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).
partial function « partial key « partially ordered
partial ordering » Partial Response Maximum
Likelihood » partition » partitioned data set