Definition:Ordering/Definition 2

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

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

Also see

 * Equivalence of Definitions of Ordering