scholarly journals The Stone spaces of Boolean algebras

2013 ◽  
pp. 11-16 ◽  
Author(s):  
A.A. Gryzlov ◽  
◽  
R.A. Golovastov ◽  
Keyword(s):  
Boolean Algebras ◽  
Stone 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.

scholarly journals Saturated boolean algebras and their stone spaces

1985 ◽  
Vol 21 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Alan Dow
Keyword(s):  
Boolean Algebras ◽  
Stone Spaces

scholarly journals Idempotent generated algebras and Boolean powers of commutative rings

10.29007/dgb4 ◽  
2018 ◽  
Author(s):  
Guram Bezhanishvili ◽  
Vincenzo Marra ◽  
Patrick J. Morandi ◽  
Bruce Olberding
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Boolean Algebras ◽  
Stone Duality ◽  
Dual Equivalence ◽  
Stone Spaces

For a commutative ring R, we introduce the notion of a Specker R-algebra and show that Specker R-algebras are Boolean powers of R. For an indecomposable ring R, this yields an equivalence between the category of Specker R-algebras and the category of Boolean algebras. Together with Stone duality this produces a dual equivalence between the category of Specker R-algebras and the category of Stone spaces.

scholarly journals Reduced coproducts of compact Hausdorff spaces

1987 ◽  
Vol 52 (2) ◽  
pp. 404-424 ◽  
Author(s):  
Paul Bankston
Keyword(s):  
Topological Structure ◽  
Boolean Algebras ◽  
Stone Space ◽  
Hausdorff Spaces ◽  
Topological Construction ◽  
Stone Spaces

AbstractBy analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

Boolean algebras, Stone spaces, and the iterated Turing jump

1994 ◽  
Vol 59 (4) ◽  
pp. 1121-1138 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Isomorphic Copy ◽  
Stone Spaces ◽  
Countable Structure

AbstractWe show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, is a countable structure with finite signature, and d is a degree, we say that has αth-jump degreed if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0(ω) is the ωth jump degree of a Boolean algebra, but that for n < ω no Boolean algebra has nth-jump degree d < 0(n). The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.

scholarly journals Enlargements of Boolean algebras and Stone spaces

1978 ◽  
Vol 100 (1) ◽  
pp. 35-39 ◽  
Author(s):  
H. Gonshor
Keyword(s):  
Boolean Algebras ◽  
Stone Spaces

scholarly journals Stone duality for first-order logic: a nominal approach to logic and topology

10.29007/tp3z ◽  
2018 ◽  
Author(s):  
Murdoch J. Gabbay
Keyword(s):  
Order Logic ◽  
Boolean Algebras ◽  
First Order Logic ◽  
Specific Class ◽  
Stone Duality ◽  
First Order ◽  
And Topology ◽  
Universal Quantification ◽  
Stone Spaces ◽  
Nominal Sets

What are variables, and what is universal quantification over a variable?Nominal sets are a notion of `sets with names', and using equational axioms in nominal algebra these names can be given substitution and quantification actions.So we can axiomatise first-order logic as a nominal logical theory.We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement.Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quantification becomes an infinite sets intersection?Given answers to these questions, we can seek notions of topology.What is the general notion of topological space of which our sets representation of predicates makes predicates into `open sets'; and what specific class of topological spaces corresponds to the image of nominal algebras for first-order logic?The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces.Nominal algebra lets us extend Boolean algebras to `FOL-algebras', and nominal sets let us correspondingly extend Stone spaces to `∀-Stone spaces'.These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets.

QUASI-TOPOLOGICAL STONE SPACES

1994 ◽  
Vol 27 (3-4) ◽  
Author(s):  
Bronislaw Tembrowski
Keyword(s):  
Stone Spaces
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