Definition:Well-Ordered Set

Let $$\left({S; \preceq}\right)$$ be a totally 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.