Compact Hausdorff topological MV-algebras and Stone MV-algebras arecompletely characterized. We obtain that compact Hausdorff topologicalMV-algebras are product (both topological and algebraic) of copies $[0,1]$ withstandard topology and finite Lukasiewicz chains with discrete topology. Goingone step further we also prove that Stone MV-algebras are product (bothtopological and algebraic) of finite Lukasiewicz chains with discrete topology.We also prove that an MV-algebra is strongly complete (isomorphic to itsprofinite completion) if and only if it is profinite and its maximal ideals offinite ranks are principal.
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