Partial order (poset)

Let $P$ be a set and let $\le$ be a relation on $P$. Then $\le$ is a partial order if:

• (Reflexive) $\forall x\in P,\; x\le x$.
• (Antisymmetric) $\forall x,y\in P,\; (x\le y \text{ and } y\le x)\Rightarrow x=y$.
• (Transitive) $\forall x,y,z\in P,\; (x\le y \text{ and } y\le z)\Rightarrow x\le z$.

A set equipped with a partial order is called a partially ordered set (or poset). A total order is a partial order in which every pair of elements is comparable.

Examples:

• $(\mathbb{R},\le)$ with the usual order is a partial order (in fact a total order).
• On the power set of $X$, the subset relation $\subseteq$ is a partial order.
• Divisibility on $\mathbb{N}$, where $a\le b$ means $a$ divides $b$, is a partial order (not total).