\mathbb{T}^{n+d}
\psset{arrows=->>,linewidth=0.2pt, labelsep=1pt
,nodesep=0pt}
\ncline{1,1}{1,2}^{\pi}
\end{psmatrix}\end{aligned}$$ This mapping also demonstrate that any pattern $\Lambda$ in the hull $\mathbb{X}$ of a model set $\Lambda _0$ is in also a model set, such that if it admits $[w,t]_\Sigma \in \mathbb{R}^{n+d}\diagup _{\Sigma }=\mathbb{T}^{n+d}$ as image then $\Lambda$ is determined, as model set, by the region $W+w$ in $\mathbb{R}^n$, next translated by the vector $t$ in $\mathbb{R}^d$. This is described in better details in first section of the main text.

The first step in determining the Ellis semigroup $E(\mathbb{X},\mathbb{R}^d)$ is to describe it as a *suspension* of another (simplier) semigroup (see section $3$). To that end we let $\Gamma$ be the subgroup of $\mathbb{R}^d$ obtained as orthogonal projection of the lattice $\Sigma$ used to construct $\Lambda_0$ as a model set. $\Gamma$ is not a lattice of $\mathbb{R}^d$ unless $\Lambda_0$ itself is a lattice, and will often be dense in $\mathbb{R}^d$, although it is always finitely generated. We now consider the collection $\Xi ^\Gamma$ of point patterns in $\mathbb{X}$ which are contained, as subsets of $\mathbb{R}^d$, into $\Gamma$. This subset of $\mathbb{X}$ remains stable under the action of any vector of $\mathbb{R}^d$ which lies in $\Gamma$, and when endowed with a suitable topology it gives 