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Loomis-Sikorski theorem for monotone σ-complete effect algebras

Published online by Cambridge University Press:  09 April 2009

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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2005

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