Definition:Well-Ordered Set

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

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

That is, if every $$T \subseteq S: T \ne \varnothing$$ has a minimal or first element.

That is, $$\exists a \in T: \forall x \in T: a \preceq x$$.

Note the every in the above.

Also see

 * Partially ordered set (poset)
 * Totally ordered set (toset)


 * Well-ordering


 * Well-Ordering is Total Ordering, which shows that every woset is in fact a toset.