Large intervals in the clone lattice
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We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones C_1, C_2, C_3 such that:
(1) the interval [C_1, O] in the lattice of local clones is (as a lattice) isomorphic to {0,1,2, ...} under the divisibility relation,
(2) the interval [C_2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice,
(3) the interval [C_3, 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.
LaTeX2e, 9 pages. Revised version contains additional comments and references
LaTeX2e, 9 pages. Revised version contains additional comments and references