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Measures on MV-algebras. (English) Zbl 1021.28012

Summary: We study sequentially continuous measures on semisimple MV-algebras. Let \(A\) be a semisimple MV-algebra and let \(I\) be the interval \([0,1]\) carrying the usual Łukasiewicz MV-algebra structure and the natural sequential convergence. Each separating set \(H\) of MV-algebra homomorphisms of \(A\) into \(I\) induces on \(A\) an initial sequential convergence. Semisimple MV-algebras carrying an initial sequential convergence induced by a separating set of MV-algebra homomorphisms into \(I\) are called \(I\)-sequential and, together with sequentially continuous MV-algebra homomorphisms, they form a category \(\mathcal{SM}(I)\). We describe its epireflective subcategory \(\mathcal{ASM}(I)\) consisting of absolutely sequentially closed objects and we prove that the epireflection sends \(A\) into its distinguished \(\sigma\)-completion \(\sigma_H(A)\). The epireflection is the maximal object in \(\mathcal{SM}(I)\) which contains \(A\) as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of \(\sigma_H(A)\) depending on the choice of \(H\). We show that the coproducts in the category of \(D\)-posets of suitable families of \(I\)-sequential MV-algebras yield a natural model of probability spaces having a quantum nature. The motivation comes from probability: \(H\) plays the role of elementary events, the embedding of \(A\) into \(\sigma_H(A)\) generalizes the embedding of a field of events \(A\) into the generated \(\sigma\)-field \(\sigma(A)\), and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras and, unlike in the Loomis-Sikorski Theorem, objects in \(\mathcal{ASM}(I)\) correspond to the generated tribes (no quotient is needed, no information about the elementary events is lost). Finally, \(D\)-poset coproducts lift fuzzy events, random functions and probability measures to events, random functions and probability measures of a quantum nature.

MSC:

28E10 Fuzzy measure theory
06D35 MV-algebras
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