Large Intervals in the Clone Lattice

We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones C1, C2, C3 such that: (1) the interval [C1, O] in the lattice of local clones is (as a lattice) isomorphic to {0, 1, 2, . . .} under the divisibility relation, (2) the interval [C2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice, (3) the interval [C3, O] in the lattice of all clones is isomorphic to the lattice of all filters on X. These examples explain the difficulty of obtaining a satisfactory analysis of the clone lattice on infinite sets. In particular, (1) shows that the lattice of local clones is not dually atomic.