Definition:Ordering/Definition 2

Definition
Let $S$ be a set. An ordering on $S$ is a relation $\RR$ on $S$ such that:
 * $(1): \quad \RR \circ \RR = \RR$
 * $(2): \quad \RR \cap \RR^{-1} = \Delta_S$

where:
 * $\circ$ denotes relation composition
 * $\RR^{-1}$ denotes the inverse of $\RR$
 * $\Delta_S$ denotes the diagonal relation on $S$.

Also see

 * Equivalence of Definitions of Ordering