We may shortly present the face semigroup $\mathfrak{T}_{W}$ associated with the polytope $W$ in $\mathbb{R}^n$ as follows: The polytope $W$ determines a finite collection of linear hyperplanes $\mathfrak{H}_{W}$ in $\mathbb{R}^n$, namely the ones which are parallel to at least one face of $W$. This collection in turns determine a stratification of $\mathbb{R}^n$ by cones, all being, for each hyperplane $H\in \mathfrak{H}_W$, included into $H$ or integrally part of one of the two possible complementary half spaces. An illustration of this construction is provided in section $7$, where $W$ is there a regular octagon of $\mathbb{R}^2$. Now the face semigroup $\mathfrak{T}_{W}$ is set-theoretically the finite collection of cones resulting from this stratification process, together with a (non-commutative) semigroup product stating that the product $C.C'$ of two cones is the cone where the head of $C$ enters after being translated by small vectors of $C'$. The elements of $\mathfrak{T}_W$ are more conveniently described as “side maps”, which consist of mappings from $\mathfrak{H}_W$ to the three symbols set $\lbrace -, 0,+\rbrace$, giving the relative position of any cone with respect to each hyperplane. This formalism has the advantage to allows for a concise and handy formulation of the product law on this semigroup 