paths from $a_0$ to $a_1$, and by $\mathrm{P}(a_0)$ the set of closed paths with endpoint $a_0$. ${\mathcal{P}}$ is said to be *pathwise connected* whenever for any pair $a_0,a_1$ of 0-simplices there exists a path $p\in \mathrm{P}(a_0,a_1)$. The *support* of the path is the collection $|p|{\equiv}\{|b_i| \ | \ i=1,\ldots,n\}$, and we will write $|p|\subseteq P$ if $P$ is a subset of ${\mathcal{P}}$ with $|b_i|\in P$ for any $i$. Furthermore, with an abuse of notation, we will write $|p|\subseteq{\mathcal{O}}$ if ${\mathcal{O}}\in{\mathcal{P}}$ with $|b_i|\leq {\mathcal{O}}$ for any $i$.

![$b$ is a 1-simplex, $c$ is a 2-simplex, $p=\{b_3,b_2,b_1\}$ is a path. The symbol $\delta$ stands for $\partial$.](simplex.eps)

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**Causal disjointness and net of local algebras -** Given a poset ${\mathcal{P}}$, a *causal disjointness relation* on ${\mathcal{P}}$ is a symmetric binary relation $\perp$ on ${\mathcal{P}}$ satisfying the following properties: $$\label{Aa:0}
\begin{array}{ll}
\mbox{(i)} &  {\mathcal{O}}_1\in{\mathcal{P}}\ \Rightarrow \  \exists{\mathcal{O}}_2\in{\mathcal{P}}\mbox{ such that }  \ {\mathcal{O}}_1\perp {\mathcal{O}}_2;\\
\mbox{(ii)} & {\mathcal{O}}_1\leq {\mathcal{O}}_2  \mbox{ and }
  {\mathcal{O}}_2\perp {\mathcal{O}}_3 \ \Rightarrow \ {\mathcal{O}}_1\perp {\mathcal{O}}_3;\\
\end{array}$$ Given a subset $P\subseteq{\mathcal{P}}$, the *causal complement* of $P$ is the subset $P^\perp$ of ${\mathcal{P}}$ defined as $$P^\perp {\equiv}\{{\mathcal{O}}\in{\mathcal{P}}\ | \ {\mathcal{O}}\perp{\mathcal{O}}_1, \ \forall {\mathcal{O}}_1\in P\}.$$ Note that if $P_1\subseteq P$, then $P^\perp\subseteq P^\perp_1$. Now, 