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Total Order

A total order satisfies the conditions for a Partial Order plus the comparability condition. A Relation $\leq$ is a partial order on a Set $S$ if

1. Reflexivity: $a\leq a$ for all $a\in S$

2. Antisymmetry: $a\leq b$ and $b\leq a$ implies $a=b$

3. Transitivity: $a\leq b$ and $b\leq c$ implies $a\leq c$,

and is a total order if, in addition,
4. Comparability: For any $a,b \in S$, either $a\leq b$ or $b\leq a$.

See also Partial Order, Relation

© 1996-9 Eric W. Weisstein