Generalizations of the Rudin - Keisler preorder and their model-theoretic applications

Abstract

Generalizing of the Rudin--Keisler preorder, we introduce relations $R_\alpha$ (and $R_{<\alpha}$) on the set $\beta\omega$ of ultrafilters on~$\omega$. They form an ordinal sequence of length~$\omega_1$ 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 $R_{<\alpha}$ is transitive (and so, a preorder) if the ordinal~$\alpha$ is multiplicatively indecomposable. The proposed constructions have several model-theoretic consequences. Generalizing significantly results of Garc{\'\i}a-Ferreira, Hindman, and Strauss concerning an interplay between ultrafilter extensions of semigroups and the Comfort preorder, we prove that for every model~$\mathfrak A$, ultrafilter~$\mathfrak u$, and ordinal~$\alpha$, the set $\{\mathfrak u:\mathfrak u\,R_{<\alpha}\,\mathfrak v\}$ forms a~submodel of the ultrafilter extension~$\beta\mathfrak A$ of~$\mathfrak A$ if the ordinal~$\alpha$ is additively indecomposable. Furthermore, generalizing Blass' characterization of the Rudin--Keisler preorder via ultrapowers, we characterize the relations $R_\alpha$, and in particular, the Comfort preorder, via a~specific version of limit ultrapowers.