Forbidden substructures for coherence of domains

Abstract

A coherent domain, in the sense of Dana Scott's domain theory, is a domain in which the intersection of every two compact saturated subsets is again compact, when the domain is equipped with the Scott topology. Coherence plays key roles in classifying Cartesian closed subcategories of domains and in characterizing Lawson compactness of domains. In this paper, we find two typical domains that fail to be coherent, and prove that a bounded algebraic domain fails to be coherent if and only if it has one of the two typical domains as its Scott-continuous retract. Similar results also generalize to bounded continuous domains, provided that the domains in consideration are hereditarily Lindelöf and weakly Hausdorff in the Scott topology.