For any regular uncountable cardinal lambda I will describe a "creature-based" \(\lambda^+\)-complete forcing notion that introduces a "wild" ultrafilter on lambda.
Assuming \(2^\lambda=\lambda^+\), we can find a sufficiently generic filter on this forcing notion; this allows us to construct a clone on lambda which is not contained in any coatom of the clone lattice, solving an old problem in clone theory. (These notions will be explained in the talk.)
I have sketched a corresponding result for \(\lambda=\omega\) in my talk in November 2002. Both results are a joint work with Saharon Shelah.
I will give a related talk (that concentrates on the algebraic rather than set-theoretic background) in the algebra seminar at TU Wien on April 7, 2006. [1]