problem, in social choice theory it is a customary convention to drop the transitivity request.

Mathematicians and economists studied this problem during the last 50 years with different approaches. For example, tournament (and, in general, graph) theory turned out to be strictly connected to voting and social choice problems, since Landau started to study this subject [@LandauI; @LandauII; @LandauIII]. In the works of Eckmann [@ecman], Eckmann, Ganea and Hilton [@ecmannb], and Weinberger [@wein] there has been an “unexpected application of algebraic topology to a different field of intellectual enterprise,”[^3] i.e. the social choice theory. Topology is also used to study social choice problems as, for instance, Chichilnisky [@cici1; @cici2; @cici3] and Baryshnikov [@bary] have done. Very recently, Saari [@saari94; @Saarib] used geometry to analyze the matter of voting. Moreover, Terao [@terao] introduced an admissible map of chambers of a real central arrangement which is a generalization of a social welfare function.

Social choice theory usually assumes that agents are faced with a set of exogenously given and mutually exclusive alternatives. These alternatives are “simple”, in the sense that are one-dimensional objects or, even when they are multidimensional, they are simply points in some portion of the homogeneous $\mathbb{R}^n$ space and 