  The study of IR evaluation metrics through axiomatic analysis enables a
better understanding of their numerical properties. Some works have modelled
the effectiveness of retrieval metrics with axioms that capture desirable
properties on the set of rankings of documents. This paper formally explores
the effect of these ranking axioms on the numerical values of some IR
evaluation metrics. It focuses on the set of ranked lists of documents with
multigrade relevance. The possible orderings in this set are derived from three
commonly accepted ranking axioms on retrieval metrics; then, they are
classified by their latticial properties. When relevant documents are
prioritised, a subset of document rankings are identified: the join-irreducible
elements, which have some resemblance to the concept of basis in vector space.
It is possible to compute the precision, recall, RBP or DCG values of any
ranking from their values in the join-irreducible elements. However this is not
the case when the swapping of documents is considered.
