Transitive matrices, strict preference order and ordinal evaluation operators

Let X = {x(1), x(2),..., x(n)} be a set of alternatives and a(ij) a positive number expressing how much the alternative xi 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 a(i), translates A in a coherent priority vector. In this paper we focus our attention on the matrix A, looking for conditions ensuring the existence of coherent priority vectors. Then, given a type of matrices, we look for ordinal evaluation operators, including OWA operators, associated to it.