Constructive Consistent Approximations in Pairwise Comparisons

In this paper we investigate groups which admit the existence of weighted consistent approximations for pairwise comparisons matrices. These approximations are defined by extending the classical matrix projection for R+ to abstract weighted projections on the non-linear sets of transitive group-valued matrices. It is of interest that all of them are represented by general explicit formulae dependent on an abstract logarithmic function. This general approach is applied to the groups Z(p) and F-2m m which are of fundamental importance in cryptography. Finally, we use our unified mathematical model of pairwise comparisons for continuous one-parameter unitary groups, which play a fundamental role in physics.