On a Marinacci uniqueness theorem for measures

Avallone, Anna; Basile, Achille

doi:10.1016/s0022-247x(03)00274-9

Cited by 34 publications

(13 citation statements)

References 8 publications

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“…The starting point of our paper is observing the key role played by Duniformities in the study of modular measures on L (see Avallone, 2001;Avallone and Basile, 2003), since every modular measure on L generates a D-uniformity. Also of importance is the role played in the study of modular functions on orthomodular lattices (see Weber, 1995) and of measures on MValgebras (see Barbieri and Weber, 1998;Graziano, 2000) by the lattice structure of filters which generate lattice uniformities making uniformly continuous the operations of these structures.…”

Section: Introductionmentioning

confidence: 99%

Lattice Uniformities on Effect Algebras

Avallone

Vitolo

2005

Int J Theor Phys

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Section: Introductionmentioning

confidence: 99%

Lattice Uniformities on Effect Algebras

Avallone

Vitolo

2005

Int J Theor Phys

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“…2.1 [1,2,5,13,14], An effect algebra (£7; φ, 0,1) is a structure consisting of a set E, two special elements 0 and 1, and a partially defined binary operation Θ on Ε χ E satisfying the following conditions for a, ft, c E E:…”

Section: Preliminaries and Basic Resultsmentioning

confidence: 99%

Valuation and Metric on a Lattice Effect Algebra

Khare¹,

Gupta²

2008

Demonstratio Mathematica

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Abstract. In the present paper we have derived some properties of the pseudo-metric introduced by Riecanová on a lattice effect algebra E corresponding to a valuation ω on E, which turns out to be a metric if ω happens to be faithful. Using these properties we have been able to prove that this metric is complete. Also it is observed that the resulting metric space is convex if and only if ω is non-atomic if and only if E is atomless. IntroductionKôpka, in 1992, defined the D-posets of fuzzy sets in [11], which is closed under the formations of differences of fuzzy sets while studying axiomatical systems of fuzzy sets. The structure of a D-poset supports a noncommutative measure theory and allows the solution of some problems of non-commutative probability theory including some problems of theory of quantum measurement. In the same sense an equivalent structure, called effect algebra, were introduced as the carriers of states or probability measure in the quantum (or fuzzy) probability theory [6-10, 15, 17]. The categorical equivalence of a difference poset and an effect algebra is discussed in [5] and some of its properties and examples are studied in [3, 4; see also 5]. It is proved in [13] that a state on a lattice effect algebra is sub-additive if and only if it is a valuation. A state on a lattice effect algebra need not be sub-additive. There are even finite effect algebras admitting no states hence also no probabilities [12]. Further, if a faithful (i.e. non-zero at non-zero elements) valuation on an effect algebra E exists then E is modular and separable [13].Section 2 of the present paper contains prerequisites and basic results on an effect algebra. In Section 3, we are concerned with the pseudo-metric ρ ω and some of its properties on the lattice effect algebra E\ it is found that

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“…The extension is U continuous by [30,6.1] since µ is exhaustive. Then µ has bounded variation iffμ has bounded variation.…”

Section: The Representation Theoremmentioning

confidence: 99%