Transitive Matrices, Strict Preference and Ordinal evaluation Operators

Let X be a set of alternatives and a_{ij} a positive number expressing how much the alternative x_{i} is preferred to the alternative x_{j}. Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A = (a_{ij}), the actual qualitative ranking on the set X is achievable. Then a coherent priority vector is a vector giving a weighted ranking agreeing with the actual ranking and an ordinal evaluation operator is a functional F that, acting on the row vectors of A provides a coherent priority vector. In the paper we focus on the matrix A, looking for conditions ensuring the existence of coherent priority vectors among the columns. Then, given a type of matrices, we look for ordinal evaluation operators, including OWA operators, associated to it.