Definition:Total Ordering/Definition 1

Definition
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. $\mathcal R$ is a total ordering on $S$ iff:
 * $(1): \quad \mathcal R$ is an ordering on $S$
 * $(2): \quad \mathcal R$ is connected

That is, $\mathcal R$ is an ordering with no non-comparable pairs:


 * $\forall x, y \in S: x \mathop {\mathcal R} y \lor y \mathop {\mathcal R} x$

Also see

 * Equivalence of Definitions of Total Ordering