Clones on regular cardinals

by Goldstern and Shelah. [GoSh:737]
Fundamenta Math, 2002
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.

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