Namespaces
Variants
Actions

Difference between revisions of "Pre-order"

From Encyclopedia of Mathematics
Jump to: navigation, search
(MSC 06A75)
(cite Caspard et al (2012))
Line 1: Line 1:
 
{{TEX|done}}{{MSC|06A75}}
 
{{TEX|done}}{{MSC|06A75}}
  
−
''quasi-order, pre-ordering, quasi-ordering''
+
''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 $M/\tilde{}$.
+
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}}

Revision as of 20:28, 27 September 2016

2020 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
Retrieved from "https://encyclopediaofmath.org/index.php?title=Pre-order&oldid=39327"