Diversity relations over menus

We study three classes of diversity relations over menus arising from an unobserved categorization of alternatives in the form of a partition. A basic diversity relation declares a menu to be more diverse than another if and only if for every alternative in the latter there is an alternative in the former which belongs to the same category. An extension of a basic diversity relation preserves its weak and strict parts and possibly makes additional diversity judgements between hitherto incomparable menus. A cardinality-based extension is an extension which ranks menus on the basis of the number of categories that exist in each menu. We characterize each class axiomatically. Two axioms satisfied by each of the three classes are Monotonicity, which says that larger menus are at least as diverse, and No Complements, which rules out certain complementarities between alternatives in generating diversity.