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# Topic: Totally ordered set

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 Total order - Wikipedia, the free encyclopedia A set paired with an associated total order on it is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain. Thus a total order is also a partial order, that is, a binary relation which is reflexive, antisymmetric and transitive. Sets ordered by inclusion (A < B iff A is a subset of B) are in general not totally ordered (neither {1} < {2} nor {2} < {1}) However, frequently special collections of sets turn out to be totally ordered by inclusion. en.wikipedia.org /wiki/Total_order   (1046 words)

 Station Information - Total order In mathematics, a total order or linear order on a set X is a binary relation that is reflexive, antisymmetric, transitive and total. Alternatively, one may define a totally ordered set as a particular kind of lattice, namely one in which we have {a v b, a ^ b} = {a, b} for all a, b. Similarly, the unique smallest totally ordered set with neither an upper nor a lower bound is the integers. www.stationinformation.com /encyclopedia/t/to/total_order.html   (347 words)

 PlanetMath: well ordered set   (Site not responding. Last check: 2007-11-07) 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   (136 words)

 Order topology - Wikipedia, the free encyclopedia The open sets in X are the sets that are a union of (possibly infinitely many) such open intervals and rays. The left order topology on X is the topology whose open sets consist of intervals of the form (a, ∞). For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff. en.wikipedia.org /wiki/Order_topology   (375 words)

 PlanetMath: total order   (Site not responding. Last check: 2007-11-07) is then called a total order or a linear order (or total ordering or linear ordering). A totally ordered set is also sometimes called a chain, especially when it is considered as a subset of some other poset. This is version 9 of total order, born on 2001-10-06, modified 2004-03-06. planetmath.org /encyclopedia/TotallyOrdered2.html   (173 words)

 Total order - FreeEncyclopedia   (Site not responding. Last check: 2007-11-07) A total order or linear order on a set X is a binary relation that is reflexive, antisymmetric, transitive and total. The totally ordered set X turns into a topological space if we define a subset to be open if and only if it is a union of (possibly infinitely many) such open intervals. The unique smallest unbounded totally ordered set which also happens to be dense in the sense that (a, b) is non-empty for every a < b, is the rational numbers. openproxy.ath.cx /to/Total_order.html   (539 words)

 Dedekind cut - Freepedia   (Site not responding. Last check: 2007-11-07) In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is closed upwards. In this way, the set of all Dedekind cuts is itself a linearly ordered set, and, moreover, it does have the least-upper-bound property, i.e., its every nonempty subset that has an upper bound has a least upper bound. More generally, in a partially ordered set S, the set of all nonempty downwardly closed subsets (also called order ideals) is a set partially ordered by inclusion, and in the same way we embed S within a larger partially ordered set that, generally unlike the original set S, does have the least-upper-bound property. en.freepedia.org /Dedekind_completion.html   (506 words)

 Study Guides for Discrete Mathematics   (Site not responding. Last check: 2007-11-07) Members of a partially ordered set (poset) may be compared in a more linear fashion. If S is a totally ordered set such that every nonempty subset has a least element then S is well ordered. And, unlike the elements of a set, the order of the components is important. cs-netlab-01.lynchburg.edu /courses/discmath/Stud66.htm   (524 words)

 Refinements Each element of this set is constructed from one element of the 78#78 disjunctive sets. The essential requirements is that one should be able to decompose the whole policy into totally ordered policy statements. The ordering in the set 83#83 determines the policy statement which is enforced if several policy statements are simultaneously satisfied. www.isi.edu /gost/info/gaaapi/doc/papers/mmm2001/node12.html   (487 words)

 Mudd Math Fun Facts: Ordinal Numbers Any (totally) ordered set which has this property is said to be well-ordered. Formally an ordinal number is any set which is (i) transitive (every member is a subset) and (ii) strictly well-ordered by the membership relation. The set is also well-ordered because % is a member of {%}, and % and {%} are both members of {%,{%}}. www.math.hmc.edu /funfacts/ffiles/30003.8.shtml   (590 words)

