The Jegorov theorem on MV algebras

Jurečková, Mária

doi:10.1016/s0165-0114(98)00330-3

Cited by 6 publications

(9 citation statements)

References 2 publications

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“…Under an additional technical condition, METH has been proved in [17]. Different proofs of METH can be found in [18] (Theorem 10 and Lemma 15) and [5] (Corollary 7.11).…”

Section: Definition 41 Let a Be A Semisimple Mv-algebra And Letmentioning

confidence: 99%

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Measures on MV-algebras

Frič¹

2002

Soft Computing

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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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“…Under an additional technical condition, METH has been proved in [17]. Different proofs of METH can be found in [18] (Theorem 10 and Lemma 15) and [5] (Corollary 7.11).…”

Section: Definition 41 Let a Be A Semisimple Mv-algebra And Letmentioning

confidence: 99%

“…In the proof, the METH in [17] has been used. According to Corollary 4.9, rðAÞ-continuity from below of p is equivalent to the sequential continuity of p with respect to the pointwise convergence.…”

Section: Definition 41 Let a Be A Semisimple Mv-algebra And Letmentioning

confidence: 99%

Measures on MV-algebras

Frič¹

2002

Soft Computing

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“…In this section we will assume that H is an MV σ-algebra ( [2], [3], [10], [11], [13], [14]). It is known that H is a D σ -lattice ( [3]).…”

Section: Measure Extension Theoremmentioning

confidence: 99%

On the Extension of D-Poset Valued Measures

Riečan

1998

Czechoslovak Mathematical Journal

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“…Following J. N o vá k [22]- [24], using some recent results on generalized random events and states [2], [3], [20], [21], [30] and basic categorical methods [1], [15], in [19] we have studied the process of extending generalized probability measures.…”

Section: Introductionmentioning

confidence: 99%

Real functions and the extension of generalized probability measures II

Havlíčková

2015

Tatra Mountains Mathematical Publications

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ABSTRACT. We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

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The Jegorov theorem on MV algebras

Cited by 6 publications

References 2 publications

Measures on MV-algebras

Measures on MV-algebras

On the Extension of D-Poset Valued Measures

Real functions and the extension of generalized probability measures II

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