Rosenberg's classification of maximal clones

Statement

Let $A$ be a finite set with $k = \abs{A} \ge 2$. A clone on $A$ is maximal if it is a coatom of the lattice of clones on $A$. For a relation $\rho$ on $A$, let $\operatorname{Pol}(\rho)$ be the clone of all polymorphisms of $\rho$. A relation $\rho \subseteq A^h$ is totally symmetric if it is invariant under all permutations of coordinates. It is totally reflexive if it contains every $h$-tuple with two equal entries. Then a clone $C$ on $A$ is maximal if and only if $C = \operatorname{Pol}(\rho)$ for a relation $\rho$ of one of the following six types.

1. A partial order on $A$ with a least and a greatest element.
2. The graph $\set{(a, \sigma(a)) : a \in A}$ of a permutation $\sigma$ of $A$ whose cycles all have the same prime length $p$.
3. The relation $\set{(a, b, c, d) \in A^4 : a + b = c + d}$, where $(A, +)$ is an elementary abelian $p$-group for a prime $p$.
4. An equivalence relation on $A$ other than equality and $A^2$.
5. A central relation $\rho \subseteq A^h$ with $1 \le h \le k - 1$. This means that $\rho \ne A^h$, that $\rho$ is totally symmetric and totally reflexive, and that some $c \in A$ satisfies $(c, a_2, \dots, a_h) \in \rho$ for all $a_2, \dots, a_h \in A$. For $h = 1$ it means a nonempty proper subset of $A$.
6. An $h$-regular relation with $3 \le h \le k$. This is the relation determined by a family $\theta_1, \dots, \theta_m$ of equivalence relations, $m \ge 1$, each with exactly $h$ blocks. The family must be such that any choice of one block of each $\theta_i$ has nonempty intersection. The relation consists of the $h$-tuples that, for every $i$, have two entries in the same $\theta_i$-block.