Francesco Parente
In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in terms of Boolean ultrapowers.
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In this paper, we provide a new characterization of Keisler's order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of 'separation of variables' recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler's order, answering a question posed by Benda in 1974.
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。
On the cofinality of ultrapowers
这里考虑的所有超滤子都是自然数集合Ω上的非主超滤子。我们关心的是这种超滤子的欧米茄超幂可能的共终性。我们表明，没有基数低于groupwise密度数g可以出现这样的共尾性，最多一个基数低于分裂数s可以发生。s的证明，当与Nyikos的结果相结合时，给出了额外的信息，即所有的P_{kappa}点超滤子，对于大于的边界数b，几乎是相干的。
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。
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 assuming the Ultrapower Axiom, much more about the Rudin-Frolik order can be determined. Our main theorem is that under the Ultrapower Axiom, a countably complete ultrafilter has at most finitely many predecessors in the Rudin-Frolik order. In other words, any wellfounded ultrapower (of the universe) is the ultrapower of at most finitely many ultrapowers.
We show that while the length $ω$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Prikry forcing, it is consistent that no iteration of length greater than $ω$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Prikry forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct li…
On products of ultrafilters
本文在广义连续统假设下刻画两个超滤子的张量积等于笛卡尔积的多项充要条件，涉及乘积为超滤子、交换性、不可分解性及超幂嵌入的可定义限制。
The Galvin property under the Ultrapower Axiom
本文研究了在超幂公理下非Galvin超滤子的存在性及其与p点超滤子的关系。
A few special ordinal ultrafilters
我们证明了各种结果的概念介绍了J.鲍姆加特纳的序数超滤。特别是，我们表明，这个概念的超滤波器的复杂性是独立的更熟悉的Rudin-Keisler排序。
Definable Coherent Ultrapowers and Elementary Extensions
本文通过可定义的协调超幂，研究了在任意$\mathbb{L}_{\infty, ω}$片段中定义Skolem函数的模型扩展，并利用足够完备的协调超滤子进行刻画。
Viva la difference I: Nonisomorphism of ultrapowers of countable models
证明了在ZFC中不能证明任意两个可数的初等等价结构相对于Ω上的某个超滤子具有同构的超幂。
Some applications of the ultrapower theorem to the theory of compacta
Keisler-Shelah的超幂定理允许诸如初等等价、初等嵌入和存在嵌入这样的模型论概念以范畴的语言（极限、态射图）表达。这反过来又允许类似的这些（和相关的）概念，以运输到不寻常的设置，主要是那些巴拿赫空间和阿博塔。我们在这里的兴趣是丰富的理论的rectata，特别是理论的连续性，所带来的移民模型理论的思想和技术。
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.
Maximal Chains in {}^ωωand Ultrapowers of the Integers
各种问题所提出的P. Nyikos有关超过滤器的$ω$和链的偏序$(ω,<^*)$回答。主要工具是Oracle链条件及其变体。
We prove a compactness theorem for full Boolean-valued models. As an application, we show that if $T$ is a complete countable theory and $\mathcal{B}$ is a complete Boolean algebra, then $λ^+$-saturated $\mathcal{B}$-valued models of $T$ exist. Moreover, if $\mathcal{U}$ is an ultrafilter on $T$ and $\mathbf{M}$ is a $λ^+$-saturated $\mathcal{B}$-valued model of $T$, then whether or not $\mathbf{M}/\mathcal{U}$ is $λ^+$-saturated just depends on $\mathcal{U}$ and $T$; we say that $\mathcal{U}$ $λ^+$-saturates $T$ in this case. We show that Keisler's order can be formulated as follows: $T_0 \tr…
A note on the Ketonen order and Lipschitz reducibility between ultrafilters
本文研究Ketonen序与Lipschitz序之间的关系，证明在弱超幂公理下两者不一致。
We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^ω/ \mathcal{U} \simeq \mathbf{N}^ω/ \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $ω,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.
Let $B(H)$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space $H$. We study the commutant of $B(H)$ in its ultrapower. We characterize the class of non-principal ultrafilters for which this commutant is non-trivial. Additionally, we extend the class of ultrafilters for which the commutant is trivial.
We show from a weak comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on certain kinds of ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. In the process we prove a generalization of Solovay's lemma to singular cardinals.
In standard construction of hyperrational numbers using an ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with selective ultrafilter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.
In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such characterization of an unstable class. In the present paper, we first establish the existence of so-ca…
We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler's order. Thus Keisler's order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler's order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a ve…
We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting.
We prove a strong dichotomy for the number of ultrapowers of a given countable model associated with nonprincipal ultrafilters on N. They are either all isomorphic, or else there are $2^{2^{\aleph_0}}$ many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II$_1$ factors, as well as their relative commutants and include several applications. We also show that the C*-algebra B(H) always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters on N.
The directedness of the Rudin-Keisler order at measurable cardinals
本文研究了可测基数上Rudin-Keisler顺序的定向性，证明了在ZFC下某些理论的等一致性和首次展示了在可测基数层面的紧致性性质的一致性强度低于强基数。
Order structure of P-point ultrafilters and their relatives
本文综述了选择性、稳定有序并及P点超滤器中Rudin-Keisler与Tukey归约的序结构，揭示不同归约间的联系及相似超滤器类之间序结构的显著差异。
We study various orders on countably complete ultrafilters on ordinals that coincide and are wellorders under a hypothesis called the Ultrapower Axiom. Our main focus is on the relationship between the Ultrapower Axiom and the linearity of these orders.
