We generalize the Loomis--Sikorski theorem for perfect
MV-algebras, Dedekind $\sigma$-complete BCK-algebra, and
Dedekind $\sigma$-complete $\ell$-rings showing that there
exist special kinds of tribes, BCK-tribes or r-tribes
consisting of fuzzy sets, or of bounded functions, defined
on a compact Hausdorff space. Tribes are closed under point
supremas of bounded sequences and they are a direct
generalization of  $\sigma$-algebras of crisp subsets.
