Pseudo MV-algebras and lexicographic product

We study algebraic conditions when a pseudo MV-algebra is an interval in the lexicographic product of an Abelian unital linearly ordered group and an ℓ-group that is not necessarily Abelian. We introduce two classes of pseudo MV-algebras which can be split into a system of comparable slices indexed by elements of an interval in an Abelian linearly ordered group. We show when such pseudo MV-algebras have a representation by a lexicographic product with an ℓ-group. Fixing a unital ℓ-group, we show that the category of such pseudo MV-algebras is categorically equivalent to the category of ℓ-groups.