Clones on regular cardinals

Martin Goldstern*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

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 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 2 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 languageEnglish
Pages (from-to)1-20
Number of pages20
JournalFundamenta Mathematicae
Volume173
Issue number1
DOIs
StatePublished - 2002
Externally publishedYes

Keywords

  • Maximal clones
  • Negative square bracket partition relation
  • Precomplete clones
  • Weakly compact cardinal

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