On Mathematical structures on pairwise comparisons matrices with coefficients in a group arising from quantum gravity

TL;DR

This paper explores the algebraic, topological, geometric, and probabilistic properties of pairwise comparison matrices with coefficients in arbitrary groups, highlighting the necessity of non-abelian groups in certain cases.

Contribution

It introduces a new framework for analyzing pairwise comparison matrices with group coefficients, extending traditional methods to non-abelian groups and exploring their mathematical properties.

Findings

Non-abelian groups are essential in certain comparison matrix scenarios.

The paper develops a comprehensive vocabulary for algebraic properties of inconsistency maps.

Multiple mathematical perspectives (algebraic, topological, geometric, probabilistic) are applied to these matrices.

Abstract

We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example where the use of a non abelian group is necessary. Algebraic, topological, geometric and probabilistic aspects are considered.