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 generalization of the Stone Duality Theorem

2017 ◽  
Vol 221 ◽  
pp. 237-261 ◽  
Author(s):  
G. Dimov ◽  
E. Ivanova-Dimova ◽  
D. Vakarelov
Keyword(s):  
Duality Theorem ◽  
Stone Duality

scholarly journals Two extensions of the stone duality to the category of zero-dimensional Hausdorff spaces

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1851-1878
Author(s):  
Georgi Dimov ◽  
Elza Ivanova-Dimova
Keyword(s):  
Duality Theorem ◽  
Hausdorff Space ◽  
Stone Duality ◽  
Hausdorff Spaces ◽  
Duality Theorems ◽  
Extremally Disconnected ◽  
Continuous Maps ◽  
Hausdorff Compactifications ◽  
Tychonoff Spaces

Extending the Stone Duality Theorem, we prove two duality theorems for the category ZHaus of zero-dimensional Hausdorff spaces and continuous maps. They extend also the Tarski Duality Theorem; the latter is even derived from one of them. We prove as well two new duality theorems for the category EDTych of extremally disconnected Tychonoff spaces and continuous maps. Also, we describe two categories which are dually equivalent to the category ZComp of zero-dimensional Hausdorff compactifications of zero-dimensional Hausdorff spaces and obtain as a corollary the Dwinger Theorem about zero-dimensional compactifications of a zero-dimensional Hausdorff space.

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.

On the Monge-Kantorovitch Duality Theorem

2001 ◽  
Vol 45 (2) ◽  
pp. 350-356 ◽  
Author(s):  
D. Ramachandran ◽  
L. Rüschendorf
Keyword(s):  
Duality Theorem

scholarly journals The Eilenberg-Borsuk duality theorem

1972 ◽  
Vol 75 (1) ◽  
pp. 68-72 ◽  
Author(s):  
J.M Aarts ◽  
T Nishiura
Keyword(s):  
Duality Theorem

scholarly journals A Fenchel-Rockafellar type duality theorem for maximization

1979 ◽  
Vol 20 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Space ◽  
Duality Theorem ◽  
Locally Convex Space ◽  
Bounded Subset ◽  
Convex Functional ◽  
Lower Semi ◽  
Locally Convex

We prove that sup(f-h)(E) = sup(h*-f*)(E*), where f is a proper lower semi-continuous convex functional on a real locally convex space E, h: E → = [-∞, +∞] is an arbitrary-functional and, f*, h* are their convex conjugates respectively. When h = δG, the indicator of a bounded subset G of E, this yields a formula for sup f(G).

Takesaki–Takai Duality Theorem in Hilbert C*-Modules

2004 ◽  
Vol 20 (6) ◽  
pp. 1079-1088
Author(s):  
Mao Zheng Guo ◽  
Xiao Xia Zhang
Keyword(s):  
Duality Theorem

On Grothendieck's local duality theorem

1979 ◽  
Vol 85 (3) ◽  
pp. 431-437 ◽  
Author(s):  
M. H. Bijan-Zadeh ◽  
R. Y. Sharp
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theorem ◽  
Great Emphasis ◽  
Triangulated Category ◽  
Elementary Approach ◽  
Commutative Noetherian Ring ◽  
Local Duality ◽  
Dualizing Complexes

In (11) and (12), a comparatively elementary approach to the use of dualizing complexes in commutative algebra has been developed. Dualizing complexes were introduced by Grothendieck and Hartshorne in (2) for use in algebraic geometry; the approach to dualizing complexes in (11) and (12) differs from that of Grothendieck and Hartshorne in that it avoids use of the concepts of triangulated category, derived category, and localization of categories, and instead places great emphasis on the concept of quasi-isomorphism of complexes of modules over a commutative Noetherian ring.

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