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Difference between revisions of "Pre-order"

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''quasi-order, pre-ordering, quasi-ordering''
 
''quasi-order, pre-ordering, quasi-ordering''
  
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A reflexive and transitive [[Binary relation|binary relation]] on a set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743201.png" /> is a pre-order on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743202.png" />, then the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743203.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743206.png" />, is an [[Equivalence|equivalence]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743207.png" />. The pre-order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743208.png" /> induces an [[Order relation|order relation]] (cf. also [[Order (on a set)|Order (on a set)]]) on the quotient set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074320/p0743209.png" />.
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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 $M/\tilde{}$.

Revision as of 20:36, 14 April 2014

quasi-order, pre-ordering, quasi-ordering

A reflexive and transitive 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 on $M$. The pre-order $\leq$ induces an order relation (cf. also Order (on a set)) on the quotient set $M/\tilde{}$.

How to Cite This Entry:
Pre-order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-order&oldid=15209
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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