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Clones on regular cardinals
by
Martin Goldstern
TU Wien
Coauthors: Saharon Shelah
Let X be an infinite set. A clone C is a set of finitary functions on X which is closed under composition (=substitution) and contains all projections. The family of all clones on X forms a complete algebraic lattice. We are interested in the co-atoms of this lattice, the ``precomplete'' clones, and in particular in those clones which contain all unary functions. On a countable set, there are exactly two precomplete clones containing all unary functions (Gavrilov, 1965). We give a new proof of this theorem and show that it generalizes to weakly compact cardinals. We also show that on many other regular cardinals the number of precomplete clones containing all unary functions is very large.
Date received: May 6, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-25.