Definition:Well-Ordering/Definition 1

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

Then the ordering $\preceq$ is a well-ordering on $S$ iff every non-empty subset of $S$ has a smallest element under $\preceq$:
 * $\forall T \subseteq S: \exists a \in T: \forall x \in T: a \preceq x$

Also see

 * Equivalence of Definitions of Well-Ordering


 * Definition:Well-Ordered Set
 * Definition:Strict Well-Ordering