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Topic: Partial ordering relation

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Binary relation

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	Partially ordered set - Wikipedia, the free encyclopedia
		In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation.
		A partial order is a binary relation R over a set P which is reflexive, antisymmetric, and transitive, i.e., for all a, b and c in P, we have that:
		Strict partial orders are also useful because they correspond more directly to directed acyclic graphs (dags): every strict partial order is a dag, and the transitive closure of a dag is both a strict partial order and also a dag itself.
	en.wikipedia.org /wiki/Partial_order (509 words)

	PlanetMath: partial order
		A partial order (often simply referred to as an order or ordering) is a relation
		A total order is a partial order that satisfies a fourth property known as comparability:
		This is version 13 of partial order, born on 2001-10-06, modified 2004-11-13.
	planetmath.org /encyclopedia/PartialOrder.html (138 words)

	Binary Relation
		Relation, a mathematical concept, is a set of ordered pairs.
		With the definition of a relation explicitly stated, one is able to not only construct examples, but also define an algebra (used vaguely, not as the mathematical concept in the branch of mathematics called algebra) on relations.
		Another interesting topic in the study of relation is the study of relation on a set X. In this case, both the domain and range of the relation are subsets of X. On these relations, mathematicians use a few adjectives to describe the different elementary types of relation on a given set.
	www.iscid.org /encyclopedia/Binary_Relation (2105 words)

	Comparison of topologies - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics comparison of topologies refers to the fact that two topological structures on a given set may stand in relation to each other.
		The set of all possible topologies on a given set forms a partially ordered set.
		This order relation can be used to compare the different topologies.
	en.wikipedia.org /wiki/Finer_topology (502 words)

	Encyclopedia :: encyclopedia : Antisymmetric relation (Site not responding. Last check: 2007-10-29)
		In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b.
		There are relations which are both symmetric and antisymmetric (equality), relations which are neither symmetric nor antisymmetric (divisibility on the integers), relations which are symmetric and not antisymmetric (congruence modulo n), and relations which are not symmetric but are anti-symmetric ("is less than").
		An antisymmetric relation that is also transitive and reflexive is a partial order.
	www.hallencyclopedia.com /Antisymmetric_relation (214 words)

	PlanetMath: ordering relation
		Warning: In some cases, an author may use the term “ordering relation” to mean a partial order instead of a
		Cross-references: total order, partial order, mean, term, totally ordered set, relation
		This is version 4 of ordering relation, born on 2001-10-21, modified 2004-03-20.
	planetmath.org /encyclopedia/OppositeOrdering.html (59 words)

	partial ordering (Site not responding. Last check: 2007-10-29)
		A relation R is a partial ordering if it is a pre-order (i.e.
		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.
		A lattice is a partial ordering where all finite subsets have a least upper bound and a greatest lower bound.
	burks.brighton.ac.uk /burks/foldoc/87/86.htm (224 words)

	CMIS Research - Image Analysis Activities - Recent Projects - Colour Morphology
		We have shown that by artificially imposing an invertible ordering of the partially ordered multi-variate space, it is possible to morphological filter a multi-variate image while introducing no new pixel values [1].
		The concept of an infimum and a supremum stem from the partial ordering relation.
		Applying a single 3 x 3 closing in conjunction with this ordering removes the salt and the pepper noise simultaneously, because the salt is considered to be as dark as the pepper.
	www.cmis.csiro.au /iap/RecentProjects/colour.htm (1997 words)

	Equivalence Relation
		The relation r is a partial ordering on the set s, or s is a partially ordered set via r, or s is a poset, if r is transitive and antisymmetric.
		A lattice is a partial ordering with a lub and a glb for every pair of points.
		A linear ordering, or total ordering, is a partial ordering where all pairs of elements are comparable.
	www.mathreference.com /set,rst.html (1118 words)

	[No title]
		It seems to me that defining a partial ordering using the symbol ">" and the word "greater" would be better than ">=" and "greater than or equal to" (by being narrower but still correct), since no two different elements of the set are allowed to be equal to each other anyway.
		Once we have a partial order, it is possible to define a maximum element of the set.
		To be sure, transitivity and anti-symmetry of the partial order means that we may have quite a nice sequence of ranked elements, with one clearly the biggest.
	www.rasmusen.org /GI/lattice.theory.notes.txt (3223 words)

