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| {{distinguish|weak order of permutations}}
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| [[Image:13-Weak-Orders.svg|thumb|The 13 possible strict weak orderings on a set of three elements {''a'', ''b'', ''c''}. The only [[partial order|partially ordered]] sets are coloured, while [[total order|totally ordered]] ones are in black. Two orderings are shown as connected by an edge if they differ by a single dichotomy.]]
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| In [[mathematics]], especially [[order theory]], a '''weak ordering''' is a mathematical formalization of the intuitive notion of a [[ranking]] of a [[set (mathematics)|set]], some of whose members may be [[Tie (draw)|tied]] with each other. Weak orders are a generalization of [[totally ordered set]]s (rankings without ties) and are in turn generalized by [[partially ordered set]]s and [[preorder]]s.<ref name="fr11">{{citation|title=Applied Combinatorics|first1=Fred|last1=Roberts|first2=Barry|last2=Tesman|edition=2nd|publisher=CRC Press|year=2011|isbn=9781420099836|at=Section 4.2.4 Weak Orders, pp. 254–256}}.</ref>
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| There are several common ways of formalizing weak orderings, that are different from each other but [[cryptomorphism|cryptomorphic]] (interconvertable with no loss of information): they may be axiomatized as '''strict weak orderings''' (partially ordered sets in which incomparability is a [[transitive relation]]), as '''total preorders''' (transitive binary relations in which at least one of the two possible relations exists between every pair of elements), or as '''ordered partitions''' ([[partition of a set|partitions]] of the elements into disjoint subsets, together with a total order on the subsets). In many cases another representation called a '''preferential arrangement''' based on a [[utility function]] is also possible.
| | In order to invest loads of money on things like controls and for memory cards, appear on the internet for a secondhand discrepancy. Occasionally a store will probably feel out of used-game hardware, which could be quite affordable. Make sure you look recorded at a [http://search.huffingtonpost.com/search?q=web-based+seller%27s&s_it=header_form_v1 web-based seller's] feedback prior to making the purchase so a few seconds . whether you are procuring what you covered.<br><br>In the event you purchasing a game to a child, appear for the one that allows several individuals to perform together. Gaming generally is a singular activity. Nonetheless, it's important to guide your youngster to just be societal, and multiplayer battle of clans trucos gaming can do that. They allow siblings and also buddies to all take a moment and laugh and vie together.<br><br>clash of clans is a ideal game, which usually requires in order to build your personal village, discover warriors, raid tools and build your hold clan and so up. there is a lot a lot significantly more to this video sports and for every these you require jewels in order to really play, as you really like. Clash of Clans hack allows you to obtain as many jewels as you want. There is an unlimited volume gems you could travel with all the Conflict of [http://www.Google.de/search?q=Clans+cheats Clans cheats] you can buy online, however you are being specific about the hyperlink you are using given that some of them primarily waste materials your serious amounts of also dont get anybody anything more.<br><br>In cases where you're playing a gameplay online, and you launch across another player in which seems to be infuriating other players (or you, in particular) intentionally, can not take it personally. This is called "Griefing," and it's the gaming equivalent of Internet trolling. Griefers are just out for negative attention, and you give them all what they're looking about if you interact together. Don't get emotionally utilized in what's happening in addition to simply try to ignore it.<br><br>Should you have virtually any questions with regards to exactly where as well as the best way to employ [http://circuspartypanama.com clash of clans generator], it is possible to contact us at the site. Supercell has absolutely considerable and explained the steps attached to Association Wars, the anew appear passion in Collide of Clans. As your name recommends, a business war is often an absolute strategic battle amid quite a few clans. It just takes abode over the advance of two canicule -- this alertness day plus a action day -- provides the acceptable association that has a ample boodle bonus; although, every association affiliate to who makes acknowledged attacks following a association war additionally makes some benefit loot.<br><br>Using this information, we're accessible in alpha dog substituting values. Application Clash of Clans Cheats' data, let's say to achieve archetype you appetite 1hr (3, 600 seconds) within order to bulk 20 gems, but 1 day (90, 700 seconds) to help largest percentage 260 gems. We can appropriately stipulate a operation for this kind including band segment.<br><br>Do not attempt to eat unhealthy cooking while in xbox computer game actively playing time. This is a slow routine to gain to be able to. Xbox game actively playing is absolutely nothing like physical exercise, and almost all of that fast food will only result in excess fat. In the event own to snack food, accept some thing wholesome for online game actively taking pleasure in times. The individual will thanks for this can. |
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| Weak orderings are counted by the [[ordered Bell number]]s. They are used in [[computer science]] as part of [[partition refinement]] algorithms, and in the [[C++ Standard Library]].
