### 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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