Complete sigma-centered Boolean algebra not adding Cohen reals

Orderings of ultrafilters on Boolean algebras

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.

Team games, hypergraph spaces, and projective Boolean algebras

We modify the game Fuchino, Koppelberg, and Shelah used to characterize the $κ$-Freese-Nation property for a given Boolean algebra $A$, replacing players I and II each with a team of $n$ players with limited information. We show that $A$ is tightly $κ$-filtered exactly when team II has a winning strategy for every finite team size. Case $κ=\aleph_0$ characterizes projective Boolean algebras and, hence, Dugundji spaces. In terms of the open-open game of Daniels, Kunen, and Zhou, this characterization is a team version of very I-favorable. We similarly characterize Cohen algebras in terms of a t…

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that the continuum c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(N)/fin has under CH and in the aleph2-Cohen model. We prove a similar result in the category of Banach spaces.

Embeddings of Cohen algebras

Complete Boolean algebras proved to be an important tool in topology and set theory. Two of the most prominent examples are B(kappa), the algebra of Borel sets modulo measure zero ideal in the generalized Cantor space {0,1}^kappa equipped with product measure, and C(kappa), the algebra of regular open sets in the space {0,1}^kappa, for kappa an infinite cardinal. C(kappa) is much easier to analyse than B(kappa) : C(kappa) has a dense subset of size kappa, while the density of B(kappa) depends on the cardinal characteristics of the real line; and the definition of C(kappa) is simpler. Indeed, C…

几乎细化、收获和超滤子数

Almost refinement, reaping, and ultrafilter numbers

本文研究了布尔代数中最大反链的组合结构，探讨了收获关系及超滤子数的性质，并通过参数化钻石原理证明了科恩代数的超滤子数等于ℵ₁。

Cohen 模型中 Calkin 代数的较小 masa

Small masas of the Calkin algebra in the Cohen model

本文在 Cohen 模型下研究 Calkin 代数的极大交换子代数，证明其密度可小于连续统，并刻画了由布尔代数构造的 masa 的同构与酉等价类型，同时给出 ZFC 中不存在密度小于 𝔭 的 masa 的结论。

Abundance of independent sequences in compact spaces and Boolean algebras

It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where i…

Convergence of measures after adding a real

We prove that if $\mathcal{A}$ is an infinite Boolean algebra in the ground model $V$ and $\mathbb{P}$ is a notion of forcing adding any of the following reals: a Cohen real, an unsplit real, or a random real, then, in any $\mathbb{P}$-generic extension $V[G]$, $\mathcal{A}$ has neither the Nikodym property nor the Grothendieck property. A similar result is also proved for a dominating real and the Nikodym property.

历史上的史诗数学对决：格罗滕迪克 vs 尼科迪姆

Epic math battle of history: Grothendieck vs Nikodym

本文通过定义σ-中心力迫概念，构造出具有格罗滕迪克性质却无尼科迪姆性质的布尔代数，证明其存在与连续统假设否定一致，且可简化塔拉格兰德在连续统假设下的构造。

Hereditary Interval Algebras and Cardinal Characteristics of the Continuum

An interval algebra is a Boolean algebra which is isomorphic to the algebra of finite unions of half-open intervals, of a linearly ordered set. An interval algebra is hereditary if every subalgebra is an interval algebra. We answer a question of M. Bekkali and S. Todorčević, by showing that it is consistent that every $σ$-centered interval algebra of size $\mathfrak{b}$ is hereditary. We also show that there is, in ZFC, an hereditary interval algebra of cardinality $\aleph_1.$

Combinatorial properties of classical forcing notions

We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number …

接近布里斯托尔模型

Approaching a Bristol model

本文介绍了布里斯托尔模型的构造，指出其中布尔素理想定理失效，并探讨了金纳-瓦格纳原理在$\mathsf{ZF}$中的潜在作用。

Orthogonal Families of Real Sequences

For x and y sequences of real numbers define the inner product (x,y) = x(0)y(0) + x(1)y(1)+ ... which may not be finite or even exist. We say that x and y are orthogonal iff (x,y) converges and equals 0. Define l_p to be the set of all real sequences x such that |x(0)|^p + |x(1)|^p + .. converges. For Hilbert space, l_2, any family of pairwise orthogonal sequences must be countable. Thm 1. There exists a pairwise orthogonal family F of size continuum such that F is a subset of l_p for every p>2. It was already known that there exists a family of continuum many pairwise orthogonal elements of r…