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Difference between revisions of "Pre-order"

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(MSC 06A75)
(cite Caspard et al (2012))
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''quasi-order, pre-ordering, quasi-ordering''
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''quasi-order, pre-ordering, quasi-ordering, weak order''
  
A reflexive and transitive [[Binary relation|binary relation]] on a set. If $\leq$ is a pre-order on a set $M$, then the relation $a\tilde{}b$ if and only if $a\leq b$ and $b\leq a$, $a,b\in M$, is an [[Equivalence|equivalence]] on $M$. The pre-order $\leq$ induces an [[Order relation|order relation]] (cf. also [[Order (on a set)|Order (on a set)]]) on the quotient set $M/\tilde{}$.
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A reflexive and transitive [[binary relation]] on a set. If $\leq$ is a pre-order on a set $M$, then the ''indifference'' relation $a\tilde{}b$ if and only if $a\leq b$ and $b\leq a$, $a,b\in M$, is an [[equivalence]] on $M$. The pre-order $\leq$ induces an [[order relation]] (cf. also [[Order (on a set)]]) on the quotient set $M/\tilde{}$.
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====References====
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* Caspard, Nathalie; Leclerc, Bruno; Monjardet, Bernard "Finite ordered sets. Concepts, results and uses".  Encyclopedia of Mathematics and its Applications '''144''' Cambridge University Press (2012) ISBN 978-1-107-01369-8 {{ZBL|1238.06001}}

Revision as of 20:28, 27 September 2016

2010 Mathematics Subject Classification: Primary: 06A75 [MSN][ZBL]

quasi-order, pre-ordering, quasi-ordering, weak order

A reflexive and transitive binary relation on a set. If $\leq$ is a pre-order on a set $M$, then the indifference relation $a\tilde{}b$ if and only if $a\leq b$ and $b\leq a$, $a,b\in M$, is an equivalence on $M$. The pre-order $\leq$ induces an order relation (cf. also Order (on a set)) on the quotient set $M/\tilde{}$.

References

  • Caspard, Nathalie; Leclerc, Bruno; Monjardet, Bernard "Finite ordered sets. Concepts, results and uses". Encyclopedia of Mathematics and its Applications 144 Cambridge University Press (2012) ISBN 978-1-107-01369-8 Zbl 1238.06001
How to Cite This Entry:
Pre-order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-order&oldid=35721
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article