Difference between revisions of "Pre-order"
From Encyclopedia of Mathematics
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''quasi-order, pre-ordering, quasi-ordering'' | ''quasi-order, pre-ordering, quasi-ordering'' | ||
A reflexive and transitive [[Binary relation|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|equivalence]] on $M$. The pre-order $\leq$ induces an [[Order relation|order relation]] (cf. also [[Order (on a set)|Order (on a set)]]) on the quotient set $M/\tilde{}$. | A reflexive and transitive [[Binary relation|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|equivalence]] on $M$. The pre-order $\leq$ induces an [[Order relation|order relation]] (cf. also [[Order (on a set)|Order (on a set)]]) on the quotient set $M/\tilde{}$. | ||
Revision as of 20:23, 19 December 2014
2020 Mathematics Subject Classification: Primary: 06A75 [MSN][ZBL]
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
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