Clones on regular cardinals

Martin Goldstern¹ ; Saharon Shelah²

¹ Algebra TU Wien Wiedner Hauptstrasse 8-10/118.2 A-1040 Wien, Austria
² Department of Mathematics Hebrew University of Jerusalem 91904 Jerusalem, Israel

Fundamenta Mathematicae, Tome 173 (2002) no. 1, pp. 1-20

Résumé

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 $2^{2^{\lambda }}$ 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 from pcf theory we show that for cardinals $ \lambda $ (in particular, for all successors of regulars) there are $2^{2^\lambda }$ such clones on a set of size $\lambda $. Finally, we show that on a weakly compact cardinal there are exactly 2 precomplete clones which contain all unary functions.

Détail

DOI : 10.4064/fm173-1-1

Keywords: investigate structure lattice clones infinite set first observe ultrafilters naturally induce clones yields simple proof rosenbergs theorem there lambda maximal precomplete clones set size lambda clones construct contain unary functions investigate clones contain unary functions using strong negative partition theorem pcf theory cardinals lambda particular successors regulars there lambda clones set size lambda finally weakly compact cardinal there exactly precomplete clones which contain unary functions