partial ordering

partial ordering

   A {relation} R is a partial ordering if it is a {pre-order}
   (i.e. it is {reflexive} (x R x) and {transitive} (x R y R z =>
   x R z)) and it is also {antisymmetric} (x R y R x => x = y).
   The ordering is partial, rather than total, because there may
   exist elements x and y for which neither x R y nor y R x.

   In {domain theory}, if D is a set of values including the
   undefined value ({bottom}) then we can define a partial
   ordering relation <= on D by

   	x <= y  if  x = bottom or x = y.

   The constructed set D x D contains the very undefined element,
   (bottom, bottom) and the not so undefined elements, (x,
   bottom) and (bottom, x).  The partial ordering on D x D is
   then

   	(x1,y1) <= (x2,y2)  if  x1 <= x2 and y1 <= y2.

   The partial ordering on D -> D is defined by

   	f <= g  if  f(x) <= g(x)  for all x in D.

   (No f x is more defined than g x.)

   A {lattice} is a partial ordering where all finite subsets
   have a {least upper bound} and a {greatest lower bound}.

   ("<=" is written in {LaTeX} as {\sqsubseteq}).

   (1995-02-03)