Loomis-Sikorski theorem for sigma-complete MV-algebras and l-groups

We show that every sigma-complete MV-algebra is an MV-sigma-homomorphic ima ge of some sigma-complete MV-algebra of fuzzy sets, called a tribe, which i s a system of fuzzy sets of a crisp set Omega containing 1(Ohm) and closed under fuzzy complementation and formation of min{Sigma(n)f(n), 1}. Since a tribe is a direct generalization of a sigma-algebra of crisp subsets, the r epresentation theorem is an analogue of the Loomis-Sikorski theorem for MV- algebras. In addition, this result will be extended also for Dedekind sigma -complete l-groups with strong unit.