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

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

scholarly journals A DUALIZING OBJECT APPROACH TO NONCOMMUTATIVE STONE DUALITY

2013 ◽  
Vol 95 (3) ◽  
pp. 383-403 ◽  
Author(s):  
GANNA KUDRYAVTSEVA
Keyword(s):  
Boolean Algebras ◽  
Left Adjoint ◽  
Stone Duality ◽  
Twisted Product ◽  
Left Handed ◽  
Boolean Spaces ◽  
Skew Boolean Algebras ◽  
Noncommutative Setting

AbstractThe aim of the present paper is to extend the dualizing object approach to Stone duality to the noncommutative setting of skew Boolean algebras. This continues the study of noncommutative 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 this paper we construct a series of dual adjunctions between the categories of left-handed skew Boolean algebras and Boolean spaces, the unital versions of which are induced by dualizing objects $\{ 0, 1, \ldots , n+ 1\} $, $n\geq 0$. We describe the categories of Eilenberg-Moore algebras of the monads of the adjunctions and construct easily understood noncommutative reflections of left-handed skew Boolean algebras, where the latter can be faithfully embedded (if $n\geq 1$) in a canonical way. As an application, we answer the question that arose in a recent paper by Leech and Spinks to describe the left adjoint to their ‘twisted product’ functor $\omega $.

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.

The connection of skew Boolean algebras and discriminator varieties to Church algebras

2015 ◽  
Vol 73 (3-4) ◽  
pp. 369-390 ◽  
Author(s):  
Karin Cvetko-Vah ◽  
Antonino Salibra
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties ◽  
Skew Boolean Algebras

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.

VARIETIES OF SKEW BOOLEAN ALGEBRAS WITH INTERSECTIONS

2016 ◽  
Vol 102 (2) ◽  
pp. 290-306
Author(s):  
JONATHAN LEECH ◽  
MATTHEW SPINKS
Keyword(s):  
Boolean Algebras ◽  
Lattice Of Varieties ◽  
Skew Boolean Algebras

Skew Boolean algebras for which pairs of elements have natural meets, called intersections, are studied from a universal algebraic perspective. Their lattice of varieties is described and shown to coincide with the lattice of quasi-varieties. Some connections of relevance to arbitrary skew Boolean algebras are also established.

Stone Duality for Nominal Boolean Algebras with И

2011 ◽  
pp. 192-207 ◽  
Author(s):  
Murdoch J. Gabbay ◽  
Tadeusz Litak ◽  
Daniela Petrişan
Keyword(s):  
Boolean Algebras ◽  
Stone Duality

Skew Boolean algebras

1990 ◽  
Vol 27 (4) ◽  
pp. 497-506 ◽  
Author(s):  
Jonathan Leech
Keyword(s):  
Boolean Algebras ◽  
Skew Boolean Algebras

scholarly journals A NONCOMMUTATIVE GENERALIZATION OF STONE DUALITY

2010 ◽  
Vol 88 (3) ◽  
pp. 385-404 ◽  
Author(s):  
M. V. Lawson
Keyword(s):  
Boolean Algebras ◽  
Inverse Monoid ◽  
Stone Duality ◽  
Inverse Monoids ◽  
Group Of Units ◽  
Boolean Spaces ◽  
Thompson Group

AbstractWe prove that the category of boolean inverse monoids is dually equivalent to the category of boolean groupoids. This generalizes the classical Stone duality between boolean algebras and boolean spaces. As an instance of this duality, we show that the boolean inverse monoid Cn associated with the Cuntz groupoid Gn is the strong orthogonal completion of the polycyclic (or Cuntz) monoid Pn. The group of units of Cn is the Thompson group Vn,1.

Skew Boolean algebras and discriminator varieties

1995 ◽  
Vol 33 (3) ◽  
pp. 387-398 ◽  
Author(s):  
R. J. Bignall ◽  
J. E. Leech
Keyword(s):  
Boolean Algebras ◽  
Discriminator Varieties ◽  
Skew Boolean Algebras
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