Profinite MV-algebras

We characterize all profinite MV-algebras, these are MV-algebras that are inverse limits of finite MV-algebras. It is shown that these are exactly direct product of finite \L ukasiewicz's chains. We also prove that the category $\mathbb{M}$ of multisets is dually equivalent to the category $\mathbb{P}$ of profinite MV-algebras and homomorphisms that reflect principal maximal ideals. Thus generalizing the corresponding result for finite MV-algebras, and finite multisets.