Editing Domain theory (section)

=== Bases of domains ===
The previous thoughts raise another question: is it possible to guarantee that all elements of a domain can be obtained as a limit of much simpler elements? This is quite relevant in practice, since we cannot compute infinite objects but we may still hope to approximate them arbitrarily closely.

More generally, we would like to restrict to a certain subset of elements as being sufficient for getting all other elements as least upper bounds. Hence, one defines a '''base''' of a poset ''P'' as being a subset ''B'' of ''P'', such that, for each ''x'' in ''P'', the set of elements in ''B'' that are way below ''x'' contains a directed set with supremum ''x''. The poset ''P'' is a '''continuous poset''' if it has some base. Especially, ''P'' itself is a base in this situation. In many applications, one restricts to continuous (d)cpos as a main object of study.

Finally, an even stronger restriction on a partially ordered set is given by requiring the existence of a base of ''finite'' elements. Such a poset is called '''[[algebraic poset|algebraic]]'''. From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones. As remarked before, not every finite element is "finite" in a classical sense and it may well be that the finite elements constitute an [[uncountable]] set.

In some cases, however, the base for a poset is [[countable]]. In this case, one speaks of an '''ω-continuous''' poset. Accordingly, if the countable base consists entirely of finite elements, we obtain an order that is '''ω-algebraic'''.