# 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 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).

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