Lexicographic Pseudo MV-algebras

Abstract

A lexicographic pseudo MV-algebra is an algebra that is isomorphic to an interval in the lexicographic product of a linear unital group with an arbitrary $\ell$-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 $[0,u]$ of some unital linearly ordered group $(H,u)$. Finally, we show that fixing $(H,u)$, the category of $(H,u)$-lexicographic pseudo MV-algebras is categorically equivalent to the category of $\ell$-groups.