A well-ordered set is normally defined as a totally ordered set in which every nonempty subset has a least member, as the parent object does.

It is possible to define well-ordered sets as follows:

a well-ordered set $X$ is a partially ordered set in which every nonempty subset of $X$ has a least member.

To justify the alternative, we prove that every partially ordered set $X$ in which every nonempty subset has a least member is total:

let $x\in X$ and $y\in X$ , $x\neq y$ . Now, $\{x, y\}$ has a least member, a fortiori, $x, y$ are comparable. Hence, $X$ is totally ordered.

The alternative has the benefit of being a stronger statement in the sense that

\begin{equation*} (partial\;order) \Longrightarrow (total\;order) \end{equation*} given that every nonempty subset has a least member.

Bibliography

1

Schechter, E., Handbook of Analysis and Its Foundations, 1st ed., Academic Press, 1997.

2

Jech, T., Set Theory, 3rd millennium ed., Springer, 2002.