Definition:Well-Ordered Set

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

Then $\struct {S, \preceq}$ is a well-ordered set if the ordering $\preceq$ is a well-ordering.

That is, if every non-empty subset of $S$ has a smallest element:
 * $\forall T \in \powerset S: \exists a \in T: \forall x \in T: a \preceq x$

where $\powerset S$ denotes the power set of $S$.

That is, such that $\preceq$ is a well-founded total ordering.

Also see

 * Definition:Ordered Set
 * Definition:Partially Ordered Set
 * Definition:Totally Ordered Set


 * Definition:Well-Ordering