Definition:Total Ordering/Definition 2

Definition
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. $\mathcal R$ is a total ordering on $S$ iff:
 * $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
 * $(2): \quad \mathcal R \cap \mathcal R^{-1} = \Delta_S$
 * $(3): \quad \mathcal R \cup \mathcal R^{-1} = S \times S$

Also see

 * Equivalence of Definitions of Total Ordering


 * Definition:Ordering/Definition 2