Viewing topic: Totally Ordered Set

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Totally Ordered Set

  • Definition: 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 in 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 in S, either a<=b or b<=a. The first three are the axioms of a partial order, while addition of the trichotomy law defines a total order. Every finite totally ordered set is well ordered. Any two totally ordered sets with k elements (for k a nonnegative integer) are order isomorphic, and therefore have the same order type (which is also an ordinal number).


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