partial ordering

partial ordering

[′pär·shəl ′ȯr·də·riŋ]
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

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).
This article is provided by FOLDOC - Free Online Dictionary of Computing (
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The monograph begins to answer the question of what sorts of degree spectra can be found in nature by proving results relative to a cone, particularly the partial ordering of degree spectra on a cone and whether or not there are fullness results for certain types of degree spectra.
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x- partial ordering is associated with closeness index, which lies between 1/2 and 1.
To compare the interval vectors in I[(R).sup.n], we define the following partial ordering [[less than or equal to].sup.n.sub.x].
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In this paper, we introduce a partial ordering on dualistic partial metric spaces utilizing partial metric function and use the same to prove a fixed point theorem for single valued nondecreasing mappings on ordered dualistic partial metric spaces.
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Their topics are mathematical chemistry, graph theory and chemistry, characteristic polynomial, structure-activity, molecular descriptors, partial ordering, novel molecular matrices, on highly similar molecules, aromaticity revisited, clar aromatic sextet, renormalization in chemistry, graphical bioinformatics, and beauties and sleeping beauties in science.