# partial ordering

## partial ordering

[′pär·shəl ′ȯr·də·riŋ]
(mathematics)
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).
References in periodicals archive ?
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.
According to this partial ordering, for any two intervals [??] and [??], closeness of [??] with b is [mathematical expression not reproducible], which always lies between 50% to 100% and also known as degree of closeness.
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].
Now, we introduce a partial ordering on the [??]- and [??]- classes of a semigroup S.
In particular, for the regular elements of S, the partial ordering is just the one which is given in .
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.
Filters can depend upon the results of other filters and these dependencies can be expressed as a partial ordering. For example, a face recognition filter (e.g., "is this George?") might require that a face detection filter (e.g., "does this image contain any human faces, and if so where?") be run as a prerequisite step.
In the denotational approach to programming semantics, this threefold process is studied by assigning types to data objects (computations) specifying the operations that may be performed on them, equipping these types with an information-based partial ordering, and creating a topology based on this partial ordering whose open sets represent the logical formulas.
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.

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