Clones on regular cardinals
  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. 
Martin Goldstern (Technische Universitaet Wien) and Saharon Shelah   (Hebrew University of Jerusalem)