On relations between Rudin–Keisler and Comfort preorders, part I

Joint work with Nikolai L. Poliakov.
Canonical ultrafilter extensions of multi-place operations naturally lead to relations on ultrafilters that generalize the classical Rudin–Keisler preorder on ultrafilters (defined by unary operations). An increasing chain of these relations is extended trafsinitely. It turns out that the union of all the obtained relations yields another well-known relation on ultrafilters, the Comfort preorder. We calculate the composition of these relations and, as a consequence, establish that such a relation is a preorder if and only if its index is a multiplicatively indecomposable ordinal. We also present some model-theoretic applications that significantly generalize previously known results.