Maximal and Minimal Closed Classes in Multiple-valued Logic

We consider classes of operations in multiple-valued logic that are closed under composition as well as under permutation of variables, identification of variables (diagonalization) and introduction of inessential variables (cylindrification). Such closed classes on a given finite set form a complete lattice that includes the lattice of clones as the principal filter above the trivial clone. We determine all maximal closed classes; it turns out that there is only one family of closed classes besides Rosenberg's six families of maximal clones. For minimal closed classes we prove an analogon of Rosenberg's five-type classification of minimal clones and we describe explicitly the unary minimal closed classes.