Loomis-Sikorski Theorems for MV-algebras and BCK-algebras

We recall that σ-complete MV-algebras are MV-algebras which are σ-complete lattices. Such MV-algebras are always semisimple algebras, and they are exactly those for which there exists an MV-isomorphism with a Bold algebra, i.e., with an algebra of fuzzy sets on a crisp set Ω which contains l_Ω, and which is closed under the fuzzy complementation and formation of min{f + g, l}. Belluce [Bel] showed that every semisimple MV-algebra M can be always represented as a Bold algebra of continuous fuzzy sets on the compact Hausdorff space of all maximal ideals of M. And this is an analogue of Stone’s representation theorem for Boolean algebras. Situation with σ-complete MV-algebras is more complicated as we will see below.