Spectral Spaces and Spectral Maps

2019 ◽  
pp. 1-47
Keyword(s):  
Spectral Spaces

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.

Revisiting Numerical Errors in Direct and Large Eddy Simulations of Turbulence: Physical and Spectral Spaces Analysis

2001 ◽  
Vol 174 (2) ◽  
pp. 816-851 ◽  
Author(s):  
Ivan Fedioun ◽  
Nicolas Lardjane ◽  
Iskender Gökalp
Keyword(s):  
Large Eddy Simulations ◽  
Numerical Errors ◽  
Large Eddy ◽  
Spectral Spaces

SUBMAXIMAL AND SPECTRAL SPACES

2008 ◽  
Vol 108A (1) ◽  
pp. 137-147
Author(s):  
M. E. Adams ◽  
Karim Belaid ◽  
Lobna Dridi ◽  
Othman Echi
Keyword(s):  
Spectral Spaces

scholarly journals Bitopological duality for distributive lattices and Heyting algebras

2010 ◽  
Vol 20 (3) ◽  
pp. 359-393 ◽  
Author(s):  
GURAM BEZHANISHVILI ◽  
NICK BEZHANISHVILI ◽  
DAVID GABELAIA ◽  
ALEXANDER KURZ
Keyword(s):  
Boolean Algebras ◽  
Distributive Lattices ◽  
Heyting Algebras ◽  
Priestley Spaces ◽  
Spectral Spaces ◽  
Algebraic Concepts ◽  
Stone Spaces

We introduce pairwise Stone spaces as a bitopological generalisation of Stone spaces – the duals of Boolean algebras – and show that they are exactly the bitopological duals of bounded distributive lattices. The category PStone of pairwise Stone spaces is isomorphic to the category Spec of spectral spaces and to the category Pries of Priestley spaces. In fact, the isomorphism of Spec and Pries is most naturally seen through PStone by first establishing that Pries is isomorphic to PStone, and then showing that PStone is isomorphic to Spec. We provide the bitopological and spectral descriptions of many algebraic concepts important in the study of distributive lattices. We also give new bitopological and spectral dualities for Heyting algebras, thereby providing two new alternatives to Esakia's duality.

Cevian properties in ideal lattices of Abelian ℓ-groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Miroslav Ploščica
Keyword(s):  
Distributive Lattice ◽  
Recent Result ◽  
Ideal Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Spectral Spaces ◽  
Abelian Lattice

Abstract We consider the problem of describing the lattices of compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. (Equivalently, describing the spectral spaces of Abelian lattice-ordered groups.) It is known that these lattices have countably based differences and admit a Cevian operation. Our first result says that these two properties are not sufficient: there are lattices having both countably based differences and Cevian operations, which are not representable by compact ℓ {\ell} -ideals of Abelian lattice-ordered groups. As our second result, we prove that every completely normal distributive lattice of cardinality at most ℵ 1 {\aleph_{1}} admits a Cevian operation. This complements the recent result of F. Wehrung, who constructed a completely normal distributive lattice having countably based differences, of cardinality ℵ 2 {\aleph_{2}} , without a Cevian operation.

scholarly journals ‘Welcome to London’: Spectral Spaces in Sherlock Holmes’s Metropolis

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Christina Lee
Keyword(s):  
Popular Culture ◽  
Sherlock Holmes ◽  
Fictional Narratives ◽  
Embodied Experiences ◽  
Spectral Spaces ◽  
The City

This article examines the burgeoning tourist trade for locations featured in fictional narratives in popular culture. Symptomatic of a postmodern, hyperlinked culture referencing a vast reservoir of texts, such tourism produces a convergence of effects which render places ambivalent. Through a case study of Sherlock Holmes tourism in London, I argue that the city is constructed as seething with the spectral in which there is tension and slippage between paratexts, past and present, history and fiction, the observable and imperceptible. The tourist seeks out embodied experiences of their own secret London(s) which reside somewhere in-between the multiplicitous topographies.

Tone color and spectral spaces for steady sounds

1986 ◽  
Vol 80 (S1) ◽  
pp. S92-S93
Author(s):  
Evi Papachristou ◽  
William J. Strong ◽  
Bruce L. Brown
Keyword(s):  
Tone Color ◽  
Spectral Spaces

Subsets of Spectral Spaces

2019 ◽  
pp. 102-140
Keyword(s):  
Spectral Spaces

An evaluation of different bi-spectral spaces for discriminating burned shrub-savannah

2001 ◽  
Vol 22 (13) ◽  
pp. 2641-2647 ◽  
Author(s):  
S. Trigg ◽  
S. Flasse
Keyword(s):  
Spectral Spaces

Group Theoretical Structure of Spectral Spaces

2005 ◽  
Vol 23 (3) ◽  
pp. 297-313 ◽  
Author(s):  
Reiner Lenz ◽  
Thanh Hai Bui ◽  
Javier Hernández-Andrés
Keyword(s):  
Theoretical Structure ◽  
Spectral Spaces
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