In other words we are ensuring that the equation (R', P') ~ (R, P) is soluble for R' provided that there exist `R, R such that (`R, P') >= (R, P') >= (R, P'). An assumption of continuity of the precision and recall factors would ensure this.

The fifth condition is not limiting in any way but needs to be stated. It requires, in a precise way, that each component is essential.

Definition 5. Component R is essential if and only if there exist R1, R2 [[propersubset]] R and P1 [[propersubset]] P such that it is not the case that (R1, P1) ~ (R2, P1). A similar definition holds for P.

Thus we require that variation in one while leaving the other constant gives a variation in effectiveness.

Finally we need a technical condition which will not be explained here, that is the Archimedean property for each component. It merely ensures that the intervals on a component are comparable. For details the reader is referred to Krantz et al.

We now have six conditions on the relational structure <R x P, >= > which in the theory of measurement are necessary and sufficient conditions* for it to be an additive conjoint structure. This is enough for us to state the main representation theorem. It is a theorem asserting that if a given relational structure satisfies certain conditions (axioms), then a homomorphism into the real numbers is often referred to as a scale. Measurement may therefore be regarded as the construction of homomorphisms for empirical relational structures of interest into numerical relational structures that are useful.

In our case we can therefore expect to find real-valued functions [[Phi]]1 on R and [[Phi]]2 on P and a function F from Re x Re into Re, 1:1 in each variable, such that, for all R, R' [[propersubset]] R and P, P' [[propersubset]] P we have:

(R, P) >= (R', P') <=> F [[[Phi]]1 (R ), [[Phi]]2 (P )] >= F [[[Phi]]1 (R' ), [[Phi]]2 (P' )]

(Note that although the same symbol >= is used, the first is a binary relation on R x P, the second is the usual one on Re, the set of reals.)

In other words there are numerical scales [[Phi]]i on the two components and a rule F for combining them such that the resultant measure preserves the qualitative ordering of effectiveness. When such a representation exists we say that the structure is decomposable. In this representation the components (R and P) contribute to the effectiveness measure independently. It is not true that all relational