The dual geometry of Boolean semirings

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…

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。

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。

Non-commutative Stone duality

We show explicitly that Boolean inverse semigroups are in duality with what we term Boolean groupoids. This generalizes the classical Stone duality, which we refer to as commutative Stone duality, between generalized Boolean algebras and locally compact Hausdorff $0$-dimensional spaces.

Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse $\wedge$-semigroups arise as completions of inverse semigroups we call pre-Boolean. An inverse $\wedge$-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where the definition of a tight filter is obtained by combining work of both Exel and Lenz. A simple necessary condition for a semi…

教义性哥德尔完备性定理与类型空间函子

The doctrinal Gödel's completeness theorem and the type space functor

研究在一阶布尔教义形式体系内自证哥德尔完备性定理，表明该定理使一阶布尔教义的纤维式斯通对偶成为类型空间函子，阐述了公式布尔代数对偶与模型斯通空间的关系。

Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categories

** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete $C^{*}$-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal $*$-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand…

A dualizing object approach to non-commutative Stone duality

The aim of the present paper is to extend the dualizing object approach to Stone duality to the non-commutative setting of skew Boolean algebras. This continues the study of non-commutative generalizations of different forms of Stone duality initiated in recent papers by Bauer and Cvetko-Vah, Lawson, Lawson and Lenz, Resende, and also the current author. In particular, we construct a series of dual adjunctions between the categories of Boolean spaces and skew Boolean algebras, unital versions of which are induced by dualizing objects ${0,1,..., n+1}$, $n\geq 0$. We describe Eilenberg-Moore cat…

Enriched Stone-type dualities

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including …

Extensions of dualities and a new approach to the Fedorchuk duality

Applying a general categorical construction for the extension of dualities, we present a new proof of the Fedorchuk duality between the category of compact Hausdorff spaces with their quasi-open mappings and the category of complete normal contact algebras with suprema-preserving Boolean homomorphisms which reflect the contact relation.

Topological Representation of Precontact Algebras and a Connected Version of the Stone Duality Theorem -- I

The notions of a {\em 2-precontact space}\/ and a {\em 2-contact space}\/ are introduced. Using them, new representation theorems for precontact and contact algebras are proved. It is shown that there are bijective correspondences between such kinds of algebras and such kinds of spaces. As applications of the obtained results, we get new connected versions of the Stone Duality Theorems for Boolean algebras and for complete Boolean algebras, as well as a Smirnov-type theorem for a kind of compact $T_0$-extensions of compact Hausdorff extremally disconnected spaces. We also introduce the notion …