Pre-order

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

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

A reflexive and transitive binary relation on a set. 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 $M/\tilde{}$.

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