 Clearing up the market cycle... best Totally Ordered Set   (Site not responding. Last check: 2007-11-07) a total order on it is called a totally ordered set, a linearly ordered set, or a chain... Totally Ordered Set -- from MathWorld Totally Ordered Set -- from MathWorld A total order (or "totally ordered set," or "linearly ordered set") is a set plus a relation on the set (called a total order) that satisfies the conditions for a... Well Ordered Set -- from MathWorld Well Ordered Set -- from MathWorld A totally ordered set (A,\leq) is said to be well ordered (or have a well-founded order) iff every nonempty subset of A has a least element (Ciesielski 1997, p. ascot.pl /th/Fourier5/Totally-Ordered-Set.htm   (874 words)

 MA522 Fall 2004 Homework   (Site not responding. Last check: 2007-11-07) A totally ordered set with neither a maximal nor a minimal element A partially ordered set with neither a maximal nor a minimal element Prove that the set of algebraic numbers (numbers that are a root of a polynomial with integer coefficients) is countable. people.clarkson.edu /~pschultz/fall04/ma522/homework.html   (209 words)

 [No title] Show that the set of rational numbers, R, is equivalent to the set of Integers, Z, i.e. Consider the relation R : "is a subset of" on the set N. Classify the relation as in question (4). Design the dfa to accept L. For each of the following languages, prove that it is a regular language by construction or cogent argument or prove that it is not, by careful application of the pumping lemma. www.bridgeport.edu /~dichter/cs520/problemSet.01.doc   (329 words)

 TOTAL Nearby terms: TOSS « total function « totally ordered « totally ordered set » total ordering » toto » touchpad A relation R on a set A which is a partial ordering; i.e. Nearby terms: total function « totally ordered « totally ordered set « total ordering » toto » touchpad » touch screen www.linuxguruz.com /foldoc/foldoc.php?TOTAL   (153 words)

 Policy Representation An EACL entry consists of a positive or negative access right and four condition blocks: a set of pre-conditions, a set of request-result conditions, a set of mid-conditions and a set of post-conditions. An ordered evaluation approach is easier to implement as it allows only partial evaluation of an EACL and resolves the authorization conflicts. The problem with this approach is that it requires total ordering among authorizations. www.isi.edu /gost/info/gaaapi/doc/papers/policy2002/node9.html   (325 words)

 Articles - Directed set   (Site not responding. Last check: 2007-11-07) Directed sets are generalizations of totally ordered sets. The set of natural numbers N with the ordinary order ≤ is a directed set (and so is every totally ordered set). Directed subsets are most commonly used in domain theory, where one studies orders for which these sets are required to have a least upper bound. www.gaple.com /articles/Directed_set   (488 words)

 The Ultimate Simplicial set - American History Information Guide and Reference Using the language of category theory, a simplicial set is a simplicial object in Alternatively, a simplicial set can be viewed as a presheaf on Δ. These definitions arise from the relationship of the conditions imposed on the face and degeneracy maps to the category of finite, totally ordered sets. www.historymania.com /american_history/Simplicial_set   (156 words)

 Total order : Totally ordered set   (Site not responding. Last check: 2007-11-07) terms defined : Total order : Totally ordered set All is still licensed under the GNU FDL. If you received it such person may choose to alternatively give you a replacement. www.termsdefined.net /to/totally-ordered-set.html   (738 words)

 Def. of Interval   (Site not responding. Last check: 2007-11-07) An Interval is a subset S of a totally ordered set T with the property that whenever then z is in S. "totally ordered" set is a set over which the Upper Bound -- definition of "upper bound" and "least upper bound" of either sets or functions mcraeclan.com /MathHelp/CalculusDefinitionInterval.htm   (125 words)

 totally ordered set - OneLook Dictionary Search We found 6 dictionaries with English definitions that include the word totally ordered set: Tip: Click on the first link on a line below to go directly to a page where "totally ordered set" is defined. Totally Ordered Set : Eric Weisstein's World of Mathematics [home, info] onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=totally+ordered+set   (108 words)