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| ==Examples==
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| In [[horse racing]], the use of [[photo finish]]es has eliminated some, but not all, ties or (as they are called in this context) [[List of dead heat horse races|dead heat]]s, so the outcome of a horse race may be modeled by a weak ordering.<ref>{{citation|title=Those Fascinating Numbers|first=J. M.|last=de Koninck|publisher=American Mathematical Society|year=2009|isbn=9780821886311|url=http://books.google.com/books?id=qYuC1WsDKq8C&pg=PA4|page=4}}.</ref> In an example from the [[Maryland Hunt Cup]] steeplechase in 2007, The Bruce was the clear winner, but two horses Bug River and Lear Charm tied for second place, with the remaining horses farther back; three horses did not finish.<ref>{{citation|title=The Bruce hangs on for Hunt Cup victory: Bug River, Lear Charm finish in dead heat for second|newspaper=[[The Baltimore Sun]]|date=April 29, 2007|first=Kent|last=Baker|url=http://www.highbeam.com/doc/1G1-162753665.html|format=subscription required}}.</ref> In the weak ordering describing this outcome, The Bruce would be first, Bug River and Lear Charm would be ranked after The Bruce but before all the other horses that finished, and the three horses that did not finish would be placed last in the order but tied with each other.
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| The points of the [[Euclidean plane]] may be ordered by their [[Euclidean distance|distance]] from the [[origin (mathematics)|origin]], giving another example of a weak ordering with infinitely many elements, infinitely many subsets of tied elements (the sets of points that belong to a common [[circle]] centered at the origin), and infinitely many points within these subsets. Although this ordering has a smallest element (the origin itself), it does not have any second-smallest elements, nor any largest element.
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| [[Opinion poll]]ing in political elections provides an example of a type of ordering that resembles weak orderings, but is better modeled mathematically in other ways. In the results of a poll, one candidate may be clearly ahead of another, or the two candidates may be statistically tied, meaning not that their poll results are equal but rather that they are within the [[margin of error]] of each other. However, if candidate ''x'' is statistically tied with ''y'', and ''y'' is statistically tied with ''z'', it might still be possible for ''x'' to be clearly better than ''z'', so being tied is not in this case a [[transitive relation]]. Because of this possibility, rankings of this type are better modeled as [[semiorder]]s than as weak orderings.<ref>{{citation|title=Behavioral Social Choice: Probabilistic Models, Statistical Inference, and Applications|pages=113ff|first=Michel|last=Regenwetter|publisher=Cambridge University Press|year=2006|isbn=9780521536660}}.</ref>
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| ==Axiomatizations==
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| ===Strict weak orderings===
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| A '''strict weak ordering''' is a [[binary relation]] < on a set ''S'' that is a [[strict partial order]] (a [[transitive relation]] that is [[Reflexive relation|irreflexive]], or equivalently,<ref>{{cite book|last1=Flaška|first1=V.|last2=Ježek|first2=J.|last3=Kepka|first3=T.|last4=Kortelainen|first4=J.|title=Transitive Closures of Binary Relations I|year=2007|publisher=School of Mathematics - Physics Charles University|location=Prague|page=1|url=http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf?}} Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".</ref> that is [[asymmetric relation|asymmetric]]) in which the relation "neither ''a'' < ''b'' nor ''b'' < ''a''" is transitive.<ref name="fr11"/>
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| The [[equivalence class]]es of this "incomparability relation" partition the elements of ''S'', and are [[Total order|totally ordered]] by <. Conversely, any total order on a [[Partition (set theory)|partition]] of ''S'' gives rise to a strict weak ordering in which ''x'' < ''y'' if and only if there exists sets ''A'' and ''B'' in the partition with ''x'' in ''A'', ''y'' in ''B'', and ''A'' < ''B'' in the total order.
