Namespaces
Variants
Actions

Difference between revisions of "Pre-order"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (better)
Line 3: Line 3:
 
''quasi-order, pre-ordering, quasi-ordering, weak order, preference''
 
''quasi-order, pre-ordering, quasi-ordering, weak order, preference''
  
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 $a\sim b$ 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 $u(a) \leq u(b)$.   

Revision as of 16:38, 30 December 2018

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=42308
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article