On a Problem of Conrad on Riesz Space Structures

Abstract

This paper is concerned with Riesz space structures on a lattice ordered abelian group, continuing a line of research conducted by the author and the collaborators Antonio Di Nola and Gaetano Vitale. First we prove a statement in a paper of Paul Conrad (given without proof) that every non-archimedean totally ordered abelian group has at least two Riesz space structures, if any. Then, as a main result, we prove that there is a non-archimedean lattice ordered abelian group with strong unit having only one Riesz space structure. This gives a solution to a problem posed in a paper of Conrad dating back to 1975. Then we combine these results and the categorial equivalence between lattice ordered abelian groups with strong unit and MV-algebras (due to Daniele Mundici) and the one between Riesz spaces with strong unit and Riesz MV-algebras (due to Di Nola and Ioana Leustean). By combining these tools, we prove that every non-semisimple totally ordered MV-algebra has at least two Riesz MV-algebra structures, if any, and that there is a non-semisimple MV-algebra with only one Riesz MV-algebra structure.