Profinite MV-algebras and Multisets

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Simple algebraic and topological characterizations of profinite MV-algebras (i.e., MV-algebras that are inverse limits of finite MV-algebras) are obtained. It is shown that these are the direct products of finite Łukasiewicz chains. We also prove that the category \(\mathbb {M}\) of multisets is dually equivalent to the category \(\mathbb {P}\) of profinite MV-algebras and complete homomorphisms. This duality extends the well-known duality between finite MV-algebras, and finite multisets on one hand, and the duality between sets with functions and atomic complete Boolean algebras with complete homomorphisms on the other hand.