A characterisation of the least - xed - point operatorby

The paper addresses the question of when the least-xed-point operator, in a cartesian closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a suucient condition on a cartesian closed full subcategory of the category of algebraic cpos for the characterisation to hold. The condition is quite mild, and the least-xed-point operator is so characterised in many of the most commonly used categories of domains. By using retractions, the charac-terisation extends to the associated cartesian closed categories of continuous cpos. However, dinaturality does not always characterise the least-xed-point operator. We show that in cartesian closed full subcategories of the category of continuous lattices the characterisation fails.