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Difference between revisions of "Pre-order"

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''quasi-order, pre-ordering, quasi-ordering''
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{{TEX|done}}{{MSC|06A75}}
  
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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''quasi-order, pre-ordering, quasi-ordering, weak order, preference''
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A [[reflexivity|reflexive]] and [[transitive relation|transitive]] [[binary relation]] on a set. A ''complete'' pre-order is one in which any two elements are comparable.  If $\leq$ is a pre-order on a set $M$, then the ''indifference'' relation $a\sim b$ 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 $M/\sim$.
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In [[mathematical economics]] and [[social choice]] theory, a complete pre-order is often termed a ''preference'' relation.  In [[utility theory]], preferences on a set $M$ are obtained from a real-valued utility function $u$ with $a \leq b$ if $u(a) \leq u(b)$. 
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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}}

Latest revision as of 06:54, 9 April 2023

2020 Mathematics Subject Classification: Primary: 06A75 [MSN][ZBL]

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

A reflexive and transitive binary relation on a set. A complete pre-order is one in which any two elements are comparable. If $\leq$ is a pre-order on a set $M$, then the indifference relation $a\sim b$ 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 $M/\sim$.

In mathematical economics and social choice theory, a complete pre-order is often termed a preference relation. In utility theory, preferences on a set $M$ are obtained from a real-valued utility function $u$ with $a \leq b$ if $u(a) \leq u(b)$.

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=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