Measures on MV-algebras

Abstract

We study sequentially continuous measures on semisimple MV-algebras. Let A be a semisimple MValgebra and let I be the interval ½0; 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 Isequential and, together with sequentially continuous MValgebra homomorphisms, they form a category S MðIÞ. We describe its epireflective subcategory AS MðIÞ consisting of absolutely sequentially closed objects and we prove that the epireflection sends A into its distinguished r-completion r H ðAÞ. The epireflection is the maximal object in S MðIÞ which contains A as a dense subobject and over which all sequentially continuous measures can be continuously extended. We discuss some properties of r H ðAÞ depending on the choice of H. We show that the coproducts in the category of D-posets [9] 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 r H ðAÞ generalizes the embedding of a field of events A into the generated r-field rðAÞ, and it can be viewed as a fuzzyfication of the corresponding results for Boolean algebras in [8,11,14]. Sequentially continuous homomorphisms are dual to generalized measurable maps between the underlying sets of suitable bold algebras [13] and, unlike in the LoomisSikorski Theorem, objects in AS MðIÞ 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.Sequences and sequential continuity of mappings are the simplest natural tools in approximation. We continue [7][8][9][10][11][12][13][14] our study of the initial sequential convergence and its applications to the mathematical foundation of probability.By a sequential convergence on a set X we understand a subset of X N Â X specifying which sequence converges to which point. As a rule, we assume all four axioms of convergence: constants, subsequences, uniqueness of limits, Frechet-Urysohn axiom. If X carries an algebraic structure, then we assume additional axioms: the sequential continuity of operations and, if X carries order, the intermediate axiom (cf. [16]). Let A be an MV-algebra and let L A N Â A be a sequential convergence on A (satisfying all the axioms mentioned before). Then ðA; LÞ is said to be a convergence MV-algebra.MV-algebras generalize Boolean algebras and hence classical probability events. In order to work with sequential approximations, we need r-completeness. Since each r-complete MV-algebra is semisimple (i.e. Archimedean), we restrict ourselves to semisimple MV-alge...

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“…As shown in Frič [12,[15][16][17]19] and Papčo [24][25][26], ID is a suitable category in which basic notions of a probability theory having quantum character can be defined in a natural way. Generalized probability has been studied also in the realm of D-posets, see [7,8,14,16,22,23]. In the next section we study D-posets in a broader context of algebraic quantum structures, cf.…”

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