to the analysis of the relation between discrete problems of social choice and their topological structure. It provides a bridge between a geometrical representation and a topological one of a social choice problem to create a more general framework in which the topological space is manipulable through object construction.

A local study is strictly connected to the geometric structure of the hyperplane arrangement and to the “local” structure of the graph, while global properties depend also on the whole graph. Therefore, in the search for global properties also combinatorial and computational problems arise.

In this paper we tie the model described in [@MarengoSette] to tournament theory. This new link allows us to get results and opens new problems. Tournaments are relevant in many fields of science, so they have been greatly studied by many mathematicians, between 1940 and 1970, and almost everything has been done. We translate the voting and social choice problems, arising in [@MarengoSette], into new tournament-theory problems interesting for mathematicians too.

Moreover, modularity plays a fundamental role in many Natural Complex Systems [@Modularity] and hence we believe that this model can be applied to other fields of science, e.g. genetics (see Stadler [@Stadler]). We plan to go 