On Countably Closed Complete Boolean Algebras

标准 Borel 空间的一个普适刻画

A universal characterization of standard Borel spaces

本文证明了标准 Borel 空间范畴在可数完备 Boolean extensive 范畴的 2-范畴中是（双）初始对象，并推广到任意无穷正则基数情形，给出了其普适刻画。

Remarks on countable tightness

Countable tightness may be destroyed by countably closed forcing. We characterize the indestructibility of countable tightness under countably closed forcing by combinatorial statements similar to the ones Tall used to characterize indestructibility of the Lindelof property under countably closed forcing. We consider the behavior of countable tightness in generic extensions obtained by adding Cohen reals. We show that certain classes of well-studied topological spaces are indestructibly countably tight. Stronger versions of countable tightness, including selective versions of separability, are…

Nagata's conjecture and countably compactifications in generic extensions

Nagata conjectured that every $M$-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. This conjecture was refuted by Burke and van Douwen, and A. Kato, independently. However, we can show that there is a c.c.c. poset $P$ of size $2^ω$ such that in $V^P$ Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space $X\in V$ is an $M$-space in $V^P$ then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in $V^P$). In fact, w…

Semi-Cohen Boolean algebras

We investigate classes of Boolean algebras related to the notion of forcing that adds Cohen reals. A >>Cohen algebra<< is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen algebras: semi-Cohen algebras, pseudo-Cohen algebras and potentially Cohen algebras. These classes of Boolean algebras are closed under completion.

射影和弱射影布尔代数的刻画

Characterising projective and weakly projective Boolean algebras by coherent Freese--Nation separations

本文利用Freese-Nation性质的变体，给出了射影和弱射影布尔代数的刻画。

Filter classes of upsets of distributive lattices

Let us say that a class of upward closed sets (upsets) of distributive lattices is a finitary filter class if it is closed under homomorphic preimages, intersections, and directed unions. We show that the only finitary filter classes of upsets of distributive lattices are formed by what we call $n$-filters. These are related to the finite Boolean lattice with $n$ atoms in the same way that filters are related to the two-element Boolean lattice: $n$-filters are precisely the intersections of prime $n$-filters and prime $n$-filters are precisely the homomorphic preimages of the prime $n$-filter …

Locally compact, $ω_1$-compact spaces

An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…

柱形正交格与柱形布尔代数的典范完备化及对偶性

Canonical completion and duality for cylindric ortholattices and cylindric Boolean algebras

本文研究柱形正交格与柱形布尔代数的典范完备化和拓扑对偶，证明其在典范完备化下封闭，并构造不依赖选择公理的谱空间对偶等价。

Generic countably infinite groups

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not fin…

实闭域中的 Ellis 包络半群

Ellis enveloping semigroups in real closed fields

本文引入 d-半代数集的布尔代数，证明其 Stone 空间与半代数集布尔代数的 Stone 空间的 Ellis 包络半群自然同构，并研究其与外部可定义集族的关系及表示函数。

Polish group actions and computability

Let G be a closed subgroup of the group of all permutations of a countably infinite set. Let X be a Polish G-space with a countable basis A of clopen sets. Each x from X defines a characteristic function f on A by f(U)=1 iff x belongs to U (where U is from A). We consider computable complexity of f and some related questions.

Completeness for the coalgebraic cover modality

We study the finitary version of the coalgebraic logic introduced by L. Moss. The syntax of this logic, which is introduced uniformly with respect to a coalgebraic type functor, required to preserve weak pullbacks, extends that of classical propositional logic with a so-called coalgebraic cover modality depending on the type functor. Its semantics is defined in terms of a categorically defined relation lifting operation. As the main contributions of our paper we introduce a derivation system, and prove that it provides a sound and complete axiomatization for the collection of coalgebraically v…

Ramsey property for Boolean algebras with ideals and $\mathcal{P}(ω_1)/\rm{fin}$

We apply the Dual Ramsey Theorem of Graham and Rothschild to prove the Ramsey property for classes of finite Boolean algebras with distinguished ideals. This allows us to compute the universal minimal flow of the group of automorphisms of $\mathcal{P}(ω_1)/\rm{fin},$ should it be isomorphic to $\mathcal{P}(ω)/\rm{fin}$ or not, and of other quotients of power set algebras. Taking Fraïssé limits of these classes, we can compute universal minimal flows of groups of homeomorphisms of the Cantor set fixing some closed subsets.

矢列式代数的对偶理论

Duality Theory for Sequent Algebras

本文研究由D. Vakarelov提出的矢列式代数，建立其与矢列式空间范畴的对偶等价，刻画矢列式同余，证明该范畴在布尔代数范畴上是拓扑的并得到相关完备性结果。

笛卡尔闭簇 II：与代数和自相似性的联系

Cartesian closed varieties II: links to algebra and self-similarity

本文证明笛卡尔闭簇中的匹配对 [B|M] 等价于布尔限制幺半群和充足源étale拓扑范畴，从而建立其与 Cuntz 代数、Leavitt 路代数等自相似结构所用广群之间的联系，并开启从逻辑视角研究这些广群性质的方向。

Extension of Boolean algebra by a Bayesian operator; application to the definition of a Deterministic Bayesian Logic

This work contributes to the domains of Boolean algebra and of Bayesian probability, by proposing an algebraic extension of Boolean algebras, which implements an operator for the Bayesian conditional inference and is closed under this operator. It is known since the work of Lewis (Lewis' triviality) that it is not possible to construct such conditional operator within the space of events. Nevertheless, this work proposes an answer which complements Lewis' triviality, by the construction of a conditional operator outside the space of events, thus resulting in an algebraic extension. In particul…

Choice-free Stone duality

The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean …

一些从属布尔代数的双拓扑对偶性

A bitopological duality for some subordination Boolean algebras

研究S4 - 从属代数，借助双拓扑空间给出其拓扑表示，用于刻画S5 - 从属代数等，揭示同余与特定闭子集对应关系，探索两种态射并给出拓扑表示。