  The axiomatic analysis of IR evaluation metrics has contributed to a better
understanding of their properties. Some works have modelled the effectiveness
of retrieval measures with axioms that capture desirable properties on the set
of ranked lists of documents. Recently, it has been shown that three of these
axioms lead to some orderings. This work formally explores the metric
properties of the set of rankings, endowed with these orderings. Based on
lattice theory, the possible metrics and pseudo-metrics, defined on these
structures, are determined. It is found that, when the relevant documents are
prioritized, precision, recall, RBP and DCG are metrics on the set of rankings,
however they are pseudo-metrics when the swapping of documents is considered.
