> [!NOTE] Definition (Partial Order) > A [[Binary Relation|binary relation]] is a partial order iff it is [[Reflexive Relation|reflexive]], [[Antisymmetric Relation|antisymmetric]], and [[Transitive Relation|transitive]]. > [!Example] Examples > The identity relation is a partial order on $\mathbb{N}.$ > The relation of $\leq$ on $\mathbb{N}$ is a partial order. > A third is the relation of divisibility on positive natural numbers. > The relation $\{ \subset \}$ is a partial order on $\mathcal{P}(X).$ > [!NOTE] Definition (Poset) > A partially ordered set # Properties A [[Total Order|total order]] is partial order that is additionally [[Total Order|total]].