  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.
