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 closed classes.