Shimon Garti, Menachem Magidor, Saharon Shelah
We show that the character spectrum $Sp_χ(λ)$, for a singular cardinal $λ$ of countable cofinality, may include any prescribed set of regular cardinals between $λ$ and $2^λ$.
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We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of infinite cardinals which are ultraproducts of finite cardinals modulo D.
We show the consistency of the set of regular cardinals which are the character of some ultrafilter on omega is not convex. We also deal with the set of pi chi-characters of ultrafilters on omega.
On Ultrapowers and Cohesive Ultrafilters
本文用超幂刻画了Tukey序与内聚超滤，借此度量超滤序间的距离，改进了Kanamori的两项定理，并在超紧基数下回答了其问题，探讨了点谱与深度谱的一致性。
For a free ultrafilter U on omega we study several cardinal characteristics which describe part of the combinatorial structure of U. We provide various consistency results; e.g. we show how to force simultaneously many characters and many pi --characters. We also investigate two ideals on the Baire space omega^omega naturally related to U and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.
We extend to singular cardinals the model-theoretical relation $λ\stackrelκ{\Rightarrow} μ$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics.
We prove that consistently there is a singular cardinal $κ$ of uncountable cofinality such that $2^κ$ is weakly inaccessible, and every regular cardinal strictly between $κ$ and $2^κ$ is the character of some uniform ultrafilter on $κ$.
We introduce the decomposability spectrum $K_D=\{λ\geq ω| D \text{is} λ\text{-decomposable}\}$ of an ultrafilter $D$, and show that Shelah's $\pcf$ theory influences the possible values $K_D$ can take. For example, we show that if $\aaa$ is a set of regular cardinals, $μ\in \pcfa$, the ultrafilter $D$ is $|\aaa |^+$-complete and $K_D \subseteq \aaa$, then $μ\in K_D$. As a consequence, we show that if $ λ$ is singular and for some $ λ' < λ$ $K_D$ contains all regular cardinals in $ [λ', λ)$ then: (a) if $\cf λ= ω$ then either $ λ\in K_D$, or $ λ^+ \in K_D$; and (b) if $D$ is $(\cf λ)^+$-complet…
We study some limitations and possible occurrences of uniform ultrafilters on ordinals without the axiom of choice. We prove an Easton-like theorem about the possible spectrum of successors of regular cardinals which carry uniform ultrafilters; we also show that this spectrum is not necessarily closed.
Applications of the Magidor Iteration to Ultrafilter Theory
研究Magidor迭代对可测基数上正常超滤子的和，并证明弱超幂公理与超幂公理不等价，同时构造了非刚性的超幂和不同基数上的相同超幂的均匀超滤子。
We show the consistency of: the set of regular cardinals which are the character of some ultrafilter on omega can be quite chaotic, in particular not only can be not convex but can have many gaps. We also deal with the set of pi-characters of ultrafilters on omega.
Combinatorics of Schur ultrafilters
本文研究了可数交换群上Schur ultrafilters的元素的组合特征，构造了一个自由Schur ultrafilter，它不是无限Schur的，并在连续假设下证明了存在一个自由Schur P-point。
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
$κ$-barely independent families and Tukey types of ultrafilters
本文引入 $λ$ 上的 $κ$-勉强独立族，结合广义收割数给出其存在条件，并据此刻画一致超滤子的 Tukey 顶性质与极大特征。
Ultrafilter characters and local $π$-bases
该研究解决了Kania的问题5.3，建立了关于空间基数与局部π特征关系的实现定理，构造了具有特定局部π-基性质的空间，属于拓扑学领域的理论成果。
In this paper, we characterize the possible cofinalities of the least $λ$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $δ$, that carries a $λ$-complete uniform ultrafilter, it is consistent, relative to the existence of a supercompact cardinal above $δ$, that the least $λ$-strongly compact cardinal has cofinality $δ$. On the other hand, provably the cofinality of the least $λ$-strongly compact cardinal always carries a $λ$-complete uniform ultrafilter.
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only $α$-many iterations for some countable ordinal $α$. Here we study such $α$-iterable cardinals. We show that the $α$-iterable cardinals form a strict hierarchy for $α\leqω_1$, that they are downward absolute to $L$ for $α<ω_1^L…
We characterize the situation of small cardinality for a product of cardinals divided by an ultrafilter. We develop the notion of weak normality. We include an application to Boolean Algebras.
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 Fréchet filter onto the ultrapower has a right inverse. The proof uses the Continuum Hypothesis. We characterize the ultrafilters $\mathcal U$ with this property, and show that consistently with ZFC such ultrafilters need not exist. We also prove a similar ZFC result sufficiently strong to obtain all concrete applications of the existence of a right inverse to the quotie…
Model Theory for a Compact Cardinal
该研究针对紧致基数下的L_θ,θ完全理论，限定θ-完全超滤子场景，通过超幂方法推进模型论研究，刻画了Keisler序下的极小理论，并建立了其与L^1_<θ逻辑的等价关联。
Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, pi-character, hereditary Lindelof number, hereditary density. It is shown to be consistent that there exists a sequence <B_i:i<kappa> of Boolean algebras and an ultrafilter D on kappa such that C(prod_{i<kappa}B_i/D)<|prod_{i<kappa}C(B_i)/D|. This answers a number of problems posed by Monk.
We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $κ$ and $\mathscr{C}(D)$ is the class of all pairs $(θ_1, θ_2),$ where $(θ_1, θ_2)$ is the cofinality of a cut in $J^κ/D$ and $J$ is some $(θ_1+θ_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(θ_1, θ_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $θ_1 + θ_2 > 2^κ,$ then $θ_1=θ_2.$
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