Definition:Well-Ordered Set

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Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

Then $\left({S, \preceq}\right)$ is a well-ordered set if the ordering $\preceq$ is a well-ordering.


That is, if every non-empty subset of $S$ has a smallest element:

$\forall T \in \mathcal P \left({S}\right): \exists a \in T: \forall x \in T: a \preceq x$

where $\mathcal P \left({S}\right)$ denotes the power set of $S$.


Or, such that $\preceq$ is a well-founded total ordering.


Also known as

The term well-ordered set is sometimes abbreviated as woset.

The term is also found unhyphenated: well ordered or wellordered.


Also see

  • Results about well-orderings and well-ordered sets can be found here.


Sources