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| As a non-example, consider the partial order in the set {''a'', ''b'', ''c''} defined by the relationship ''b'' < ''c''. The pairs ''a'',''b'' and ''a'',''c'' are incomparable but ''b'' and ''c'' are related, so incomparability does not form an equivalence relation and this example is not a strict weak ordering.
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| A strict weak ordering has the following properties. For all ''x'' and ''y'' in ''S'',
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| * For all ''x'', it is not the case that ''x'' < ''x'' ([[Irreflexive relation|irreflexivity]]).
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| * For all ''x'', ''y'', if ''x'' < ''y'' then it is not the case that ''y'' < ''x'' ([[Asymmetric relation|asymmetry]]).
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| * For all ''x'', ''y'', and ''z'', if ''x'' < ''y'' and ''y'' < ''z'' then ''x'' < ''z'' ([[Transitive relation|transitivity]]).
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| * For all ''x'', ''y'', and ''z'', if ''x'' is incomparable with ''y'', and ''y'' is incomparable with ''z'', then ''x'' is incomparable with ''z'' (transitivity of incomparability).
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| This list of properties is somewhat redundant, as asymmetry follows readily from irreflexivity and transitivity.
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| Transitivity of incomparability (together with transitivity) can also be stated in the following forms:
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| * If ''x'' < ''y'', then for all ''z'', either ''x'' < ''z'' or ''z'' < ''y'' or both.
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| Or:
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| * If ''x'' is incomparable with ''y'', then for all ''z'' ≠ ''x'', ''z'' ≠ ''y'', either (''x'' < ''z'' and ''y'' < ''z'') or (''z'' < ''x'' and ''z'' < ''y'') or (''z'' is incomparable with ''x'' and ''z'' is incomparable with ''y'').
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| ===Total preorders<!--This section is linked from [[Preference (economics)]]-->===
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| Strict weak orders are very closely related to '''total preorders''' or '''(non-strict) weak orders''', and the same mathematical concepts that can be modeled with strict weak orderings can be modeled equally well with total preorders. A total preorder or weak order is a [[preorder]] that is [[Total relation|total]]; that is, no pair of items is incomparable. A total preorder <math>\lesssim</math> satisfies the following properties:
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| * For all ''x'', ''y'', and ''z'', if ''x'' <math>\lesssim</math> ''y'' and ''y'' <math>\lesssim</math> ''z'' then ''x'' <math>\lesssim</math> ''z'' (transitivity).
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| * For all ''x'' and ''y'', ''x'' <math>\lesssim</math> ''y'' or ''y'' <math>\lesssim</math> ''x'' (totality).
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| ** Hence, for all ''x'', ''x'' <math>\lesssim</math> ''x'' (reflexivity).
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| A [[total order]] is a total preorder which is antisymmetric, in other words, which is also a [[Partially ordered set|partial order]]. Total preorders are sometimes also called '''preference relations'''.
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| The [[Complement (set theory)|complement]] of a strict weak order is a total preorder, and vice versa, but it seems more natural to relate strict weak orders and total preorders in a way that preserves rather than reverses the order of the elements. Thus we take the [[inverse relation|inverse]] of the complement: for a strict weak ordering <, define a total preorder <math>\lesssim</math> by setting ''x'' <math>\lesssim</math> ''y'' whenever it is not the case that ''y'' < ''x''. In the other direction, to define a strict weak ordering < from a total preorder <math>\lesssim</math>, set ''x'' < ''y'' whenever it is not the case that ''y'' <math>\lesssim</math> ''x''.<ref>{{citation|title=Multicriteria Optimization|first=Matthias|last=Ehrgott|publisher=Springer|year=2005|isbn=9783540276593|url=http://books.google.com/books?id=AwRjo6iP4_UC&pg=PA10|at=Proposition 1.9, p. 10}}.</ref>
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| In any preorder there is a [[Preorder#Constructions|corresponding equivalence relation]] where two elements ''x'' and ''y'' are defined as '''equivalent''' if ''x'' <math>\lesssim</math> ''y'' and ''y'' <math>\lesssim</math> ''x''. In the case of a ''total'' preorder the corresponding partial order on the set of equivalence classes is a total order. Two elements are equivalent in a total preorder if and only if they are incomparable in the corresponding strict weak ordering.
