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Total order
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{{Short description|Order whose elements are all comparable}} {{redirect-distinguish|Linear order|Linear order (linguistics)}} {{More footnotes|date=February 2016}} {{Use dmy dates|date=August 2021}} In [[mathematics]], a '''total order''' or '''linear order''' is a [[partial order]] in which any two elements are comparable. That is, a total order is a [[binary relation]] <math>\leq</math> on some [[Set (mathematics)|set]] <math>X</math>, which satisfies the following for all <math>a, b</math> and <math>c</math> in <math>X</math>: # <math>a \leq a</math> ([[Reflexive relation|reflexive]]). # If <math>a \leq b</math> and <math>b \leq c</math> then <math>a \leq c</math> ([[Transitive relation|transitive]]). # If <math>a \leq b</math> and <math>b \leq a</math> then <math>a = b</math> ([[Antisymmetric relation|antisymmetric]]). # <math>a \leq b</math> or <math>b \leq a</math> ([[Connected relation|strongly connected]], formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders.{{sfn|Halmos|1968|loc=Ch.14}} Total orders are sometimes also called '''simple''',{{sfn|Birkhoff|1967|p=2}} '''connex''',{{sfn|Schmidt|Ströhlein|1993|p=32}} or '''full orders'''.{{sfn|Fuchs|1963|p=2}} A set equipped with a total order is a '''totally ordered set''';{{sfn|Davey|Priestley|1990|p=3}} the terms '''simply ordered set''',{{sfn|Birkhoff|1967|p=2}} '''linearly ordered set''',{{sfn|Schmidt|Ströhlein|1993|p=32}}{{sfn|Davey|Priestley|1990|p=3}} '''toset'''<ref name="Young 2016">{{cite conference|vauthors=Young AP, Modgil S, Rodrigues O|title=Prioritised Default Logic as Rational Argumentation|conference=Proceedings of the 15th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2016)|url=https://kclpure.kcl.ac.uk/ws/portalfiles/portal/54484626/Prioritised_Default_Logic_as_YOUNG_Published_May2016_GREEN_AAM.pdf|access-date=2025-01-16}}</ref> and '''loset'''<ref>{{Cite journal|last1=Strohmeier|first1=Alfred|last2=Genillard|first2=Christian|last3=Weber|first3=Mats|date=1990-08-01|title=Ordering of characters and strings|journal=ACM SIGAda Ada Letters|language=EN|issue=7|pages=84|doi=10.1145/101120.101136|s2cid=38115497|doi-access=free}}</ref><ref>{{Cite journal|last=Ganapathy|first=Jayanthi|title=Maximal Elements and Upper Bounds in Posets|date=1992|journal=Pi Mu Epsilon Journal|volume=9|issue=7|pages=462–464|jstor=24340068|issn=0031-952X}}</ref> are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'',{{sfn|Davey|Priestley|1990|p=3}} but generally refers to a totally ordered subset of a given partially ordered set. An extension of a given partial order to a total order is called a [[linear extension]] of that partial order.
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