Every state on semisimple MV-algebra is integral

Kroupa, Tomáš

doi:10.1016/j.fss.2006.06.015

Cited by 104 publications

(55 citation statements)

References 16 publications

(26 reference statements)

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“…The following theorem independently proved by Kroupa [20,Theorem 28] and Panti [28, Proposition 1.1], provides an integral representation of states by Borel probability measures defined on the σ-algebra B(M ax(M )) of Borel subsets of M ax(M ), where M ax(M ) is the maximal spectral space of M (see Appendix A for further details on M ax(M )). 7 Theorem 2.1.…”

Section: ] Satisfying the Followingmentioning

confidence: 99%

Coherence in the aggregate: A betting method for belief functions on many-valued events

Flaminio

Godo

Hosni

2015

International Journal of Approximate Reasoning

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Betting methods, of which de Finetti's Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a two-fold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of coherence. The main purpose of this paper is to put forward a betting method for belief functions on MV-algebras of many-valued events which allows us to isolate the corresponding coherence criterion, which we term coherence in the aggregate. Our framework generalises the classical Dutch Book method.

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Section: ] Satisfying the Followingmentioning

confidence: 99%

Coherence in the aggregate: A betting method for belief functions on many-valued events

Flaminio

Godo

Hosni

2015

International Journal of Approximate Reasoning

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“…Moreover, it is easy to see that, for every MValgebra A and for every s ∈ S(A), the restriction of s to the Boolean skeleton B(A) of A is a finitely additive probability measure. The following theorem, independently proved in [40] and [47], shows an integral representation of states by Borel probability measures defined on the σ-algebra B(X) of Borel subsets of X, where X is any compact Hausdorff topological space.…”

Section: States On Mv-algebrasmentioning

confidence: 99%

Belief Functions on MV-Algebras of Fuzzy Sets: An Overview

Flaminio

Godo

Kroupa

2014

Non-Additive Measures

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Belief functions are the measure theoretical objects Dempster-Shafer evidence theory is based on. They are in fact totally monotone capacities, and can be regarded as a special class of measures of uncertainty used to model an agent's degrees of belief in the occurrence of a set of events by taking into account different bodies of evidence that support those beliefs. In this chapter we present two main approaches to extending belief functions on Boolean algebras of events to MV-algebras of events, modelled as fuzzy sets, and we discuss several properties of these generalized measures. In particular we deal with the normalization and soft-normalization problems, and on a generalization of Dempster's rule of combination.

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“…In addition, the compact subsets of extremal Bosbach space are also homeomorphic under this mapping. Due to [25], on the Borel σ-algebra B(BS(A)), there is a unique Borel probability measure µ such that s(a) =ŝ(a/F 0 ) = SM(A/F 0 )ã dµ.…”

Section: Coherence De Finetti Maps and Borel Statesmentioning

confidence: 99%

“…The result of [25] and formula (6.4) say that whenever s is a Bosbach state, it generates a σ-additive probability such that s is in fact an integral over this Borel probability measure. Thus formula (6.4) joins de Finetti's "finitely additive probabilities" with σ-additive measures on an appropriate Borel σ-algebra.…”

Section: Coherence De Finetti Maps and Borel Statesmentioning

confidence: 99%

Measures, states and de Finetti maps on pseudo-BCK algebras

Ciungu

Dvurečenskij

2010

Fuzzy Sets and Systems

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Abstract. In this paper, we extend the notions of states and measures presented in [12] to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. Kühr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean ℓ-group. Additionally, we introduce a Borel state and using results by J. Kühr and D. Mundici from [28], we prove a relationship between de Finetti maps, Bosbach states and Borel states.

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Every state on semisimple MV-algebra is integral

Cited by 104 publications

References 16 publications

Coherence in the aggregate: A betting method for belief functions on many-valued events

Coherence in the aggregate: A betting method for belief functions on many-valued events

Belief Functions on MV-Algebras of Fuzzy Sets: An Overview

Measures, states and de Finetti maps on pseudo-BCK algebras

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