|Abstract:||For each OT tableau, there trivially exists an associated set of rankings that includes all and only rankings compatible with the tableau. Moreover, many tableaux are such that no single ranking, be it a total ranking or a stratified hierarchy, can capture correctly the restrictions on constraint domination relations the tableau provides information about; only a set of rankings can do that. Yet the structure of the domain of such sets and their correspondence with tableaux have not been studied before. The current paper fills this gap, developing the theory of sets of OT rankings. In particular, we design techniques for working with manageable representations of sets of rankings (as opposed to full listings of total rankings in a set), provide a recipe for computing the associated set of rankings for an arbitrary tableau, prove that the resulting set contains exactly the same information about rankings as the comparative form of the input tableau, and distinguish those ``proper'' sets of rankings which are associated with some OT tableau from ``improper'' sets which are not. As a technical result enabling the investigation in the main part of the paper, Appendix A presents a theory of equivalence classes of OT tableaux, including a functionally complete set of equivalence-preserving transformations for OT tableaux, a criterion for testing tableau equivalency, and the definition of a normal form for tableaux which can be used as the representative, or the name, of its equivalence class.