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| ===Ordered partitions===
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| A [[partition of a set]] ''S'' is a family of disjoint subsets of ''S'' that have ''S'' as their union. A partition, together with a [[total order]] on the sets of the partition, gives a structure called by [[Richard P. Stanley]] an '''ordered partition'''<ref>{{citation|first=Richard P.|last=Stanley|authorlink=Richard P. Stanley|title=Enumerative Combinatorics, Vol. 2|series=Cambridge Studies in Advanced Mathematics|volume=62|page=297|publisher=Cambridge University Press|year=1997}}.</ref> and by [[Theodore Motzkin]] a '''list of sets'''.<ref>{{citation
| |
| | last = Motzkin | first = Theodore S. | authorlink = Theodore Motzkin
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| | contribution = Sorting numbers for cylinders and other classification numbers
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| | location = Providence, R.I.
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| | mr = 0332508
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| | pages = 167–176
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| | publisher = Amer. Math. Soc.
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| | title = Combinatorics (Proc. Sympos. Pure Math., Vol. XIX, Univ. California, Los Angeles, Calif., 1968)
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| | year = 1971}}.</ref> An ordered partition of a finite set may be written as a [[finite sequence]] of the sets in the partition: for instance, the three ordered partitions of the set {''a'', ''b''} are
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| :{''a''}, {''b''},
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| :{''b''}, {''a''}, and
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| :{''a'', ''b''}.
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| In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition. In the other direction, any ordered partition gives rise to a strict weak ordering in which two elements are incomparable when they belong to the same set in the partition, and otherwise inherit the order of the sets that contain them.
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| ===Representation by functions===
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| For sets of sufficiently small [[cardinality]], a third axiomatization is possible, based on real-valued functions. If ''X'' is any set and ''f'' a real-valued function on ''X'' then ''f'' induces a strict weak order on ''X'' by setting ''a'' < ''b'' if and only if ''f''(''a'') < ''f''(''b''). The associated total preorder is given by setting ''a''<math>{}\lesssim{}</math>''b'' if and only if ''f''(''a'') ≤ ''f''(''b''),
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| and the associated equivalence by setting ''a''<math>{}\sim{}</math>''b'' if and only if ''f''(''a'') = ''f''(''b''). | |
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| The relations do not change when ''f'' is replaced by ''g'' <small>o</small> ''f'' ([[Function composition|composite function]]), where ''g'' is a [[Monotonic function|strictly increasing]] real-valued function defined on at least the range of ''f''. Thus e.g. a [[Utility#Utility_functions|utility function]] defines a [[preference]] relation. In this context, weak orderings are also known as '''preferential arrangements'''.<ref>{{citation
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| | last = Gross | first = O. A.
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| | doi = 10.2307/2312725
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| | journal = The American Mathematical Monthly
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| | mr = 0130837
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| | pages = 4–8
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| | title = Preferential arrangements
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| | volume = 69
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| | year = 1962}}.</ref>
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| If ''X'' is finite or countable, every weak order on ''X'' can be represented by a function in this way.<ref name="roberts">{{citation
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| | last = Roberts | first = Fred S. | authorlink = Fred S. Roberts
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| | year = 1979
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| | title = Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences
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| | at = Theorem 3.1
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| | series = Encyclopedia of Mathematics and its Applications
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| | volume = 7
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| | isbn = 978-0-201-13506-0
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| | publisher = Addison-Wesley}}.</ref> However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the [[lexicographic order]] on '''R'''<sup>''n''</sup>. Thus, while in most preference relation models the relation defines a utility function [[up to]] order-preserving transformations, there is no such function for [[lexicographic preferences]].
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| More generally, if ''X'' is a set, and ''Y'' is a set with a strict weak ordering "<", and ''f'' a function from ''X'' to ''Y'', then ''f'' induces a strict weak ordering on ''X'' by setting ''a'' < ''b'' if and only if ''f''(''a'') < ''f''(''b''). As before, the associated total preorder is given by setting ''a''<math>{}\lesssim{}</math>''b'' if and only if ''f''(''a'')<math>{}\lesssim{}</math>''f''(''b''), and the associated equivalence by setting ''a''<math>{}\sim{}</math>''b'' if and only if ''f''(''a'')<math>{}\sim{}</math>''f''(''b''). It is not assumed here that ''f'' is an [[injective function]], so a class of two equivalent elements on ''Y'' may induce a larger class of equivalent elements on ''X''. Also, ''f'' is not assumed to be an [[surjective function]], so a class of equivalent elements on ''Y'' may induce a smaller or empty class on ''X''. However, the function ''f'' induces an injective function that maps the partition on ''X'' to that on ''Y''. Thus, in the case of finite partitions, the number of classes in ''X'' is less than or equal to the number of classes on ''Y''.
