Confidence relations and ordinal information 1

We define confidence relations on Boolean lattices, which can be interpreted as ordinal representations of uncertainty or information. The set of the confidence relations on a given Boolean lattice can be ordered by set inclusion and thus is shown to form a complete meet-semilattice. We investigate and identify the maximal elements of this structure. Moreover, we prove that it is in particular an algebraic semilattice (or domain), and that its finite elements are precisely the finitely generated confidence relations. We also investigate the relationship with information systems. We define duality and self-duality for confidence relations and show that similar conclusions can be reached if we restrict ourselves to confidence relations which are in particular self-dual. Finally, we discuss the possible incompleteness of confidence relations, and the relation between the abstract mathematical structures studied here, and other existing uncertainty models.