About a consistency index for pairwise comparison matrices over a divisible alo-group

The Pairwise Comparison Matrices (PCMs) over an abelian linearly ordered (alo)-group G = (G, ⊙, ≤) have been introduced in order to gener- alize multiplicative, additive and fuzzy ones and remove some consistency drawbacks. Under the assumption of divisibility of G, for each PCM A = (aij), a ⊙-mean vector wm(A) can be associated to A and a consis- tency measure IG(A), expressed in terms of ⊙-mean of G-distances, can be provided. In this paper, we focus on the consistency index IG(A). By using the notion of rational power and the related properties, we establish a link between wm(A) and IG(A). The relevance of this link is twofold because it gives more validity to IG(A) and more meaning to wm(A); in fact, it ensures that if IG(A) is close to the identity element then, from a side A is close to be a consistent PCM and from the other side wm(A) is close to be a consistent vector; thus, it can be chosen as a priority vector for the alternatives.