# partial ordering

noun, Mathematics
1.
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 of partial ordering
1940-1945
1940-45
Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2015.
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partial ordering in Technology

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)

The Free On-line Dictionary of Computing, © Denis Howe 2010 http://foldoc.org
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