\; \; & \; \;  \Gamma 
\psset{arrows=->>,linewidth=0.4pt, labelsep=1.5pt
,nodesep=0pt}
\ncline{1,2}{1,3}
\psset{arrows=<->,linewidth=0.4pt, labelsep=1.5pt
,nodesep=0pt}
\ncline{3,2}{3,3}
\psset{arrows=<<-,linewidth=0.4pt, labelsep=1.5pt
,nodesep=0pt}
\ncline{1,1}{1,2}
\ncline{3,1}{3,2}
\end{psmatrix}  \end{aligned}$$ where $\mathsf{H}$ is a locally compact Abelian group, with:\
- $\Sigma$ is a countable lattice of $\mathsf{H}\times\mathbb{R}^{d}$, that is, a countable discrete and co-compact subgroup.\
- the canonical projection $\pi _{\mathbb{R}^d}$ onto $\mathbb{R}^d$ is bijective from $\Sigma$ onto its image $\Gamma $.\
- the image $\Gamma ^*$ of $\Sigma$ under the canonical projection $\pi _{\mathsf{H}}$ is a dense subgroup of $\mathsf{H}$.\
Hence such an environnement consists of an Euclidean space $\mathbb{R}^d$ embbedded into $\mathsf{H}\times\mathbb{R}^{d}$ in an ’irrationnal position’ with respect to the lattice $\Sigma$. There is a well established formalism for these different ingredients: the space $\mathbb{R}^d$ is often called the *physical space* whereas the space $\mathsf{H}$ is called the *internal space*. Moreover the morphism $\Gamma \longrightarrow \mathsf{H}$ which maps any $\gamma $ onto $\gamma ^*:= \pi _{\mathsf{H}}(\pi _{\mathbb{R}^d}^{-1}(\gamma ))\in \Gamma ^*$ is the *\*-map* of the cut $\&$ project scheme, whose graph is the lattice $\Sigma$. We will say that a cut $\&$ project scheme is *real* whenever the internal space $\mathsf{H}$ is a finite dimentionnal real vector space $\mathbb{R}^n$.

We shall in addition consider a certain type 