Definition:Well-Ordering/Definition 1

Definition
Let $\struct {S, \preceq}$ be an ordered set or ordered class.

The ordering $\preceq$ is a well-ordering on $S$ every non-empty subset or subclass of $S$ has a smallest element under $\preceq$:
 * $\forall T \subseteq S, T \ne \O: \exists a \in T: \forall x \in T: a \preceq x$

Also see

 * Equivalence of Definitions of Well-Ordering


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