A clone on a set X is a set of functions (on any finite arity) which is closed under composition (e.g., f(x,y) and g(x,y) are in the clone, then also g(f(x,z), f(z, y)) is in the clone.) The set of clones on X forms a complete algebraic lattice. I will present some results about this lattice for two cases:
* |X| is a weakly compact cardinal (or aleph_0) * |X| is a successor of a regular cardinal. In this case we can use a strong negative partition relation to get a "nonstructure" result.
These are results from a joint paper with Shelah, GoSh:747