Clones on regular cardinals

• Autorzy: Martin Goldstern , Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

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 $2^{2^λ}$ 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 $2^{2^λ}$ 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.

Kategorie tematyczne

• 08A40: Operations, polynomials, primal algebras
• 03B50: Many-valued logic
• 03E05: Other combinatorial set theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2002
• Tom: 173
• Numer: 1
• Strony: 1-20

Daty

wydano

2002

Twórcy

autor

Martin Goldstern

• Algebra, TU Wien, Wiedner Hauptstrasse 8-10/118.2, A-1040 Wien, Austria

autor

Saharon Shelah

• Department of Mathematics, Hebrew University of Jerusalem, 91904 Jerusalem, Israel

Identyfikatory

DOI

10.4064/fm173-1-1