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 maximal (= “precomplete”) clones on a set of size λ. 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 from pcf theory we show that for cardinals λ (in particular, for all successors of regulars) there are such clones on a set of size λ. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-1,
author = {Martin Goldstern and Saharon Shelah},
title = {Clones on regular cardinals},
journal = {Fundamenta Mathematicae},
volume = {173},
year = {2002},
pages = {1-20},
zbl = {0997.08004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-1}
}
Martin Goldstern; Saharon Shelah. Clones on regular cardinals. Fundamenta Mathematicae, Tome 173 (2002) pp. 1-20. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm173-1-1/