Definition:Well-Ordered Set

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 $T \subseteq S: T \ne \varnothing$ has a smallest element:


 * $\exists a \in T: \forall x \in T: a \preceq x$

Note the every in the above.

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

Also known as
A well-ordered set is also found abbreviated as woset.

The term is also found unhyphenated: well ordered.

Also see

 * Definition:Poset
 * Definition:Totally Ordered Set


 * Definition:Well-Ordering