Generalizations of pseudo MV-algebras and generalized pseudo effect algebras

Abstract

We deal with unbounded dually residuated lattices that generalize pseudo MV-algebras in such a way that every principal order-ideal is a pseudo MV-algebra. We describe the connections of these generalized pseudo MV-algebras to generalized pseudo effect algebras, which allows us to represent every generalized pseudo MV-algebra A by means of the positive cone of a suitable ℓ-group G A . We prove that the lattice of all (normal) ideals of A and the lattice of all (normal) convex ℓ-subgroups of G A are isomorphic. We also introduce the concept of Archimedeanness and show that every Archimedean generalized pseudo MV-algebra is commutative.