# Sh:747

• Goldstern, M., & Shelah, S. (2009). Large intervals in the clone lattice. Algebra Universalis, 62(4), 367–374.
• 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.

• Current 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}
}