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In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ) on some set X which is transitive, antisymmetric, and total. A set paired with a total order is called a totally ordered set, a linearly ordered set, a simply ordered set, or a chain.

If X is totally ordered under ≤, then the following statements hold for all a, b and c in X:

If ab and ba then a = b (antisymmetry);
If ab and bc then ac (transitivity);
ab or ba (totality).

Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the property of "totality" means that any pair of elements in the set of the relation are comparable under the relation. This also means that the set can be diagrammed as a line of elements, giving it the name linear. Totality also implies reflexivity, i.e., aa. Therefore, a total order is also a partial order. The partial order has a weaker form of the third condition. (It requires only reflexivity, not totality.) An extension of a given partial order to a total order is called a linear extension of that partial order.


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