Definition:Total Ordering/Definition 1

Definition
Let $\RR \subseteq S \times S$ be a relation on a set $S$.

$\RR$ is a total ordering on $S$ :
 * $(1): \quad \RR$ is an ordering on $S$
 * $(2): \quad \RR$ is connected

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


 * $\forall x, y \in S: x \mathop \RR y \lor y \mathop \RR x$

Also see

 * Equivalence of Definitions of Total Ordering