Proposition 10

In a right directed ordered set E, a maximal element a is the greatest element of E.

For every $x \in \mathrm{E}$ there exists by hypothesis $y \in \mathrm{E}$ such that $x \leqslant y$ and $a \leqslant y$; since $a$ is maximal, $y = a$.

¶ A right directed preordered set is left directed with respect to the opposite ordering. Any product of right directed sets is right directed. On the other hand, a subset of a right directed set is not necessarily right directed. However, a cofinal subset F of a right directed set E is always right directed; for given $x$, $y \in \mathrm{F}$ there exists $z \in \mathrm{E}$ such that $x \leqslant z$ and $y \leqslant z$, and then $t \in \mathrm{F}$ such that $z \leqslant t$.