Loomis-Sikorski theorem for monotone $\sigma$-complete effect algebras

J. Aust. Math. Soc. 79 (2005), 305-318

Loomis-Sikorski theorem for monotone $\sigma$ -complete effect algebras

Anatolij Dvurecenskij
Mathematical Institute
Slovak Academy of Sciences
Stefánikova 49
SK-814 73 Bratislava
Slovakia
dvurecen@mat.savba.sk

Abstract

We show that monotone $\sigma$ -complete effect algebras under some conditions are $\sigma$ -homomorphic images of effect-tribes (as monotone $\sigma$ -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 $\sigma$ -algebras of subsets, we present a generalization of the Loomis-Sikorski theorem for such effect algebras. We show that we can choose an effect-tribe to be a system of affine fuzzy sets. In addition, we present a new version of the Loomis-Sikorski theorem for $\sigma$ -complete MV-algebras.

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