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Clones on regular cardinals

Martin Goldstern·Saharon Shelah
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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^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.

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Martin Goldstern, Saharon Shelah. 2000-05-29. Clones on regular cardinals. https://arxiv.org/abs/math/0005273

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