- For all , . (Reflexivity)
- For all , if and , then . (Anti-symmetry)
- For all , if and , then . (Transitivity)
We use to denote .
One example of an ordering is the relation on the natural numbers.
A set with a partial order relation on is also called a partially ordered set (or poset). Note that it under some partial orderings, there can exist elements in , such that , , and . For instance, we could define to mean , in which case we can only write or if . For a more substantial example, we can let be the power set of another set , and define to mean " is a subset of ." In this case, and are not related in either direction in many cases (e.g., when and are disjoint).
We say that a partial order on a set which also satisfies the axiom
- For all , or (Comparability, or trichotomy)
is a total order. For instance, our first example, the relation on the natural numbers, is a total order. A set with a total order is called a totally ordered set.
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