Generalizations of the Rudin - Keisler preorder and their model-theoretic applications - Polyakov, Nikolai; Saveliev, Denis (Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series, 2025)
Generalizing of the Rudin–Keisler preorder, we introduce relations R_α (and R_<α) on the set βω of ultrafilters on ω. They form an ordinal sequence of length ω_1 which is strictly increasing by inclusion and lies between the Rudin–Keisler preorder and the Comfort preorder. We show that the composition of these relations is expressed via a multiplication-like operation on ordinals. Explicit calculations of this operation show that R_<α is transitive (and so, a preorder) if the ordinal α is multiplicatively indecomposable. The proposed constructions have several model-theoretic consequences. Generalizing significantly results of Garc\’ıa-Ferreira, Hindman, and Strauss concerning an interplay between ultrafilter extensions of semigroups and the Comfort preorder, we prove that for every model A, ultrafilter u, and ordinal α, the set {u:u R_<α v} forms a submodel of the ultrafilter extension βA of A if the ordinal α is additively indecomposable. Furthermore, generalizing Blass’ characterization of the Rudin–Keisler preorder via ultrapowers, we characterize the relations R_α, and in particular, the Comfort preorder, via a specific version of limit ultrapowers.