# Difference between revisions of "Pre-order"

From Encyclopedia of Mathematics

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− | ''quasi-order, pre-ordering, quasi-ordering'' | + | ''quasi-order, pre-ordering, quasi-ordering, weak order'' |

− | A reflexive and transitive [[ | + | 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==== | ||

+ | * 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