A partially ordered set is a set $X$ together with a homogeneous relation $\leq$ such that for all $x,y,z\in X$,
- $\leq$ is reflexive, i.e. $x\leq x$;
- $\leq$ is antisymmetric, i.e. if $x\leq y$ and $y\leq x$ then $x=y$;
- $\leq$ is transitive, i.e. if $x\leq y$ and $y\leq z$ then $x\leq z$.