# partial ordering

## partial ordering

[′pär·shəl ′ȯr·də·riŋ]
(mathematics)

## 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 ?
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.
The conclusion of a framework agreement does not result in the award of a public contract, but only precedes the investigation procedure to conclude the agreement between the partial ordering and a contractor, which means that the framework agreement does not result in financial liability; financial liability occurs only with the award of the contract by partial orders.
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.
A sampling of topics: a simple framework for natural animation of digitized models, color mathematical morphology based on partial ordering of spectra, an exact and efficient algorithm for the orthogonal art gallery problem, improved reversible mapping from color to gray, bipartite graph matching for video clip localization, improved real-time shadow mapping for CAD models, automatic eye localization in color images, interactive 3D scanning without tracking, unsupervised identification of coherent motion in video, and automated segmentation of the corpus callosum midsagittal surface area.
Filters can depend upon the results of other filters and these dependencies can be expressed as a partial ordering.
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.
Deliveries will take place successively in the quantity and range, according to the orders partial ordering in a period of up to 4 working days of receipt of order.

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