A Geometric Framework for the Inconsistency in Pairwise Comparisons
  In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group $G$, non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extended to the class of group $G$) are proposed. A geometric interpretation of pairwise comparisons matrices in terms of connections to a simplex is given. Approximate reasoning is more effective when inconsistency in data is reduced. 
Waldemar W. Koczkodaj and Jean-Pierre Magnot
math.LO cs.IT math.DG math.IT