2000/05/29 by Martin Goldstern, Saharon Shelah, Goldstern, Martin +1
Computer Science · Mathematics · #03E05 #08A05 #08A40 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.math/0005273
openalex publication_date 2000/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 22kappa 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 22kappa 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.