downward closed

closure
downward closed
upward closure

   1. <programming> In a {reduction system}, a closure is a data
   structure that holds an expression and an environment of
   variable bindings in which that expression is to be evaluated.
   The variables may be local or global.  Closures are used to
   represent unevaluated expressions when implementing
   {functional programming languages} with {lazy evaluation}.  In
   a real implementation, both expression and environment are
   represented by pointers.

   A {suspension} is a closure which includes a flag to say
   whether or not it has been evaluated.  The term "{thunk}" has
   come to be synonymous with "closure" but originated outside
   {functional programming}.

   2. <theory> 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).

   (1994-12-16)