# Sh:737

- Goldstern, M., & Shelah, S. (2002).
*Clones on regular cardinals*. Fund. Math.,**173**(1), 1–20. arXiv: math/0005273 DOI: 10.4064/fm173-1-1 MR: 1899044 -
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. - 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} }