A characterisation of the least-fixed-point operator by dinaturality

The paper addresses the question of when the least-fixed-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 sufficient 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-fixed-point operator is so characterised in many of the most commonly used categories of domains. By using retractions, the characterisation extends to the associated cartesian closed categories of continuous cpos. However, dinaturality does not always characterise the least-fixed-point operator. We show that in cartesian closed full subcategories of the category of continuous lattices the characterisation fails. 1 Introduction Mulry [7] showed that, under general conditions on a category of domains, the least-fixed-point operator, lfp D : D D ! D, is a dinatural transformation ..

Logical presentations of domains

Bibliography: pages 168-174.This thesis combines a fairly general overview of domain theory with a detailed examination of recent work which establishes a connection between domain theory and logic. To start with, the theory of domains is developed with such issues as the semantics of recursion and iteration; the solution of recursive domain equations; and non-determinism in mind. In this way, a reasonably comprehensive account of domains, as ordered sets, is given. The topological dimension of domain theory is then revealed, and the logical insights gained by regarding domains as topological spaces are emphasised. These logical insights are further reinforced by an examination of pointless topology and Stone duality. A few of the more prominent categories of domains are surveyed, and Stone-type dualities for the objects of some of these categories are presented. The above dualities are then applied to the task of presenting domains as logical theories. Two types of logical theory are considered, namely axiomatic systems, and Gentzen-style deductive systems. The way in which these theories describe domains is by capturing the relationships between the open subsets of domains

Disjunctive Systems and L-Domains

. Disjunctive systems are a representation of L-domains. They use sequents of the form X ` Y , with X finite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an Ldomain. On the other hand, via the notion of stable neighborhoods, every L-domain can be represented as a disjunctive system. More generally, we have a categorical equivalence between the category of disjunctive systems and the category of L-domains. A natural classification of domains is obtained in terms of the style of the entailment: when jXj = 2 and jY j = 0 disjunctive systems determine coherent spaces; when jY j 1 they represent Scott domains; when either jXj = 1 or jY j = 0 the associated cpos are distributive Scott domains; and finally, without any restriction, disjunctive systems give rise to L-domains. 1 Introduction Discovered by Coquand [Co90] and Jung [Ju90] independently, L-domains form one of the maximal cartesian closed categories of algebraic cpos. Tog..