Disjunctive Systems and L-Domains

Disjunctive systems are a representation of L-domains. They use sequents of the form X ` Y , with X nite and Y pairwise disjoint. We show that for any disjunctive system, its elements ordered by inclusion form an L-domain. 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 classiication 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 nally, without any restriction, disjunctive systems give rise to L-domains.