Definition:Strict Well-Ordering

Definition

Definition 1

Let $\struct {S, \prec}$ be a relational structure such that $\prec$ is a strict total ordering.

Then $\prec$ is a strict well-ordering on $S$ if and only if $\prec$ is a strictly well-founded relation on $S$.

Definition 2

Let $\struct {S, \prec}$ be a relational structure.

Then $\prec$ is a strict well-ordering of $S$ if and only if:

$\prec$ is a connected relation on $S$
$\prec$ is strictly well-founded on $S$.