Probability on MV-Algebras

Riečan, Beloslav; Mundici, Daniele

doi:10.1016/b978-044450263-6/50022-1

Cited by 66 publications

(45 citation statements)

References 40 publications

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“…More information concerning MV-algebras and probability on MV-algebras can be found in [29]. If a bold algebra X ⊆ [0, 1] X is sequentially closed in [0, 1] X (with respect to the coordinatewise sequential convergence), then X is a Lukasiewicz tribe (X is closed not only with respect to finite, but also with respect to countable Lukasiewicz sums, cf.…”

Section: ì óö ñ 24ºmentioning

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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Section: ì óö ñ 24ºmentioning

confidence: 99%

Real functions and the extension of generalized probability measures II

Havlíčková

2015

Tatra Mountains Mathematical Publications

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“…However, the σ-additivity of s is recovered via the Kroupa-Panti theorem (see e. g., [15]) and Riesz representation. The second notion of state is used in the present paper, similarly as in [28], and it corresponds to σ-states in von Neumann algebras. In this case a state s : M → [0, 1] is assumed to be a normalized σ-additive functional.…”

Section: Introductionmentioning

confidence: 99%

“…The possibilities of application of MV-algebras for the description of quantum mechanical systems with infinitely many degrees of freedom were discussed in [28]. In the probability theory of MV-algebras, being a generalization of the boolean algebraic probability theory, the notions of state and observable were introduced, similarly to the algebraic probability case, by abstracting the properties of probability measure and classical random variable.…”

Section: Introductionmentioning

confidence: 99%

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Generalized versions of MV-algebraic central limit theorems

Nowak¹,

Hryniewicz²

2015

Kybernetika

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“…In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13,12].…”

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confidence: 99%

Effect algebras with state operator

Pulmannová¹

EPiC Series in Computing

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This paper is a joint work with Anna Jenčová.Effect algebras have been introduced by Foulis and Bennett [2] (see also [3,4] for equivalent definitions) for modeling unsharp measurements in quantum mechanical systems [5]. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects [7]), noncommutative measure theory and probability (orthomodular lattices and posets, [6]), fuzzy measure theory and many-valued logic 9]).A state, as an analogue of a probability measure, is a basic notion in algebraic structures used in the quantum theories (see e.g., [8]), and properties of states have been deeply studied by many authors.In MV-algebras, states as averaging the truth value were first studied in [11]. In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13,12].Another approach to the state theory on MV-algebras has been presented recently in [15]. Namely, a new unary operator was added to the MV-algebras structure as an internal state (or so-called state operator). MV-algebras with the added state operator are called state MValgebras. The idea is that an internal state has some properties reminiscent of states, but, while a state is a map from an MV-algebra into [0, 1], an internal state is an operator of the algebra. State MV-algebras generalize, for example, Hájek's approach [14] to fuzzy logic with modality P r (interpreted as probably with the following semantic interpretation: The probability of an event a is presented as the truth value of P r(a). For a more detailed motivation of state MV-algebras and their relation to logic, see [15].In [1], the notion of a state operator was extended from MV-algebras to the more general frame of effect algebras. A state operator is there defined as an additive, unital and idempotent operator on E. A state operator on E is called strong, if it satisfies the additional condition(1)Since MV-algebras form a special subclass of effect algebras, so-called MV-effect algebras, it was shown that the definition of a state operator on effect algebras coincides with the original definition on MV-algebras if and only if the state operator is strong. Moreover, if τ is faithful, i.e., τ (a) = 0 implies a = 0, then property (1) is automatically satisfied.In the present paper, we show that state operators on an effect algebra E are related with states on E in the following way: (1) every state on E induces a state operator on the tensor product [0, 1] ⊗ E. (2) If E admits an ordering set of states, then every state operator on E induces a state on E. We study state operators mainly on convex effect algebras. Convex effect algebras as effect algebras with an additional convexity structure were introduced and studied in [17,16]. It was proved in [17], that every convex effect algebra is isomorphic with the interval [0, u] in an ordered real linear space (V, V + ), where u is an order unit. Moreover, (V, u) is an order unit space, i.e. u is an archimedean order unit, if and only if E ad...

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Probability on MV-Algebras

Cited by 66 publications

References 40 publications

Real functions and the extension of generalized probability measures II

Real functions and the extension of generalized probability measures II

Generalized versions of MV-algebraic central limit theorems

Effect algebras with state operator

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