Related papers: Initial Tukey structure below a stable ordered-uni…
In this paper, we study some new examples of ideals on $\omega$ with maximal Tukey type (that is, maximal among partial orders of size continuum). This discussion segues into an examination of a refinement of the Tukey order -- known as the…
We investigate the structure of ultrafilters on Boolean algebras in the framework of Tukey reducibility. In particular, this paper provides several techniques to construct ultrafilters which are not Tukey maximal. Furthermore, we connect…
We investigate the Tukey type of the generic ultrafilter added by the quotient $\mathcal{P}(\omega \times \omega) / (\mathrm{FIN} \times \mathrm{FIN})$. We prove that this ultrafilter is not basically generated and yet does not have the…
We present three models concerning Tukey types of ultrafilters on $\omega$. The first model is built via a countable support iteration, and we show there is no basically generated ultrafilter in such model. The second and third models are…
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass…
We extend the class of ultrafilters $U$ over countable sets for which $U\cdot U\equiv_T U$, extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal $\alpha\geq 2$, the generic…
This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$,…
We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs…
The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in $\beta \kappa\setminus \kappa$, where $\kappa$ is a regular cardinal. This generalizes the problem posed by Peter Simon more than…
We prove that bold N, the standard model of arithmetic, has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of bold N represented in N which is minimal (i.e. as in Rudin-Keisler…
Associated to each ultrafilter $\mathcal{U}$ on $\omega$ and each map $p:\omega\rightarrow \omega$ is a Dedekind cut in the ultrapower $\omega^{\omega}/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when…
There exist two known canonical types of ultrafilter extensions of first-order models; one comes from modal logic and universal algebra, another one from model theory and algebra of ultrafilters, with ultrafilter extensions of semigroups as…
We show that there is a set of $2^{2^{\kappa}}$ ultrafilters incomparable in Rudin-Frol\'ik order of $\beta \kappa \setminus \kappa$, where $\kappa$ is regular, for which no subset with more than one element has an infimum.
We prove that there exists a nonprincipal ultrafilter $\mathcal U$ on $\mathbb N$ such that for every countable (or separable) structure $B$ in a countable language the quotient map from the reduced product associated with the Fr\'echet…
We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom, holds in all known canonical inner models. It turns out that…
We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories $T_\mathfrak{m}$ reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the…
We like to develop model theory for $T$, a complete theory in $\mathbb{L}_{\theta,\theta}(\tau)$ when $\theta$ is a compact cardinal. By [Sh:300a] we have bare bones stability and it seemed we can go no further. Dealing with ultrapowers…
We study the strength of well-founded ultrafilters on ordinals above choiceless large cardinals and their associated Prikry forcings. Gabriel Goldberg showed that all but boundedly many regular cardinals above a rank Berkeley cardinal carry…
The ultrafilters on the partial order $([\omega]^{\omega},\subseteq^*)$ are the free ultrafilters on $\omega$, which constitute the space $\omega^*$, the Stone-Cech remainder of $\omega$. If $U$ is an upperset of this partial order (i.e., a…
An earlier paper, entitled "P-hierarchy on $\beta\omega$", investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. This study is continued in the present paper and focuses on the aspects of characterization of…