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Topic: Ordered sets

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	PlanetMath: well ordered set
		A well-ordered set is a totally ordered set in which every nonempty subset has a least member.
		An example of well-ordered set is the set of positive integers with the standard order relation
		This is version 11 of well ordered set, born on 2001-10-17, modified 2005-07-26.
	planetmath.org /encyclopedia/WellOrderedSet.html (208 words)

	Ordered Sets : Software Foundations : Thomas Alspaugh : UCI
		It is usually clear by context whether "order" refers literally to an order (an order relation) or by synechdoche to an ordered set.
		Then the ordered set of the positive integers to 15 ordered by the converse of divides (now with the divident considered "higher" than the dividend), is the dual Q∂ of Q.
		It is often useful for an ordered set to have a bottom, but not all ordered sets have one (for example, the set in Figure 6).
	www.ics.uci.edu /~alspaugh/foundations/orderedSet.html (2579 words)

	PlanetMath: properties of well-ordered sets
		Assume a bijective order morphism as stated by the theorem does not exist.
		Cross-references: empty set, maps, restrictions, equation, minimal element, sequences, monotonically decreasing, maximal element, image, surjective, injective, contradiction, mapping, inclusion, structure, order morphism, bijective, poset, well-ordered, least element, strictly, subset, totally ordered set, easy to see, properties, orderings
		This is version 3 of properties of well-ordered sets, born on 2005-07-16, modified 2005-11-08.
	planetmath.org /encyclopedia/Section4.html (450 words)

	Ordinal Numbers and Their Arithmetic
		Theorem 2: Suppose A is a well-ordered set and B is a subset of A such that there exists a similarity mapping f of A into B. Then for every element x of A, x≤f(x).
		Definition: The order type of a well-ordered set is the family of well-ordered sets it is similar to.
		is the set of all functions from A to B. The Burali-Forti Paradox: The concept of the set of all ordinal numbers is contradictory.
	www2.sjsu.edu /faculty/watkins/ordinals.htm (1606 words)

	06: Order, lattices, ordered algebraic structures
		Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field.
		Linear orderings especially on infinite sets is the study of Ordinals in Set Theory; these are traditionally considered in 03: Mathematical Logic, especially 03G: Algebraic Logic.
		Ordered sets may be viewed as topological spaces; see 54: General Topology, especially 54F05: Ordered topological spaces, for more detail.
	www.math.niu.edu /~rusin/known-math/index/06-XX.html (606 words)

	Comprehension Of Visualized Ordered Sets
		Ordered sets provide a non-numerical data structure, whose mathematical properties are very general.
		In this study, mathematical properties of ordered sets and their drawings are reported, as far as they are relevant for the subsequent empirical investigations.
		Psychological assumptions on the cognitive processes involved in the course of comprehension of visualized ordered sets are considered.
	wundt.kfunigraz.ac.at /koerner/diss-abstract.html (807 words)

	Partially ordered set (Site not responding. Last check: 2007-10-09)
		In mathematics, a partially ordered set (or poset for short) is a set equipped with a partial order relation, formalizing the intuitive concept of a (not necessarily total) ordering.
		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.
	partially-ordered-set.iqnaut.net (417 words)

	Sets and Sentence Analysis
		A set may be specified by listing its members, or by specifying particular properties or attributes the members must have in order that they be considered members of the set.
		A set also may be specified by citing an attribute or property comprising a criterion on the basis of which things are collected together as a set.
		Sets, however, include no relationship of order among their elements, and in languages such as English, the order the words in a sentence is important.
	web.uvic.ca /~ling48x/ling484/notes/sets.html (6222 words)

	Sets
		Sets are of course just collections of values and there is no question of a key (we can perhaps think of the value as being its own key).
		Thus in the case of an ordered set the values are stored in order whereas in the case of a map, it is the keys that are stored in order.
		We will finish this discussion of sets by briefly considering the additional facilities in the two sets packages (and their inner generic keys packages) just as we did for the two maps packages (the discussion is almost identical).
	www.adaic.org /standards/05rat/html/Rat-8-4.html (2507 words)

	SICStus Prolog - Ordered Set Operations (Site not responding. Last check: 2007-10-09)
		Ordered sets are sets represented as lists with the elements ordered in a standard order.
		is the ordered representation of the set denoted by the unordered representation
		The two ordered sets have at least one element in common.
	www.cl.cam.ac.uk /ailanguages/sicstus3.7/sicstus_21.html (182 words)

	Partially ordered set - Wikipedia, the free encyclopedia
		In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation.
		The set of subsets of {x,y,z}, ordered by inclusion
		The set of natural numbers equipped with the lesser than or equal to relation (this is also a total order).
	en.wikipedia.org /wiki/Partially_ordered_set (558 words)

