# Sh:747

- Goldstern, M., & Shelah, S. (2009).
*Large intervals in the clone lattice*. Algebra Universalis,**62**(4), 367–374. arXiv: math/0208066 DOI: 10.1007/s00012-010-0047-6 MR: 2670171 -
Abstract:

We give three examples of large intervals in the lattice of (local) clones on an infinite set X, by exhibiting clones {\mathcal C}_1, {\mathcal C}_2, {\mathcal C}_3 such that:(1) the interval [{\mathcal C}_1,{\mathcal O}] in the lattice of local clones is (as a lattice) isomorphic to \{0,1,2,\ldots\} under the divisibility relation,

(2) the interval [{\mathcal C}_2, {\mathcal O}] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice,

(3) the interval [{\mathcal C}_3,{\mathcal 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.

- Version 2009-04-30_11 (7p) published version (8p)

Bib entry

@article{Sh:747, author = {Goldstern, Martin and Shelah, Saharon}, title = {{Large intervals in the clone lattice}}, journal = {Algebra Universalis}, fjournal = {Algebra Universalis}, volume = {62}, number = {4}, year = {2009}, pages = {367--374}, issn = {0002-5240}, mrnumber = {2670171}, mrclass = {08A40 (03E05 03E20 08A05)}, doi = {10.1007/s00012-010-0047-6}, note = {\href{https://arxiv.org/abs/math/0208066}{arXiv: math/0208066}}, arxiv_number = {math/0208066} }