Clones on regular cardinals
Volume 173 / 2002
Fundamenta Mathematicae 173 (2002), 1-20
MSC: Primary 08A40; Secondary 03B50, 03E05.
DOI: 10.4064/fm173-1-1
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 $2^{2^{\lambda }}$ 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 from pcf theory we show that for cardinals $ \lambda $ (in particular, for all successors of regulars) there are $2^{2^\lambda }$ such clones on a set of size $\lambda $. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.