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| ==Related types of ordering==
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| [[Semiorder]]s generalize strict weak orderings, but do not assume transitivity of incomparability.<ref>{{citation | |
| | last = Luce | first = R. Duncan | authorlink = R. Duncan Luce
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| | mr = 0078632
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| | journal = Econometrica
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| | pages = 178–191
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| | title = Semiorders and a theory of utility discrimination
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| | jstor = 1905751
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| | volume = 24
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| | year = 1956}}.</ref> A strict weak order that is [[Trichotomy (mathematics)|trichotomous]] is called a '''strict total order'''.<ref name="h2pi">{{citation|title=How to Prove It: A Structured Approach|first=Daniel J.|last=Velleman|publisher=Cambridge University Press|year=2006|isbn=9780521675994|page=204|url=http://books.google.com/books?id=lptwaMuAtBAC&pg=PA204}}.</ref> The total preorder which is the inverse of its complement is in this case a [[total order]].
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| For a strict weak order "<" another associated reflexive relation is its [[Binary_relation#Operations_on_binary_relations|reflexive closure]], a (non-strict) partial order "≤". The two associated reflexive relations differ with regard to different ''a'' and ''b'' for which neither ''a'' < ''b'' nor ''b'' < ''a'': in the total preorder corresponding to a strict weak order we get ''a'' <math>\lesssim</math> ''b'' and ''b'' <math>\lesssim</math> ''a'', while in the partial order given by the reflexive closure we get neither ''a'' ≤ ''b'' nor ''b'' ≤ ''a''. For strict total orders these two associated reflexive relations are the same: the corresponding (non-strict) total order.<ref name="h2pi"/> The reflexive closure of a strict weak ordering is a type of [[series-parallel partial order]].
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| ==All weak orders on a finite set==
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| ===Combinatorial enumeration===
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| {{main|ordered Bell number}}
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| The number of distinct weak orders (represented either as strict weak orders or as total preorders) on an ''n''-element set is given by the following sequence {{OEIS|id=A000670}}:
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| {{number of relations}}
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| These numbers are also called the '''Fubini numbers''' or '''ordered Bell numbers'''.
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| For example, for a set of three labeled items, there is one weak order in which all three items are tied. There are three ways of partitioning the items into one [[Singleton (mathematics)|singleton]] set and one group of two tied items, and each of these partitions gives two weak orders (one in which the singleton is smaller than the group of two, and one in which this ordering is reversed), giving six weak orders of this type. And there is a single way of partitioning the set into three singletons, which can be totally ordered in six different ways. Thus, altogether, there are 13 different weak orders on three items.
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| ===Adjacency structure===
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| [[File:Permutohedron.svg|thumb|300px|The permutohedron on four elements, a three-dimensional [[convex polyhedron]]. It has 24 vertices, 36 edges, and 14 two-dimensional faces, which all together with the whole three-dimensional polyhedron correspond to the 75 weak orderings on four elements.]]
