Definition:Total Ordering/Definition 2

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

Also see

 * Equivalence of Definitions of Total Ordering


 * Definition:Ordering/Definition 2