total order

Definition

A total order is a relation on a totally ordered set.

A totally ordered set is a set satisfying conditions for a partial order in addition to the comparability condition (trichotomy law). Hence a relation $\leq$ is a total order on a set $S$ if the following hold

• reflexivity: $a \leq a$ for all $a \in S$
• antisymmetry: $a \leq b$ and $b \leq a$ implies $a = b$
• transitivity: $a \leq b$ and $b \leq c$ implies $a \leq c$
• comparability (trichotomy law): for any $a,b \in S$, either $a \leq b$ or $b \leq a$

References

1. https://mathworld.wolfram.com/TotalOrder.html
2. https://mathworld.wolfram.com/TotallyOrderedSet.html
3. https://www.bananaspace.org/wiki/全序集