 [No title] Thus a well ordering should satisfy Reflexivity : a EMBED Equation.3 a for all a in A. Anti-symmetry: a EMBED Equation.3 b and b EMBED Equation.3 a implies a=b. Example 1: Z is a totally ordered set w.r.t standard ordering  EMBED Equation.3 , but it is not well-ordered by  EMBED Equation.3 . Example 3: R is not a well-ordered set w.r.t standard ordering  EMBED Equation.3 . www.math.metu.edu.tr /~berkman/choice/anil.doc   (829 words)

 TOTAL from FOLDOC   (Site not responding. Last check: 2007-11-07) Next: totally ordered, totally ordered set, total ordering, toto, touchpad Next: totally ordered set, total ordering, toto, touchpad, touch screen Next: total ordering, toto, touchpad, touch screen, tourist, touristic www.instantweb.com /foldoc/foldoc.cgi?TOTAL   (148 words)

 Comparing models of computation We give a denotational framework (a meta model) within which certain properties of models of computation can be understood and compared. Each event is a value-tag pair, where the tags can come from a partially ordered or totally ordered set. Timed models are where the set of tags is totally ordered. csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/iccad/1996/7597/00/7597toc.xml&DOI=10.1109/ICCAD.1996.569613   (247 words)

 The Ultimate Ordered ring - American History Information Guide and Reference In abstract algebra, an ordered ring is a ring that is a totally ordered set. In analogy with ordinary numbers, we call an element c of an ordered ring positive if c > 0 and negative if c < 0. The order on a ring is required to satisfy certain conditions that are familiar properties of ordinary numbers: www.historymania.com /american_history/Ordered_ring   (125 words)

 [No title]   (Site not responding. Last check: 2007-11-07) After we have done this, we can remove the arrows on our edges since we already know that all of them must point upwards. [Rearrange nodes]¡V 2Uª ,ð¢ ð  ð0ì[k¿¿Àÿðä`ðÐð,$ñ Dÿÿÿ 0 ðû¨§Since a partial order must be reflexive, we can dispense with the loops that we know must be present at every vertex in order to unclutter the diagram. Then the relation          .-ð ûßÿ@Symbol-.  2 E Í¡.-ð ûßÿ@Times New Roman-. 42 E? is a partial ordering on P(A). www.cs.virginia.edu /~cmt5n/cs202/slides/chap7/chap7sec6.ppt (402 words)  Total order Life of Total Prayer: Selected Writings of Catherine of Siena The Community Charge Benefits (Permitted Total) Order 1990: Social Security (Statutory Instruments: 1990: 533) The Housing Benefit (Permitted Totals) Amendment Order 1995: Social Security (Statutory Instruments: 1995: 3151) news-server.org /t/to/total_order.html (446 words)  Finding the$\alpha N\$-Th Largest Element (ResearchIndex)   (Site not responding. Last check: 2007-11-07) 1), from a totally ordered set of n elements, using at most (1 + (1 + o(1))H(ff)) \Delta n comparisons, where H(ff) is the binary entropy function and the o(1) stands for a function that tends to 0 as ff tends to 0. For small values of ff this is almost the best possible as there is a lower bound of about (1 + H(ff)) \Delta n comparisons. 9 On the selection of the k-th element of an ordered set by pa.. citeseer.ist.psu.edu /dor95finding.html   (453 words)

 FPM 1998, vol. 4, no. 2, pp. 717-723   (Site not responding. Last check: 2007-11-07) On the global dimension of the abelian group valued diagrams category over a totally ordered set In this paper we shall calculate the global dimension of the category of functors from a totally ordered set to the category of abelian groups. We prove that for the totally ordered set of real numbers this global dimension equals 3. www.emis.ams.org /journals/FPM/eng/98/982/98217h.htm   (85 words)

 totally ordered set : Definition from the Online Dictionary at Datasegment.com   (Site not responding. Last check: 2007-11-07) totally ordered set : Definition from the Online Dictionary at Datasegment.com Online Dictionary : T : totally ordered set 1 definition found totally ordered set - Free On-line Dictionary of Computing (19 Sep 2003) : totally ordered set A set with a total ordering. onlinedictionary.datasegment.com /word/totally+ordered+set   (45 words)

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