Definition:Preordering/Definition 2

Definition
Let $\mathcal R \subseteq S \times S$ be a relation on a set $S$. $\mathcal R$ is a preordering on $S$ iff:
 * $(1): \quad \mathcal R \circ \mathcal R = \mathcal R$
 * $(2): \quad \Delta_S \subseteq \mathcal R$

where:
 * $\circ$ denotes relation composition
 * $\Delta_S$ denotes the diagonal relation on $S$.

Also see

 * Equivalence of Definitions of Preordering


 * Definition:Ordering/Definition 2