New tools are presented for reasoning about properties of recursively defined domains. We work within a general, category-theoretic framework for various notions of `relation' on domains and for actions of domain constructors on relations. Freyd's analysis of recursive types in terms of a property of mixed initiality/finality is transferred to a corresponding property of invariant relations. The existence of invariant relations is proved under completeness assumptions about the notion of relation. We show how this leads to simpler proofs of the computational adequacy of denotational semantics for functional programming languages with user-declared datatypes. We show how the initiality/finality property of invariant relations can be specialized to yield an induction principle for admissible subsets of recursively defined domains, generalizing the principle of structural induction for inductively defined sets. We also show how the initiality /finality property gives rise to the co-induct...

Or-sets were introduced by Imielinski, Naqvi and Vadaparty for dealing with limited forms of disjunctive information in database queries. Independently, Rounds used a similar notion for representing disjunctive and conjunctive information in the context of situation theory. In this paper we formulate a query language with adequate expressive power for or-sets. Using the notion of normalization of or-sets, queries at the "structural" and "conceptual" levels are distinguished. Losslessness of normalization is established for a large class of queries. We have obtained upper bounds for the cost of normalization. An approach related to that of Rounds is used to provide semantics for or-sets. We also treat or-sets in the context of partial information in databases.