In this section we introduce on X, the set of alternatives, the main structure of interest in this monograph. We shall assume throughout the sequel that X is a Cartesian product \({\prod _{{\text{i}} \in {\text{I}}}}{\Gamma _{\text{i}}}\), with I an index set. We shall nearly always, with Chapter V excepted, assume that I is a finite set (1,...,n), for n ∈ IN. Many definitions and results of this chapter are as well applicable to infinite index sets I.