On the relation of order theory and computation in terms of denumerability

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we aim to combine the theories of computability with order theory in order to study how the usual countability restrictions in these approaches are related to order density properties and functional characterizations of the order structure in terms of multi-utilities.