|
|
%
|
|
\item $c'(x) = c(x)$ for every $x \in X'$.
|
|
%
|
|
\item $C' \subset C$
|
|
%
|
|
\end{enumerate}
|
|
\end{Def}
|
|
|
|
\begin{The}
|
|
Let $Z = (X, c, C)$ be a ZigZag space. For every $d \in X$, there
|
|
exists a maximal $X' \subset X$ and a unique $\Dim{d} : X'
|
|
\rightarrow X$ such that for every $x \in X'$ it holds that $(d, x,
|
|
\Dim{d}(x)) \in C$. Additionally, $\Dim{d}$ is an injection.
|
|
\end{The}
|
|
|
|
\begin{Def}[clone dimension]
|
|
Let $Z = (X, c, C)$ be a ZigZag space. We say that $d \in X$ is a
|
|
clone dimension if, for every $x \in \Dom d$ it holds that $c(x) =
|
|
c(\Dim{d}(x))$.
|
|
\end{Def}
|
|
|
|
\end{document}
|