Abstract
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.
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