Topological characterization of semisimple MV-algebras

In this paper, we will explain the reason that filter topological characterization of 2-divisible abelian lattice ordered groups cannot be easily translated into MV-algebras. Then we prove that a 2-divisible MV-algebra A is semisimple if and only if it is Hausdorff with respect to the filter topology induced by the lattice filter of the form F=⋃n∈NAa,n for any a∈A which exceeds zero. We also show that an MV-algebra is simple if and only if it is Hausdorff with respect to any filter topology. Furthermore, an MV-algebra A is semisimple if and only if for any lattice filter F, which satisfies F∩I=∅ for any proper MV-algebra ideal I, A is Hausdorff with respect to F.