The sequential topology on complete Boolean algebras

Sequential topologies and Dedekind finite sets

It is consistent with ZF set theory that the Euclidean topology on the real line is not sequential, yet every infinite set of reals contains a countably infinite subset. This answers a question of Gutierres.

Embeddings of metric Boolean algebras in $\mathbb{R}^{N}$

A Boolean algebra $\A$ equipped with a (finitely-additive) positive probability measure $m$ can be turned into a metric space $(\A , d_{m})$, where $d_{m}(a,b)= m ((a\wedge\neg b)\vee(\neg a\wedge b))$, for any $a,b\in A$, sometimes referred to as \emph{metric Boolean algebra}. In this paper, we study under which conditions the space of atoms of a finite metric Boolean algebra can be isometrically embedded in $\mathbb{R}^{N}$ (for a certain $N$) equipped with the Euclidean metric. In particular, we characterize the topology of the positive measures over a finite algebra $\A$ such that the metr…

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 …

The algebra of conditional sets and the concepts of conditional topology and compactness

The concepts of a conditional set, a conditional inclusion relation and a conditional Cartesian product are introduced. The resulting conditional set theory is sufficiently rich in order to construct a conditional topology, a conditional real and functional analysis indicating the possibility of a mathematical discourse based on conditional sets. It is proved that the conditional power set is a complete Boolean algebra, and a conditional version of the axiom of choice, the ultrafilter lemma, Tychonoff's theorem, the Borel-Lebesgue theorem, the Hahn-Banach theorem, the Banach-Alaoglu theorem an…

关于各种出现在点free拓扑和代数逻辑中的范畴缺乏余限

On the lack of colimits in various categories arising in pointfree topology and algebraic logic

研究证明McKinsey-Tarski代数范畴不等价于代数品种，反驳了Peter Jipsen的问题。各类BAO、Heyting代数及框架的范畴不完整，因此不等价于预品种。

Ultrafilter selection and Corson compacta

We study the question which Boolean algebras have the property that for every generating set there is an ultrafilter selecting maximal number of its elements. We call it the ultrafilter selection property. For cardinality aleph-one the property is equivalent to the fact that the space of ultrafilters is not Corson compact. We also consider the pointwise topology on a Boolean algebra, proving a result on the Lindelöf number in the context of the ultrafilter selection property. Finally, we discuss poset Boolean algebras, interval algebras, and semilattices in the context of ultrafilter selection…

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。

有限MV效应代数中弱化序列乘积的第一个致命公理：局部障碍、精确低秩分类及一维边界情况

The first fatal axiom for weakened sequential products on finite MV-effect algebras: Local obstruction, exact low-rank classification, and the rank-one boundary case

本文研究了通过截断Gudder-Greechie公理包得到的效果代数二元运算，确定了有限MV效应代数首次失效的公理，并通过构造性方法分类了低秩运算。

Idempotent generated algebras and Boolean powers of commutative rings

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker R-algebras. We also show that these algebras are characterized in terms of a functorial construction having roots in t…

First-Order Logical Duality

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boole…