Rosenberg’s Characterization of Maximal Clones

We will give a proof of I. G. Rosenberg’s characterization of maximal clones, first published in [11]. The theorem lists six types of relations on a finite set such that a clone over this set is maximal if and only if it contains just the functions preserving one of the relations of the list. In Universal Algebra, this translates immediately into a characterization of the finite preprimal algebras: A finite algebra is preprimal if and only if its term operations are exactly the functions preserving a relation of one of the six types listed in the theorem. The difficult part of the proof is to show that all maximal clones or preprimal algebras respectively are of that form. This follows from, and, as we will also demonstrate, is indeed equivalent to, a characterization of primal algebras: We will show that the primal algebras are exactly those whose term operations do not preserve any of the relations on the list.