A binary relation $\leq$ on $X$ is a preorder if:
Reflexivity. For all $x\in X$,
\[x\leq x.\]Transitivity. For all $x$, $y$, $z\in X$,
\[x\leq y \,\land\, y\leq z \enspace\rimp\enspace x\leq z.\]
If $\leq$ is a preorder, then $<$ refers to the relation defined by
\[x<y \enspace\lrimp\enspace x\leq y \,\land\, x\neq y.\]