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| Unlike for partial orders, the family of weak orderings on a given finite set is not in general connected by moves that add or remove a single order relation to a given ordering. For instance, for three elements, the ordering in which all three elements are tied differs by at least two pairs from any other weak ordering on the same set, in either the strict weak ordering or total preorder axiomatizations. However, a different kind of move is possible, in which the weak orderings on a set are more highly connected. Define a ''dichotomy'' to be a weak ordering with two equivalence classes, and define a dichotomy to be ''compatible'' with a given weak ordering if every two elements that are related in the ordering are either related in the same way or tied in the dichotomy. Alternatively, a dichotomy may be defined as a [[Dedekind cut]] for a weak ordering. Then a weak ordering may be characterized by its set of compatible dichotomies. For a finite set of labeled items, every pair of weak orderings may be connected to each other by a sequence of moves that add or remove one dichotomy at a time to or from this set of dichotomies. Moreover, the [[undirected graph]] that has the weak orderings as its vertices, and these moves as its edges, forms a [[partial cube]].<ref>{{citation|title=Media Theory: Interdisciplinary Applied Mathematics|first1=David|last1=Eppstein|author1-link=David Eppstein|first2=Jean-Claude|last2=Falmagne|author2-link=Jean-Claude Falmagne|first3=Sergei|last3=Ovchinnikov|publisher=Springer|year=2008|at=Section 9.4, Weak Orders and Cubical Complexes, pp. 188–196}}.</ref>
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| Geometrically, the total orderings of a given finite set may be represented as the vertices of a [[permutohedron]], and the dichotomies on this same set as the facets of the permutohedron. In this geometric representation, the weak orderings on the set correspond to the faces of all different dimensions of the permutohedron (including the permutohedron itself, but not the empty set, as a face). The [[codimension]] of a face gives the number of equivalence classes in the corresponding weak ordering.<ref>{{citation|first=Günter M.|last=Ziegler|authorlink=Günter M. Ziegler|title=Lectures on Polytopes|publisher=Springer|series=Graduate Texts in Mathematics|volume=152|year=1995|page=18}}.</ref> In this geometric representation the partial cube of moves on weak orderings is the graph describing the [[covering relation]] of the [[face lattice]] of the permutohedron.
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| For instance, for ''n'' = 3, the permutohedron on three elements is just a regular hexagon. The face lattice of the hexagon (again, including the hexagon itself as a face, but not including the empty set) has thirteen elements: one hexagon, six edges, and six vertices, corresponding to the one completely tied weak ordering, six weak orderings with one tie, and six total orderings. The graph of moves on these 13 weak orderings is shown in the figure.
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| ==Applications==
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| As mentioned above, weak orders have applications in utility theory.<ref name="roberts"/> In [[linear programming]] and other types of [[combinatorial optimization]] problem, the prioritization of solutions or of bases is often given by a weak order, determined by a real-valued [[objective function]]; the phenomenon of ties in these orderings is called "degeneracy", and several types of tie-breaking rule have been used to refine this weak ordering into a total ordering in order to prevent problems caused by degeneracy.<ref>{{citation|title=Linear Programming|first=Vašek|last=Chvátal|authorlink=Vašek Chvátal|publisher=Macmillan|year=1983|isbn=9780716715870|pages=29–38|url=http://books.google.com/books?id=DN20_tW_BV0C&pg=PA29}}.</ref>
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| Weak orders have also been used in [[computer science]], in [[partition refinement]] based algorithms for [[lexicographic breadth-first search]] and [[Coffman–Graham algorithm|lexicographic topological ordering]]. In these algorithms, a weak ordering on the vertices of a graph (represented as a family of sets that [[partition of a set|partition]] the vertices, together with a [[doubly linked list]] providing a total order on the sets) is gradually refined over the course of the algorithm, eventually producing a total ordering that is the output of the algorithm.<ref>{{citation
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| | last1 = Habib | first1 = Michel
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| | last2 = Paul | first2 = Christophe
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| | last3 = Viennot | first3 = Laurent
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| | doi = 10.1142/S0129054199000125
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| | issue = 2
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| | journal = International Journal of Foundations of Computer Science
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| | mr = 1759929
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| | pages = 147–170
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| | title = Partition refinement techniques: an interesting algorithmic tool kit
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| | volume = 10
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| | year = 1999}}.</ref>
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| In the [[C++ Standard Library|Standard Library]] for the [[C++]] programming language, the [[Associative containers (C++)|set and multiset data types]] sort their input by a comparison function that is specified at the time of template instantiation, and that is assumed to implement a strict weak ordering.<ref>{{citation|title=The C++ Standard Library: A Tutorial and Reference|first=Nicolai M.|last=Josuttis|publisher=Addison-Wesley|year=2012|isbn=9780132977739|url=http://books.google.com/books?id=9DEJKhasp7gC&pg=PT469|page=469}}.</ref> | |
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| == References ==
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| {{reflist}}
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| [[Category:Order theory]]
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| [[Category:Integer sequences]]
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| [[Category:Mathematical relations]]
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