# Sh:737

• Goldstern, M., & Shelah, S. (2002). Clones on regular cardinals. Fund. Math., 173(1), 1–20.
• Abstract:
We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 2^{2^{\lambda}} many maximal (= “precomplete”) clones on a set of size \lambda. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many cardinals \lambda (in particular, for all successors of regulars) there are 2^{2^\lambda } many such clones on a set of size \lambda. Finally, we show that on a weakly compact cardinal there are exactly 2 maximal clones which contain all unary functions.
• Current version: 2001-09-08_11 (19p) published version (20p)
Bib entry
@article{Sh:737,
author = {Goldstern, Martin and Shelah, Saharon},
title = {{Clones on regular cardinals}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {173},
number = {1},
year = {2002},
pages = {1--20},
issn = {0016-2736},
mrnumber = {1899044},
mrclass = {08A40 (03E04 03E05)},
doi = {10.4064/fm173-1-1},
note = {\href{https://arxiv.org/abs/math/0005273}{arXiv: math/0005273}},
arxiv_number = {math/0005273}
}