Partial orders

Definition. $\le$ is a partial order on $A$ if:
- *Reflexive*: $x\le x$ for all $x$;
- *Antisymmetric*: $x\le y$ and $y\le x$ implies $x=y$;
- *Transitive*: $x\le y$ and $y\le z$ implies $x\le z$.

Example. On $\mathbb{R}$, $\le$ is a partial order (actually a total order). On $\mathcal{P}(A)$, $\subseteq$ is a partial order but not total (e.g. $\{1\}\not\subseteq\{2\}$ and vice versa).

Conclusion. Partial orders allow "incomparable" elements; total orders are a special case.