Clones on regular cardinals
环与代数
2016-09-07 v2 逻辑
摘要
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^kappa many maximal (=precomplete) clones on a set of size kappa." The clones we construct here 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 kappa there are 2^2^kappa many such clones on a set of size kappa. Finally, we show that on a weakly compact cardinal there are exactly 2 maximal clones which contain all unary functions.
引用
@article{arxiv.math/0005273,
title = {Clones on regular cardinals},
author = {Martin Goldstern and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0005273},
year = {2016}
}