A total order (or "totally ordered set, " or "linearly ordered set") is a set plus a relation on the set (called a total order) that satisfies the conditions for a partial order plus an additional condition known as the comparability condition. A relation <= is a total order on a set S ("<= totally orders S") if the following properties hold. 1. Reflexivity: a<=a for all a element S. 2. Antisymmetry: a<=b and b<=a implies a = b. 3. Transitivity: a<=b and b<=c implies a<=c. 4. Comparability (trichotomy law): For any a, b element S, either a<=b or b<=a.
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