	Special Ordered Sets
		All such sets are mutually exclusive of each other, the members are not subject to any other discrete conditions and each set is grouped together consecutively in the data.
		A specially ordered set of degree N is a collection of variables where at most N variables may be non-zero.
		In lp_solve, specially ordered sets may be of any cardinal type 1, 2, and higher, and may be overlapping.
	www.mockus.us /optimum/lp2jdk/guide/SOS.htm (1093 words)

	Springer Online Reference Works
		is ordered by the principle of first difference (as words are ordered in a dictionary).
		The lexicographic order is a special case of an ordered product of partially ordered sets (see [3]).
		is not necessarily an order in the usual sense (cf.
	eom.springer.de /l/l058330.htm (273 words)

	Jay's Corner 4
		In an ordered set, there is a first element, a second element, a third element and so on.
		Here I've indicated two such paths, the red and the green ones, giving the ordered sets A C B and C A B. It is easy to count the number of paths.
		In general, the number of ordered sets of size k that can be made from a set of size n, is denoted by P(n,k) and is equal to the number n(n-1)(n-2)...
	www-math.cudenver.edu /~wcherowi/jcorn4.html (829 words)

	Math 3309_10639
		Hence, start with a collection of linearly ordered sets and define two of these sets to be related provided they are order-isomorphic: the collection is then partitioned into equivalence classes and to each set in the same equivalence class assign a symbol called its order type.
		Note: (1) problem 7.17 uses the Principle of Mathematical Induction II, page 13; (2) the predecessor set of an element is defined in problem 7.31 and used in 7.32; (3) from problems 7.47 and 7.52, we may conclude that an ordered set with one maximal (minimal) element may not have a last (first) element.
		In section 6.4, study: (1) definition of a set that has the power of the continuum; (2) any interval of numbers has the power of the continuum; (3) the set of real numbers has the power of the continuum-verify that the given function is one-to-one and onto.
	cms.dt.uh.edu /faculty/BecerraL/Spring2004/3312_calendar.htm (2054 words)

	Fully Ordered Finite Sets for ACL2 - Nested Sets
		When I presented the set library at the ACL2 meeting on February 11, 2004, a great deal of confusion ensued as to whether or not nested sets were supported by the library.
		An objection was raised that the ordered sets library would be unable to represent the set containing as its elements: the set {1, 2}, and the list (1 2).
		In the ordered sets library, this distinction does not exist, because the representation of sets and lists is exactly the same.
	www.cs.utexas.edu /users/jared/osets/Web/nested-sets.html (880 words)

	Ordered set - Wikipedia, the free encyclopedia
		Ordered set is used with distinct meanings in order theory.
		A set with a binary relation R on its elements that is reflexive (for all a in the set aRa), antisymmetric (if aRb and bRa then a=b) and transitive (if aRb and bRc then aRc) is described as a partially ordered set or poset.
		If the binary relation is antisymmetric, transitive and also total (for all a and b in the set, aRb or bRa) then the set is a totally ordered set.
	en.wikipedia.org /wiki/Ordered_set (167 words)

	Ordinals Represent Well Ordered Sets (Site not responding. Last check: 2007-10-09)
		If a well ordered set is isomorphic to two ordinals then these ordinals are isomorphic, hence equal.
		Declare a well ordered set s bad if it is not isomorphic to an ordinal, and declare it "very bad" if every proper initial segment of s is good.
		If you can well order the members of a set, that ordered set is represented by a specific ordinal.
	www.mathreference.com /set-zf,ordrep.html (403 words)

	Mathematical Structures: Partially ordered sets (Site not responding. Last check: 2007-10-09)
		A partially ordered set (also called ordered set or poset for short) is a structure P = (P, ≤) such that P is a set and ≤ is a binary relation on P that is
		A strict partial order is a structure (P, <) such that P is a set and < is a binary relation on P that is
		Any poset is order-isomorphic to a poset of subsets of some set, ordered by inclusion.
	math.chapman.edu /cgi-bin/structures.pl?Partially_ordered_sets (215 words)

	Ordered Logic Program Solver (Site not responding. Last check: 2007-10-09)
		Such a rule order induces a natural order on extended answer sets, the minimal elements of which are call preferred answer sets.
		The preferred answer set semantics for ordered logic programs has been implemented in the olp (ordered logic program solver) system which is available for download under the GPL license.
		partially ordered sets of clauses where smaller rules carry more preference, inconsistencies, which appear as conflicts between applicable rules, are handled by satisfying more preferred rules, at the expense of defeating lesser rules.
	tinf2.vub.ac.be /olp (1237 words)

