Difference between revisions of "Pre-order"
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A [[reflexivity|reflexive]] and [[transitivity|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 $a\tilde{}b$ 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 $M/\tilde{}$. | A [[reflexivity|reflexive]] and [[transitivity|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 $a\tilde{}b$ 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 $M/\tilde{}$. | ||
| − | In [[mathematical economics]] and [[social choice]] theory, a complete pre-order is often termed a ''preference'' | + | 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==== | ====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}} | * 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 18:29, 17 November 2017
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 $a\tilde{}b$ 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 $M/\tilde{}$.
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
Pre-order. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pre-order&oldid=42308