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.
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
(1995-02-03)