latter. For the sake of simplicity, we will use the same symbol $x$ for the nodes of ${\mathcal{Y}}_{\succ}$ and $(x,y)$ for its arcs. We decided to use different notations for graphs in regard of the relevance of different theories. Note that the completeness assumption on social rules guarantees that the graph ${\mathcal{Y}}_{\succ}$ is connected.

A cycle $$(x_1,x_2),(x_2,x_3),\dotsc ,(x_h,x_1)$$ in the graph ${\mathcal{Y}}_{\succ}$ corresponds to a cycle [*à la*]{} Condorcet-Arrow [@Condorcet-Arrow], i.e. to the sequence $$x_1 \succ x_2 \succ \dotsb \succ x_h \succ x_1.$$

#### Features

Let $F=\{f_1,\dotsc ,f_n\}$ be a bundle of elements, said [*features*]{}, the $i$-th of which takes $m_i$ values, i.e. $f_i \in \{0,1,2,\dotsc, m_i-1\}$ with $i=1,\dotsc,n$. Denote by $m=(m_1,\dotsc,m_n)$ the multi-index of the numbers of values of the features. From now on, a [*social outcome*]{} (or [*configuration*]{}) will be an $n$-uple $(v_1,\dotsc,v_n)$ of values such that $0\leqslant v_i<m_i$. For the sake of shortness, it will be also denoted by $v_1\dotsm v_n$. The set of all social outcomes will be denoted by ${X}$. The cardinality of ${X}$ is $\prod_{i=1}^n m_i$ and will be denoted by $M$.

#### The hyperplane arrangement

There is a correspondence [@MarengoSette] between the set $X$ of social outcomes and the set ${\mathcal{C}}$ of the 