2. | (theory) | closure - In domain theory, given a partially ordered set, D and a subset, X of D, the upward closure of X in D is
the union over all x in X of the sets of all d in D such that
x <= d. Thus the upward closure of X in D contains the
elements of X and any greater element of D. A set is "upward
closed" if it is the same as its upward closure, i.e. any d
greater than an element is also an element. The downward
closure (or "left closure") is similar but with d <= x. A
downward closed set is one for which any d less than an
element is also an element.
("<=" is written in LaTeX as \subseteq and the upward
closure of X in D is written \uparrow_\D X). | |