Representation theorems for directed completions of consistent algebraic \(L\)-domains (Q818709)

The authors consider algebraic domains, in particular algebraic \(L\)-domains, with the weakened completeness condition that directed sets that are bounded above have a least upper bound, what they call consistent algebraic domains. They consider the dcpo obtained by taking the specialization order of the sobrification of the Scott topology, what they call the directed completion, and give various characterizations for it, e.g., it agrees with the ideal completion of the poset of compact elements. It is shown that eliminating a set of maximal elements with empty interior from an algebraic \(L\)-domain yields a consistent algebraic \(L\)-domain whose directed completion returns the original \(L\)-domain. They also show that cartesian closedness extends to the category of consistent algebraic \(L\)-domains.