Lexicographic pseudo MV-algebras

A lexicographic pseudo MV-algebra is an algebra that is isomorphic to an interval in the lexicographic product of a unital linearly ordered group with an arbitrary ℓ  -group. We present conditions when a pseudo MV-algebra is lexicographic. We show that a key condition is the existence of a lexicographic ideal, or equivalently, a case when the algebra can be split into comparable slices indexed by elements of the interval [0,u][0,u] of some unital linearly ordered group (H,u)(H,u). Finally, we show that fixing (H,u)(H,u), the category of (H,u)(H,u)-lexicographic pseudo MV-algebras is categorically equivalent to the category of ℓ-groups.