Source : Free On-Line Dictionary of Computing

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)