An Algebraic Approach to the Interpretation of Recursive Types

We present an algebraic framework for the interpretation of type systems including type recursion, where types are interpreted as partial equivalence relations over a Scott domain. We use the notion of iterative algebra, introduced by J. Tiuryn [26] as a counterpart to the categorical notion of iterative algebraic theory by C.C. Elgot [15]. We show that a suitable collection of partial equivalence relations is closed under type constructors and forms an iterative algebra. The existence of type interpretations follows from the initiality, in the class of iterative algebras, of the algebra of regular infinite trees obtained by infinitely unfolding recursive types.