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Loomis-Sikorski theorem for monotone σ-complete effect algebras

Part of: Boolean algebras (Boolean rings) General logic Algebraic logic Distributive lattices

Published online by Cambridge University Press:  09 April 2009

Anatolij Dvurečenskij
Anatolij Dvurečenskij
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia, e-mail: dvurecen@mat.savba.sk
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Abstract

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We show that monotone σ -complete effect algebras under some conditions are σ - homomorphic images of effect-tribes (as monotone σ -complete effect algebras), which are nonempty systems of fuzzy sets closed under complements, sums of fuzzy sets less than 1, and containing all pointwise limits of nondecreasing fuzzy sets. Because effect-tribes are generalizations of Boolean σ -algebras of subsets, we present a generlization of the Loomis-Sikorski theorem for such effect algebras. We show that we can choose an effect-tribe to be a system of affin fuzzy sets. In addition, we present a new version of the Loomis-Sikorski theorem for σ-complete MV-algebras.

Keywords

effect algebramonotone σ-complete effect algebraMV-algebraeffect-tribetribeRiesz decomposition propertyLoomis-Sikorski theorempo-groupℓ-groupaffine function

MSC classification

Secondary: 06D35: MV-algebras 03G12: Quantum logic 03B50: Many-valued logic
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 79 , Issue 3 , December 2005 , pp. 305 - 318
Copyright
Copyright © Australian Mathematical Society 2005

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