of relatively compact open sets codifying the topological and the causal properties of ${\mathcal{M}}$. Secondly, they take a reference representation $\pi_o$ of $\mathscr{R}_{{\mathcal{K}_{\diamond}}}$ on a Hilbert space ${\mathcal{H}}_o$ such that the net ${\mathscr{A}}_{{\mathcal{K}_{\diamond}}}:{\mathcal{K}_{\diamond}}\ni{\mathcal{O}}\rightarrow {\mathcal{A}}({\mathcal{O}}){\equiv}\pi_o(\mathcal{R}({\mathcal{O}}))''$ satisfies Haag duality and the Borchers property (see Section \[C\]). The reference representation $\pi_o$ plays for the theory the same role that the vacuum representation plays in the case of Minkowski space. Examples of physically meaningful nets of local algebras indexed by regular diamonds have been given in [@Ve]. Finally, the DHR-sectors are singled out from the representations of the net $\mathscr{R}_{{\mathcal{K}_{\diamond}}}$ by generalizing, in a suitable way, the criterion (\[Int:1\]). As in Minkowski space, the physical content of DHR-sectors is contained in the $\mathrm{C}^*-$category ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}_{\diamond}}})}$ of 1-cocycles of ${\mathcal{K}_{\diamond}}$ with values in ${\mathscr{A}}_{{\mathcal{K}_{\diamond}}}$, and when ${\mathcal{K}_{\diamond}}$ is directed under inclusion, there exist a tensor product, a symmetry and conjugated 1-cocycles. The analogy with the theory in the Minkowski space breaks down when the ${\mathcal{K}_{\diamond}}$ is not directed. In this situation only the introduction of a tensor product on ${\mathcal{Z}^1_t({\mathscr{A}}_{{\mathcal{K}_{\diamond}}})}$ and the existence of a symmetry have been achieved in [@GLRV], although the definition of the tensor product is not completely discussed (see below).\
There are 