Notes on ultrafilter extensions of almost bounded structures
本文针对度数无全局有限上界或存在无穷度数元素的结构研究难点，提出几乎有界结构概念，证明其仍为超滤扩张的初等子结构、初等嵌入可提升且可数情形下扩张与特定超幂同构，并评述相关模态逻辑。
Highly saturated models are a fundamental part of the model-theoretic machinery of nonstandard analysis. Of the two methods for producing them, ultrapowers constructed with the aid of $κ^+$-good ultrafilters seems by far the less popular. Motivated by the hypothesis that this is partly due to the standard exposition being somewhat dense, a presentation is given which is designed to be easier to digest.
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 $\de$, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected…
Boolean ultrapowers extend the classical ultrapower construction to work with ultrafilters on any complete Boolean algebra, rather than only on a power set algebra. When they are well-founded, the associated Boolean ultrapower embeddings exhibit a large cardinal nature, and the Boolean ultrapower construction thereby unifies two central themes of set theory---forcing and large cardinals---by revealing them to be two facets of a single underlying construction, the Boolean ultrapower.
Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $λ$ (i.e. a point after which all antichains of $P(λ)/D$ have cardinality less than $λ$) essentiall…
We introduce the notion of "functional extension" of a set X, by means of two natural algebraic properties of the operator * on unary functions. We study the connections with ultrapowers of structures with universe X, and we give a simple characterization of those functional extensions that correspond to limit ultrapower extensions. In particular we obtain a purely algebraic proof of Keisler's characterization of nonstandard (= complete elementary) extensions.
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…
Some simple theories from a Boolean algebra point of view
本文从布尔代数视角出发，通过手工构造布尔代数与超滤子，证明了Keisler序中两类简单秩一理论族之间的强分离，推进了Kunen关于好超滤子的工作，并给出灵活超滤子定理及开放问题。
The logic L^1_θintroduced in [Sh:797]; it is the maximal logic below L_theta theta in which a well ordering is not definable. We investigate it for theta a compact cardinal. We prove it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are L^1_theta-equivalent iff for some omega-sequence of theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.
We study types that appear in ultraproducts that have distributions which can be thought of as a sequence of graphs. The property of having distributions that are captured by graphs is motivated by a commonality of $\mathrm{SOP}_2$-types and types corresponding to pre-cuts in ultrapowers of linear orders. Through this study, we come to a simple set-theoretic description of the distributions for these types and to a new description of good ultrafilters in terms of the difference between "small" external complete subgraphs and internal complete subgraphs in ultraproducts of particular families o…
In earlier work of the second and third author the equivalence of a finite square principle square^fin_{lambda,D} with various model theoretic properties of structures of size lambda and regular ultrafilters was established. In this paper we investigate the principle square^fin_{lambda,D}, and thereby the above model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular filters D if we assume GCH. In this paper we prove in ZFC that for certain regular filters that we call "doubly^+ regular", square^fin_{la…
Amalgamation and Keisler's Order
本文统一了超滤子构造，证明了在可数低理论满足独立合并条件时，Keisler序中$T_{k,k-1}$不优于$T$，从而分离了所有$T_{k+1,k}$类。
I isolate a combinatorial property of a poset $\mathbb{P}$ that I call the strong Prikry property, which implies the existence of an ultrafilter on the complete Boolean algebra $\mathbb{B}$ of $\mathbb{P}$ such that one inclusion of the Boolean ultrapower version of the so-called \Bukovsky-Dehornoy phenomenon holds with respect to $\mathbb{B}$ and $U$. I show that in all cases that were previously studied, and for which it was shown that they come with a canonical iterated ultrapower construction whose limit can be described as a single Boolean ultrapower, the posets in question satisfy this p…
Motivated by Keisler's order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters $D$ on $I$, $|I| = λ> \aleph_0$, the fact that $P(I)/\de$ has little freedom (as measured by the fact that any maximal antichain is of size $<λ$, or even countable) does not prevent extending $D$ to an ultrafilter $D_1$ on $I$ which saturates ultrapowers of the random graph. "Saturates" means that $M^I/\de_1$ is $λ^+$-saturated whenever M is a model of the theory of the random graph. This was known to be true…
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 studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin's axiom for $σ$-centered po…
Ultrafilters over Successor Cardinals and the Tukey Order
该研究聚焦正则不可数基数（尤其$\omega_1$）上的超滤与Tukey序，证明了相关断言的ZFC独立性，给出新型超滤构造，分析了特定公理下超滤的序性质并得到多个相关结果。
We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal $λ$ for which there is $μ< λ\leq 2^μ$, we construct a regular ultrafilter D on $λ$ such that (i) for any model $M$ of a stable theory or of the random graph, $M^λ/D$ is $λ^+$-saturated but (ii) if $Th(N)$ is not simple or not low then $N^λ/D$ is not $λ^+$-saturated. The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called…
Ultrafilter Equivalence and Asymptotic Types of Five Classical t-Norms
从超滤浓度角度研究单位区间上五种经典t-范数，通过超滤等价关系分类其渐近行为，得到近1和低值两种浓度机制下的等价类，并建立离散商范畴和超度量模型。
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。
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