# partial ordering

## partial ordering

noun Mathematics.
a relation defined on a set, having the properties that each element is in relation to itself, the relation is transitive, and if two elements are in relation to each other, the two elements are equal.

Origin:
1940–45

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Computing Dictionary

### partial ordering definition

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)

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Example sentences
Ranking of sampling stations is obtained by partial ordering the vectors representing each station.