Lexicographic MV-algebras and lexicographic states

In this paper we introduce lexicographic MV-algebras and prove that they are the counterpart of unital abelian lattice-ordered groups defined via lexicographic products. The Di Nola–Lettieri categorical equivalence between perfect MV-algebras and abelian lattice-ordered groups is extended to lexicographic MV-algebras. We also investigate lexicographic states of lexicographic MV-algebras. These are additive and normalized maps from any lexicographic MV-algebra into an ad hoc   defined MV-subalgebra of a non-principal ultraproduct [⁎0,1] of the real unit interval [0,1][0,1]. For lexicographic states we prove a representation theorem which can be regarded as the measure-theoretical analog of the representation theorem for lexicographic MV-algebras. We also provide necessary and sufficient conditions for a lexicographic state to be faithful.