# Pre-order

From Encyclopedia of Mathematics

*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=31681

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