Baryshnikov presented a remarkable algebraic topology proof of Arrow's impossibility theorem trying to understand the underlying reason behind the numerous proofs of this fundamental result of social choice theory. We present here a novel combinatorial topology approach that does not use advanced mathematics, while giving a geometric intuition of the impossibility. This exposes a remarkable connection with distributed computing techniques.

Starting from the notion of a multidistance, we formalize, through a suitable system of axioms, the concept of an inequality measure defined on a nonempty set with no additional structure implemented a priori. Among inequality measures, apart from multidistances we pay special attention to dispersions, and study their main features. Classical concepts will be generalized to this abstract setting. Multidistances are then revisited, and some new methods to generate them are implemented. A wide spectrum of interdisciplinary applications is outlined in the final section.

Throughout this article, our main idea is to analyze from a theoretical and normative point of view different methods to aggregate individual rankings. To do so, first, we introduce the concept of a general mean on an abstract set. This new concept conciliates the social choice—where well-known impossibility results as the Arrovian ones are encountered—and the decision-making approaches—where the necessity of fusing rankings is unavoidable. Moreover, it gives rise to a reasonable definition of the concept of a ranking fusion function that does indeed satisfy the axioms of a general mean. Then, we will introduce some methods to build ranking fusion functions, paying a special attention to the use of score functions, and pointing out the equivalence between ranking and scoring. To conclude, we prove that any ranking fusion function introduces a partial order on rankings implemented on a finite set of alternatives. Therefore, this allows us to compare rankings and different methods of aggregation, so that in practice, one should look for the maximal elements with respect to such orders defined on rankings.