Partial order

A partial order on a set $X$ is a relation $\preceq\ \subseteq X\times X$ such that for all $x,y,z\in X$:

• (Reflexive) $x\preceq x$.
• (Antisymmetric) If $x\preceq y$ and $y\preceq x$, then $x=y$.
• (Transitive) If $x\preceq y$ and $y\preceq z$, then $x\preceq z$.

A set equipped with a partial order is a partially ordered set (poset). Partial orders capture “comparison” that may leave some pairs incomparable.

Examples:

• $(\mathcal{P}(X),\subseteq)$ is a poset for any set $X$.
• On $\mathbb{N}$, define $a\preceq b$ iff $a$ divides $b$; this is a partial order.
• On $\mathbb{R}^2$, define $(x_1,y_1)\preceq(x_2,y_2)$ iff $x_1\le x_2$ and $y_1\le y_2$ (coordinatewise order); many pairs are incomparable.