Novosibirsk State University（新西伯利亚国立大学）；Kazan (Volga Region) Federal University（喀山（伏尔加沿岸）联邦大学）；Sobolev Institute of Mathematics SB RAS（俄罗斯科学院西伯利亚分院索博列夫数学研究所）
Nikolay A. Bazhenov, Iskander Sh. Kalimullin, Marina V. Schwidefsky
The paper studies computability-theoretic aspects of topological $T_0$-spaces. We introduce effective versions of the notions of a countable $c$-poset and a (second-countable) topological space with base. Based on this, we prove an effective version of the known Stone-type duality between the category $\mathbf{AS}$ (whose objects are almost semispectral spaces with base and whose morphisms are spectral mappings) and the category $\mathbf{DP}$ (whose objects are distributive $c$-posets and whose morphisms are strict mappings). Namely, we show that for an arbitrary set $Z\subseteq ω$, this duality is preserved when one restricts to objects which have $Z$-computably enumerable presentations only. Following this approach, we establish several results in computable topology. We prove that every degree spectrum of a countable algebraic structure can be realized as the degree spectrum of a topological space with base. We show that for any non-zero natural number $N$, there is a computable topological space with base that has precisely $N$-many computable copies, up to effective spectral homeomorphisms.
De Vries Duality generalizes Stone duality between Boolean algebras and Stone spaces to a duality between de Vries algebras (complete Boolean algebras equipped with a subordination relation satisfying some axioms) and compact Hausdorff spaces. This duality allows for an algebraic approach to region-based theories of space that differs from point-free topology. Building on the recent choice-free version of Stone duality developed by Bezhanishvili and Holliday, this paper establishes a choice-free duality between de Vries algebras and a category of de Vries spaces. We also investigate connection…
The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander's Subbase Theorem, which asserts that a topological space $X$ is compact if every subbasic open cover of $X$ admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem - whose proof depends upon Zorn's Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-fr…
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 …
In this paper we provide a Stone style duality for monotone semilattices by using the topological duality developed in \cite{Celani2020} for semilattices together with a topological description of their canonical extension. As an application of this duality we obtain a characterization of the congruences of monotone semilattices by means of monotone lower-Vietoris-type topologies.
We exhibit Pontryagin duality as a special case of Stone duality in a continuous logic setting. More specifically, given an abelian topological group $A$, and $\mathcal F$ the family (group) of continuous homomorphisms from $A$ to the circle group $\mathbb T$, then, viewing $(A,+)$ equipped with the collection $\mathcal F$ as a continuous logic structure $M$, we show that the local type space $S_\mathcal F(M)$ is precisely the Pontryagin dual of the group $\mathcal F$ where the latter is considered as a discrete group. We conclude, using Pontryagin duality (between compact and discrete abelian…
A duality theoretic view on limits of finite structures
Nesetril和Ossona de Mendez发展了有限模型结构极限的系统理论。它是基于这样的见解，即有限结构的集合可以通过他们称为Stone配对的映射嵌入到测量空间中，在那里可以计算出所需的限制。我们表明，一个密切相关的，但更细粒度的措施空间出现--通过斯通-普里斯特利对偶和模型理论的类型的概念--通过丰富的表达能力的一阶逻辑与某些“概率算子”。我们提供了一个健全的和完整的演算，这个扩展的逻辑，并揭示了这个建设的函子性质。其后果是双重的。一方面，我们确定了结构极限理论的逻辑依据。另一方面，我们的建设表明，二重性理论的斯通配对的变体捕获添加一层量词，从而使一个强有力的链接到最近的工作半环量词在逻辑上的话。在这个过程中，我们将类型的模型论概念确定为这一联系背后的统一概念。这些结果有助于弥合链的逻辑在计算机科学侧重于语义和算法和复杂性相关的领域，分别。
