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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 M of multisets is dually equivalent to the category 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.

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Abstract. The article has two main objectives: characterize compact Hausdorff topological MV-algebras and Stone MV-algebras on one hand, and characterize strongly complete MV-algebras on the other hand. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies [0, 1] with the interval topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. In the second part we prove that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and its only maximal ideals of finite ranks are principal.To the memory of a great mentor, John V. Leahy (1937Leahy ( -2015.

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Residuated lattices play an important role in the study of fuzzy logic based ont-norms. In this paper, we introduce some notions ofn-fold filters in residuated lattices, study the relations among them, and compare them with prime, maximal and primary, filters. This work generalizes existing results in BL-algebras and residuated lattices, most notably the works of Lele et al., Motamed et al., Haveski et al., Borzooei et al., Van Gasse et al., Kondo et al., Turunen et al., and Borumand Saeid et al., we draw diagrams summarizing the relations between different types ofn-fold filters andn-fold residuated lattices.

The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola [1] to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Lukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.