We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. Inspired by the well-known mathematical structures of quantum gravity and lattice gauge theories in physics and by the application of this framework in economy by other authors, we describe how the same structures arise in pairwise comparisons. First, we provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, and give a full description of elementary properties that extend to our generalized setting. Then, we show a geometric picture where inconsistency in pairwise comparisons can be interpreted as the non-trivial Holonomy on a loop. We continue with probabilistic aspects, again adapting ideas from quantum physics where probabilities modelize uncertainty. We finish by examples where the use of a non-abelian group is necessary.
Keywords: Applied mathematics; Approximate reasoning; Graph; Group; Holonomy; Inconsistency; Matrix; Pairwise comparisons; Simplex.