	EE290N Lecture 9 Notes
		In an ordered signal process network, T(s) is totally ordered for each signal s but the set of all tags T(ss) is typically partially ordered.
		The output signal preserves the ordering of the tokens of the input signals, but there is no ordering between tokens of different input signals.
		Therefore there are many possible output signals, each corresponding to a linearization of the partial order of the tokens of all the input signals.
	ptolemy.eecs.berkeley.edu /~eal/ee290n/lecture9/lec9.html (568 words)

	PART from FOLDOC
		A PER is an equivalence relation on the subset for which it is defined, i.e.
		Partial evaluation might change the termination properties of the program if, for example, the expression (x * 0) was reduced to 0 it would terminate even if x (and thus x * 0) did not.
		Partial evaluation of built-in functions applied to constant arguments is known as constant folding.
	ftp.sunet.se /foldoc/foldoc.cgi?PART (1222 words)

	Know it all Inc. (P) (Site not responding. Last check: 2007-10-29)
		The processors may either communicate in order to be able to cooperate in solving a problem or they may run completely independently, possibly under the control of another processor which distributes work to the others and collects results from them (a "{processor farm}").
		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}.
		A semi-closed party thrown for hackers at a science-fiction convention (especially the annual Worldcon); one must have a {network address} to get in, or at least be in company with someone who does.
	artikbre.synchro.net /docs/P.html (18136 words)

	Abstract of NTZ 20/00 (Site not responding. Last check: 2007-10-29)
		Abstract: The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space-time dimension d<=4 is investigated.
		For that purpose, a description of orbit types in terms of cohomology elements of space-time, derived earlier, is used.
		It is shown that, on the level of these cohomology elements, the partial ordering relation is characterized by a system of algebraic equations.
	www.uni-leipzig.de /~ntz/abs/abs2000.htm (108 words)

	Examples of Partially Ordered Sets
		is a partial ordering by proving that the properties (1), (2) and (3) above all hold.
		Another example of a partial ordering which arises in the real world is the building of a new house in which there are certain tasks such as digging the foundations, laying the floor, which must be completed before other phases of the construction such as erecting walls and building the roof can be undertaken.
		is called a partially ordered set or poset.
	scom.hud.ac.uk /scomtlm/book/node44.html (306 words)

	Partial Ordering
		Such a partial order is also referred to as a weak partial order.
		A weak partial order is characterised by the reflexivity property, that is the statement
		A set on which there is a partial ordering is called a partially ordered set or poset.
	scom.hud.ac.uk /scomtlm/book/node43.html (133 words)

	June 2000 : Making topic maps more colourful
		R is anpartial ordering relation :R is reflexive, anti-symmetric, and transitive.
		R is atotal order relation :R is reflexive, anti-symmetric, transitive, and connex.
		R is astrong order relation :R is anti-reflexive, anti-symmetric, and transitive.
	www.infoloom.com /gcaconfs/WEB/paris2000/S29-01.HTM (5576 words)

	QUANTUM LOGIC WITH IMPLICATION
		The success or lack of success of this effort as related to actual application of this formalism in the foundations of quantum theory is, I think, a matter for the physicists to decide; the empirical judgment involved in this application is not in the scope of this paper.
		If a formalism is to have a reasonable interpretation as a theory of deduction, it seems clear that it must include a relation which, when it holds between two elements of the formalism, may be reasonably interpreted as expressing that one of the elements 'follows from' the other, in some sense of 'follows from'.
		We note that the properties of the accessibility relation as determined by the sequence of experiments include at least transitivity and reflexivity--which are the properties of this relation in frames characteristic of the system S4.
	www.clas.ufl.edu /users/jzeman/quantumlogic/quantum_logic_with_implication.htm (1715 words)

	PlanetMath: tree (set theoretic)
		The partial ordering represents distance from the root, and the well-ordering requirement prohibits any loops or splits below a node (that is, each node has at most one parent, and therefore at most one grand-parent, and so on).
		Since there is generally no requirement that the tree be connected, the null ordering makes any set into a tree, although the tree is a trivial one, since each element of the set forms a single node with no children.
		Since the set of ancestors of any node is well-ordered, we can associate it with an ordinal.
	planetmath.org /encyclopedia/Parent2.html (408 words)