	Isomorphisms Between Ordered Sets (Site not responding. Last check: 2007-10-09)
		A bijection that equates the members of two partialy ordered sets is an isomorphism if it respects order.
		When the sets are linearly ordered, one is mapped onto the other in a "continuous" fashion, although some sections may be stretched and others shrunk.
		If s and t are two well ordered sets, and there is an isomorphism between them, it is unique.
	www.mathreference.com /set-zf,iso.html (288 words)

	Previous Results and Research Plans
		We resolve the complexity of FPFMAP with a construction of length six, yet FPFMAP is polynomial for ordered sets of length one.
		It is of interest, therefore, to resolve the complexity of FPFMAP for ordered sets of length three, in particular.
		A cutset is a subset of an ordered set that meets every maximal chain (dually, a fibre is a subset that meets every maximal antichain).
	www.mathcs.emory.edu /~goddard/plans.html (724 words)

	InterOperability Laboratory: Services: Testing: Gigabit Ethernet: Knowledge Base
		The notation used for ordered sets is similar to that used for code groups.
		Ordered sets are written in the form of /XY/ where X is a letter and Y is sometimes used and contains a number.
		The defined ordered sets are: /C/, /C1/, /C2/, /I/, /I1/, /I2/, /R/, /S/, /T/ and /V/.
	www.iol.unh.edu /training/ge/orderedSets.html (508 words)

	Recitation 15: Implementing ordered sets
		Here is an ordered set signature that is designed to support implementation of both set and map abstractions.
		If key is string and elem is int, * a set could be {("elephant", 2), ("rhino", 25), ("zebra", 2)} ) type key type elem type set* (* compare(k1,k2) reports the ordering of k1 and k2.
		This implementation represents a set as a red-fl tree plus an integer that keeps track of the total number of elements in the set.
	www.cs.cornell.edu /courses/cs312/2004sp/lectures/rec15.html (1142 words)

	DIMACS Workshop on Applications of Lattices and Ordered Sets to Computer Science (Site not responding. Last check: 2007-10-09)
		These range from lattices as models for logics, which are fundamental to understanding computation, to the ordered sets as models for computation, to the role both lattices and ordered sets play in combinatorics, a fundamental aspect of computation.
		These include such areas as knowledge representation, text categorization and data mining, where order plays a fundamental organizing principle, to the use of lattices and ordered sets to analyze crypto-protocols in security, to inductive logic programming, where ordered sets form basic models.
		The talks by leaders in lattice theory and ordered sets will focus on applications to computer science, while the talks by leading researchers in these areas will describe how techniques from lattice theory and ordered sets have recently led to results and applications to problems in computation.
	dimacs.rutgers.edu /Workshops/Lattices/announcement.html (395 words)

	2.2 Ordinals
		limits the diversity of well-ordered sets, and produces a canonical representation for all well-ordered sets.
		To prove some theorems regarding ordinals, it is necessary to discuss, more generally, well-ordered sets and order isomorphisms.
		The first elements of a well-ordered set, with respect to its ordering, have special importance, and are useful in proving theorems about well-ordered sets.
	www.u.arizona.edu /~miller/thesis/node7.html (575 words)

	Partially Ordered Sets
		This states that, in any partially ordered set that is not a linear order, there is some pair x,y of elements such that the proportion of linear extensions in which x is above y lies between 1/3 and 2/3.
		The dual of a circle order is not necessarily a circle order
		The dimension of suborders of the Boolean lattice
	www.maths.lse.ac.uk /Personal/graham/research-pos.html (2126 words)

	Analysis WebNotes: Chapter 02, Class 04
		The remaining axiom only involves the order structure of the real numbers, so in this section we will concentrate on linearly ordered sets in general.
		We should start by looking at a couple of examples to see how different sets can be distinguished by properties that have only to do with the ordering.
		In order to describe it we need to have a number of definitions, working towards the definition of supremum and the least upper bound property
	www.math.unl.edu /~webnotes/classes/class04/class04.htm (266 words)

	On greedy algorithms, partially ordered sets, and submodular functions IBM Journal of Research and Development - Find ... (Site not responding. Last check: 2007-10-09)
		Greedy algorithms are widely used in many contexts as a heuristic, with a hope and a prayer (sometimes even a proof) that the rapid calculation will yield an acceptable answer.
		Our aim in this paper is to review an interesting development in greedy theory in a series of related papers appearing in the last few years, to describe the common themes, to offer generalizations of two of these themes, and to comment on weaknesses or lacunae remaining in our treatment.
		Central to this work being summarized and extended are the concepts (to be defined below) of partially ordered set and submodular functions.
	www.findarticles.com /p/articles/mi_qa3751/is_200301/ai_n9171590 (745 words)

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