Editing Domain theory (section)

=== Special types of domains ===
A simple special case of a domain is known as an '''elementary''' or '''flat domain'''.  This consists of a set of incomparable elements, such as the integers, along with a single "bottom" element considered smaller than all other elements.

One can obtain a number of other interesting special classes of ordered structures that could be suitable as "domains". We already mentioned continuous posets and algebraic posets. More special versions of both are continuous and algebraic [[complete partial order|cpos]]. Adding even further [[completeness (order theory)|completeness properties]] one obtains [[Lattice (order)#Continuity and algebraicity|continuous lattices]] and [[algebraic lattices]], which are just [[complete lattice]]s with the respective properties. For the algebraic case, one finds broader classes of posets that are still worth studying: historically, the [[Scott domain]]s were the first structures to be studied in domain theory. Still wider classes of domains are constituted by [[SFP-domain]]s, [[L-domain]]s, and [[bifinite domain]]s.

All of these classes of orders can be cast into various [[category (mathematics)|categories]] of dcpos, using functions that are monotone, Scott-continuous, or even more specialized as [[morphism]]s. Finally, note that the term ''domain'' itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.