### Large Intervals in the Clone Lattice

by Goldstern and Shelah. [GoSh:747]

Algebra Universalis, 2010

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.

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