On convex directed subgroups of pseudo lattice-ordered groups

We show that all convex directed subgroups of a $pl$-group form a distributive lattice under inclusions that is a Brouwer lattice. We succeeded in extending some $l$-group results concerning rectifying and regular subgroups to the class of $\mathcal{AO}$-groups. Necessary and sufficient conditions are given for an element of a $pl$-group to be an element with a unique value. In order to prove this, some properties of lexicographic extensions of $\mathcal{AO}$-groups and $pl$-groups are investigated.