# Difference between revisions of "Pre-order"

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

− | A [[reflexivity|reflexive]] and [[ | + | 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 $M/\sim$. |

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 $u(a) \leq u(b)$. | 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 $u(a) \leq u(b)$. |

## Latest revision as of 12:29, 6 February 2021

2010 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=43595