partial ordering

partial ordering

[′pär·shəl ′ȯr·də·riŋ] (mathematics) ordering

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 mayexist 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 theundefined value (bottom) then we can define a partialordering 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 isthen

(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 subsetshave a least upper bound and a greatest lower bound.

("<=" is written in LaTeX as \\sqsubseteq).