Definition. Well-order (well ordering) [002k]
Given a poset \((X, \leq )\), we say that the relation \(\leq \) is a well-order and that \(X\) is well-ordered by \(\leq \) iff. every non-empty subset \(S \subseteq X\) has a least element; that is, there exists an element \(m \in S\) such that for all \(s \in S\), \(m \leq s\). So, formally we have: (1)
\[
\forall (S : \mathcal {P}(X)).\ S \ \mathrlap {\,/}{=}\ \emptyset \implies (\exists m \in S.\ \forall s \in S.\ m \leq s)
\]
When X is nonempty, if we pick out any two-element subset, \(\{a, b\}\) of \(X\), since the subset must have a least element, either \(a \leq b\) or \(b \leq a\). Thus, a well-order is a total order.