Editing Domain theory (section)

=== Computations and domains ===
Now that we have some basic formal descriptions of what a domain of computation should be, we can turn to the computations themselves. Clearly, these have to be functions, taking inputs from some computational domain and returning outputs in  some (possibly different) domain. However, one would also expect that the output of a function will contain more information when the information content of the input is increased. Formally, this means that we want a function to be '''[[monotonic]]'''.

When dealing with '''[[complete partial order|dcpos]]''', one might also want computations to be compatible with the formation of limits of a directed set. Formally, this means that, for some function ''f'', the image ''f''(''D'') of a directed set ''D'' (i.e. the set of the images of each element of ''D'') is again directed and has as a least upper bound the image of the least upper bound of ''D''. One could also say that ''f'' ''preserves directed suprema''. Also note that, by considering directed sets of two elements, such a function also has to be monotonic. These properties give rise to the notion of a '''[[Scott-continuous]]''' function. Since this often is not ambiguous one also may speak of ''continuous functions''.