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 22λ maximal (= "precomplete") clones on a set of size λ. 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 λ (in particular, for all successors of regulars) there are 22λ such clones on a set of size λ. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.
Original language | English |
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Pages (from-to) | 1-20 |
Number of pages | 20 |
Journal | Fundamenta Mathematicae |
Volume | 173 |
Issue number | 1 |
DOIs | |
State | Published - 2002 |
Externally published | Yes |
Keywords
- Maximal clones
- Negative square bracket partition relation
- Precomplete clones
- Weakly compact cardinal