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)

A set with a partial ordering.

In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory.