Editing Weak ordering (section)

==Examples==

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=https://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|archive-url=https://web.archive.org/web/20150329105940/http://www.highbeam.com/doc/1G1-162753665.html|url-status=dead|archive-date=March 29, 2015|url-access=subscription }}.</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.

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.

[[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 <math>x</math> is statistically tied with <math>y,</math> and <math>y</math> is statistically tied with <math>z,</math> it might still be possible for <math>x</math> to be clearly better than <math>z,</math> 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=[https://archive.org/details/behavioralsocial00rege/page/113 113ff]|first=Michel|last=Regenwetter|publisher=Cambridge University Press|year=2006|isbn=9780521536660|url=https://archive.org/details/behavioralsocial00rege/page/113}}.</ref>