On Convex Directed Subgroups of Pseudo Lattice-Ordered Groups

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