From Sazonov's Non-Dcpo Natural Domains to Closed Directed-Lub Partial Orders

TL;DR

This paper advances the theory of natural domains by introducing lub-rules and classes, focusing on closed directed lub partial orders, and explores their completion to restricted dcpos, enriching domain theory.

Contribution

It develops the theory of natural domains by defining lub-rules, classifying axiom systems, and analyzing the structure and completion of closed directed lub partial orders.

Findings

Defined lub-rules for natural lubs

Identified a canonical subcategory of natural domains

Showed how cdlubpos can be completed to dcpos

Abstract

Normann proved that the domains of the game model of PCF (the domains of sequential functionals) need not be dcpos. Sazonov has defined natural domains for a theory of such incomplete domains. This paper further develops that theory. It defines lub-rules that infer natural lubs from existing natural lubs, and lub-rule classes that describe axiom systems like that of natural domains. There is a canonical proper subcategory of the natural domains, the closed directed lub partial orders (cdlubpo), that corresponds to the complete lub-rule class of all valid lub-rules. Cdlubpos can be completed to restricted dcpos, which are dcpos that retain the data of the incomplete cdlubpo as a subset.