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.

We show that every σ-complete MV-algebra is an MV-σ-homomorphic image of some σ-complete MV- algebra of fuzzy sets, called a tribe, which is a system of fuzzy sets of a crisp set Ω containing 1Ω and closed under fuzzy complementation and formation of min {∑nfn, 1}. Since a tribe is a direct generalization of a σ-algebra of crisp subsets, the representation theorem is an analogue of the Loomis-Sikorski theorem for MV-algebras. In addition, this result will be extended also for Dedekind σ-complete ℓ-groups with strong unit.