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 x 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) The partial ordering on D -">= y2.
The partial ordering on D - > D is defined by
f (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 .
(" LaTeX as \sqsubseteq ).
(1995-02-03)