scholarly journals Inside Every Model of Abstract Stone Duality Lies an Arithmetic Universe

2005 ◽  
Vol 122 ◽  
pp. 247-296 ◽  
Author(s):  
Paul Taylor
Keyword(s):  
Stone Duality

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe 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 article, 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 algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

scholarly journals Boolean sets, skew Boolean algebras and a non-commutative Stone duality

2015 ◽  
Vol 75 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Ganna Kudryavtseva ◽  
Mark V. Lawson
Keyword(s):  
Boolean Algebras ◽  
Stone Duality ◽  
Skew Boolean Algebras

Stone duality for stable functions

2005 ◽  
pp. 1-15 ◽  
Author(s):  
Thomas Ehrhard ◽  
Pasquale Malacaria
Keyword(s):  
Stone Duality

Some generalizations of the Stone Duality Theorem

2012 ◽  
Vol 80 (3-4) ◽  
pp. 255-293 ◽  
Author(s):  
GEORGI D. DIMOV
Keyword(s):  
Duality Theorem ◽  
Stone Duality

scholarly journals A refinement of Stone duality to skew Boolean algebras

2012 ◽  
Vol 67 (4) ◽  
pp. 397-416 ◽  
Author(s):  
Ganna Kudryavtseva
Keyword(s):  
Boolean Algebras ◽  
Stone Duality ◽  
Skew Boolean Algebras

Stone duality for lattices

1997 ◽  
Vol 37 (3) ◽  
pp. 391-401 ◽  
Author(s):  
C. Hartonas ◽  
J.M. Dunn
Keyword(s):  
Stone Duality

scholarly journals SPACES OF TYPES IN POSITIVE MODEL THEORY

2019 ◽  
Vol 84 (02) ◽  
pp. 833-848
Author(s):  
LEVON HAYKAZYAN
Keyword(s):  
Model Theory ◽  
Stone Space ◽  
Distributive Lattices ◽  
Stone Duality ◽  
Order Model ◽  
First Order ◽  
Countable Models

AbstractWe introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

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