A cpo is consistently complete (or bounded complete) just if every subset that has an upper bound (in ) has a least upper bound.
Let with . An upper bound for is an element (not necessarily in !) such that and . This intuitively means that and are "consistent": they represent pieces of information that do not contradict each other. In these circumstances, the least upper bound exists.
Thus, the intuitive idea is that consistent information can be stuck together.
We can alternatively define a consistently complete cpo through the existence if binary consistent suprema: we can ask that the supremum exists whenever and have an upper bound. This implies that suprema of finite consistent sets exist.
However, this is equivalent to the above; this is by the usual trick of turning a colimit into a directed one. The general case obviously implies the finite case. But if we have a consistent set , define . Each finite is consistent, because is, so the supremum exists. But then is directed, so exists, and it's equal to .
Recall that a complete lattice can be given either by having all suprema, or having all infima: if you have one, then you can give the other.
Similarly, it is possible to characterise consistently complete cpos as cpos having all non-empty infima. The argument is simple: the supremum of a consistent set is just the infimimum of all its upper bounds. Thus, having such infima proves consistent completeness. Conversely, to find infima in a consistently complete set, take the set of all lower bounds; this set is consistent, so it has a supremum.