	2.5.1 Searching in a Space of Partial Plans
		The idea is to iteratively achieve required subgoals in a partial plan while ensuring that no conflicts arise that could destroy the solution developed so far.
		A vertex in the partial-plan search graph is a partial plan, and an edge is constructed by extending one partial plan to obtain another partial plan that is closer to completion.
		One straightforward generalization is to develop multiple partial plans and decide which one to refine in each iteration.
	msl.cs.uiuc.edu /planning/node62.html (371 words)

	Ordering
		Definition 35 A partial ordering on a set A is a relation on A which is antisymmetric and transitive.
		Definition 36 A ordering on a set A is a relation on A which is reflexive, antisymmetric and transitive.
		Definition 37 A linear ordering on a set A is a relation on A which is reflexive, antisymmetric, transitive, and total.
	www.iwu.edu /~lstout/theoremlist/node10.html (223 words)

	GNU Smalltalk User's Guide
		Before approaching the problem of the syntax to be implemented and of its very implementation, we have to point out which structural relations are going to be established between environments.
		Let us call the partial ordering relation which occurs between the two environments to be inheritance.
		Not only that "inheritance" is the standard term for the partial ordering relation in the lattice theory but the feature of inheritance in the meaning of object-orientation is associated with this relation.
	www.delorie.com /gnu/docs/smalltalk/gst_11.html (401 words)

	Ordered and Well-Ordered Set
		A set S is called partially ordered if there exists a relation r (usually denoted by the symbol
		A set S is called ordered if it is partially ordered and every pair of elements x and y from the set S can be compared with each other via the partial ordering relation.
		A set S is called well-ordered if it is an ordered set for which every non-empty subset contains a smallest element.
	pirate.shu.edu /projects/reals/infinity/defs/ordering.html (85 words)

	Eager Derivation Replay
		In order to understand the validation process, we must first describe the decision steps that the planner takes in arriving at a solution to a planning problem.
		Semantically, a partial plan is a shorthand notation for the set of ground operator sequences that are consistent with its constraints.
		A threat is resolved by adding a step ordering which either promotes or demotes the clobberer relative to the causal link.
	www.cs.cmu.edu /afs/cs.cmu.edu/project/jair/pub/volume7/ihrig97a-html/node3.html (930 words)

	jelia.html
		The novel element of the paper is that it presents two general principles on the preference relation between belief sets (answer sets in logic programming, extensions in default logic) that should be satisfied by any system based on prioritized defeasible rules.
		In the most original contribution of the paper the authors demonstrate that in order to properly capture updates, the principle of inertia} should be applied not just to individual literals but to entire clauses of the program representing the knowledge base being updated.
		Disjunctive ordered logic or DOL, can be viewed as an extension of disjunctive DATALOG with negation by allowing negation in the heads of clauses, modularization (clauses can be grouped into modules), and inheritance (modules can be ordered by a partial ordering relation).
	www.cs.ucy.ac.cy /compulog/dec98update/areakrr/kr98.htm (1361 words)

	The Dimensions of Context-Space
		There is no fixed order in which the dimensions are evaluated; rather, that order is specified for a context by (earlierDimension C d1 d2), which defines a partial order of the dimensions di such as Time, GeoLocation, etc. However, our expectation is that the order of dimension-constraints will rarely matter.
		The relation "subtopic" is much more complicated than "subset." There are many cases where a topic's subtopics are not subsets of it; e.g., "FreshwaterAlgae" and "Dam-building" are sub-topics of "Rivers", though they aren't subsets, elements, etc. of #$River.
		This will be a poset28 (partially ordered set) of posets of heuristics, the latter posets ordered by preference and the former ordered by refinement.
	casbah.org /resources/cycContextSpace.shtml (16228 words)

	SRFI 101: FTL (Function Template Library)
		In order for an algorithm parameterized by these four operations to work as expected, all operations should conform to a certain set of requirements, explicitly specified for each interface.
		Order and Equality interface is an extension of the Equality interface; in addition to the equivalence procedure, it includes a two-argument procedure implementing a partial ordering relation, compatible with the equivalence relation.
		This means that the ordering relation must be transitive, irreflexive, and obey the trichotomy law with regard to the equivalence relation.
	www.malgil.com /esl/ftl.html (3423 words)

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