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Totally Ordered Set


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).


See also

Order Isomorphic, Order Type, Partial Order, Relation, Total Order, Trichotomy Law, Well Ordered Set

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References

Séroul, R. Programming for Mathematicians. Berlin: Springer-Verlag, p. 23, 2000.

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Totally Ordered Set

Cite this as:

Weisstein, Eric W. "Totally Ordered Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TotallyOrderedSet.html

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