Partial order

Definition

Let $A$ be a set. A partial order on $A$ is a preorder $\le$ on $A$ that is antisymmetric. For all $a, b, c \in A$:

1. $a \le a$;
2. $a \le b$ and $b \le a$ imply $a = b$;
3. $a \le b$ and $b \le c$ imply $a \le c$.

The pair $(A, \le)$ is a partially ordered set.

Remarks

Examples are $\le$ on $\mathbb{R}$, divisibility on $\mathbb{N}$, and $\subseteq$ on the power set of a set. "Partial" means that two elements need not be comparable: neither $\{1\} \subseteq \{2\}$ nor $\{2\} \subseteq \{1\}$. The strict order $a < b$ means $a \le b$ and $a \neq b$. A finite poset can be drawn as a Hasse diagram.

Explanations

L1 human

A partial order ranks some pairs of elements, but not necessarily all pairs. In the family tree, "is an ancestor of or equal to" is a partial order: two cousins are not comparable.