outcomes $x,y\in{X}$.

Let $(x,y) \in {\mathcal{Y}}_{1,\succ}$ be an arc given by a formal sum with coefficients $1$ of arcs that are in ${\mathcal{Y}}_{1,\succ}$. If the social rule is transitive the arc can be deleted, because it can be reconstructed by means of the other arcs.

Saari has greatly contributed to establishing general geometric representations of voting models and voting paradoxes [@saari94; @saari00a; @saari00b]. Salvetti’s complex is a CW-complex in ${\mathbb{C}}^n$, but it has an underlying real structure which is a purely simplicial complex. Moreover, vertices in this complex can be freely chosen inside each chamber. This structure can be used in order to recast and generalize some existing geometric models of voting such as those provided by Saari.

#### Objects schemes

Given a non-empty subset $I\subseteq\{1,\dotsc, n\}$, the [*object*]{} ${\mathcal{A}}_I$ is the subset $${\mathcal{A}}_I=\{H_{i,j}\}_{{i \in I} \atop {0 \leqslant j < m_i-1}}$$ of the arrangement ${\mathcal{A}}_{n,m}$. The cardinality of ${\mathcal{A}}_I$ is called [*size*]{} of the object ${\mathcal{A}}_I$ and is denoted by $|{\mathcal{A}}_I|$. The complement of a set $I$ in $\{1,\dotsc, n\}$ will be denoted (as usual) by $I^c$, hence the complement ${\mathcal{A}}_I^c={\mathcal{A}}_{n,m} \setminus {\mathcal{A}}_I$ of the arrangement ${\mathcal{A}}_I$ in ${\mathcal{A}}_{n,m}$ turns out to equal ${\mathcal{A}}_{I^c}$. The [*object instantiation*]{} $x({\mathcal{A}}_I)$ of 