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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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independent families in Boolean Algebras with some separation properties
Algebra Universalis, 2013Co-Authors: Piotr Koszmider, Saharon ShelahAbstract:We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size \({\mathfrak{c}}\), the size of the continuum. This improves a result of Argyros from the 1980s, which asserted the existence of an uncountable independent family. In fact, we prove it for a bigger class of Boolean Algebras satisfying much weaker properties. It follows that the Stone space \({K_\mathcal{A}}\) of all such Boolean Algebras \({\mathcal{A}}\) contains a copy of the Cech–Stone compactification of the integers \({\beta\mathbb{N}}\) and the Banach space \({C(K_\mathcal{A})}\) has l∞ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.
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depth and length of Boolean Algebras
arXiv: Rings and Algebras, 2013Co-Authors: Shimon Garti, Saharon ShelahAbstract:Suppose � = cf(�),� > cf(�) = � + and � = � � . We prove that there exist a sequence hBi : i < �i of Boolean Algebras and an ultra- filter D onso that � = Q i<� Depth + (B i)/D < Depth + ( Q i<� B i/D) = � + . An identical result holds also for Length + . The proof is carried in ZFC, and it holds even above large cardinals.
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independent families in Boolean Algebras with some separation properties
arXiv: Logic, 2012Co-Authors: Piotr Koszmider, Saharon ShelahAbstract:We prove that any Boolean algebra with the subsequential completeness property contains an independent family of size continuum. This improves a result of Argyros from the 80ties which asserted the existence of an uncountable independent family. In fact we prove it for a bigger class of Boolean Algebras satisfying much weaker properties. It follows that the Stone spaces of all such Boolean Algebras contains a copy of the Cech-Stone compactification of the integers and the Banach space of contnuous functions on them has $l_\infty$ as a quotient. Connections with the Grothendieck property in Banach spaces are discussed.
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the number of openly generated Boolean Algebras
Journal of Symbolic Logic, 2008Co-Authors: Stefan Geschke, Saharon ShelahAbstract:This article is devoted to two difierent generalizations of projective Boolean Algebras: openly generated Boolean Algebras and tightly ae-flltered Boolean Algebras. We show that for every uncountable regular cardinalthere are 2 • pairwise non-isomorphic openly generated Boolean Algebras of size • > @1 provided there is an almost free non-free abelian group of size •. The openly generated Boolean Algebras constructed here are almost free. Moreover, for every inflnite regular cardinalwe construct 2 • pairwise non-isomorphic Boolean Algebras of sizethat are tightly ae-flltered and c.c.c. These two results contrast nicely with Koppelberg's theorem in (13) that for every uncountable regular cardinalthere are only 2 <• isomorphism types of projective Boolean Algebras of size •.
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The number of openly generated Boolean Algebras
Journal of Symbolic Logic, 2008Co-Authors: Stefan Geschke, Saharon ShelahAbstract:This article is devoted to two difierent generalizations of projective Boolean Algebras: openly generated Boolean Algebras and tightly ae-flltered Boolean Algebras. We show that for every uncountable regular cardinalthere are 2 • pairwise non-isomorphic openly generated Boolean Algebras of size • > @1 provided there is an almost free non-free abelian group of size •. The openly generated Boolean Algebras constructed here are almost free. Moreover, for every inflnite regular cardinalwe construct 2 • pairwise non-isomorphic Boolean Algebras of sizethat are tightly ae-flltered and c.c.c. These two results contrast nicely with Koppelberg's theorem in (13) that for every uncountable regular cardinalthere are only 2
Weiru Liu - One of the best experts on this subject based on the ideXlab platform.
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Rough operations on Boolean Algebras
Information Sciences, 2004Co-Authors: Weiru LiuAbstract:In this paper, we introduce two pairs of rough operations on Boolean Algebras. First we define a pair of rough approximations based on a partition of the unity of a Boolean algebra. We then propose a pair of generalized rough approximations on Boolean Algebras after defining a basic assignment function between two different Boolean Algebras. Finally, some discussions on the relationship between rough operations and some uncertainty measures are given to provide a better understanding of both rough operations and uncertainty measures on Boolean Algebras.
Sabine Koppelberg - One of the best experts on this subject based on the ideXlab platform.
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Densities of ultraproducts of Boolean Algebras
Canadian Journal of Mathematics, 1995Co-Authors: Sabine Koppelberg, Saharon ShelahAbstract:We answer three problems by J. D. Monk on cardinal invariants of Boolean Algebras. Two of these are whether taking the algebraic density …A resp. the topological density dA of a Boolean algebra A commutes with formation of ul- traproducts; the third one compares the number of endomorphisms and of ideals of a Boolean algebra.
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Pseudo-trees and Boolean Algebras
Order, 1992Co-Authors: Sabine Koppelberg, J. Donald MonkAbstract:We consider Boolean Algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean Algebras.
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On L∞κ-free Boolean Algebras
Annals of Pure and Applied Logic, 1992Co-Authors: Sakaé Fuchino, Sabine Koppelberg, Makoto TakahashiAbstract:Abstract We study L ∞κ -freeness in the variety of Boolean Algebras. It is shown that some of the theorems on L ∞κ -free Algebras which are known to hold in varieties such as groups, abelian groups etc. are also true for Boolean Algebras. But we also investigate properties such as the ccc of L ∞κ -free Boolean Algebras which have no counterpart in the varieties above.
Karin Cvetkovah - One of the best experts on this subject based on the ideXlab platform.
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the connection of skew Boolean Algebras and discriminator varieties to church Algebras
Algebra Universalis, 2015Co-Authors: Karin Cvetkovah, Antonino SalibraAbstract:We establish a connection between skew Boolean Algebras and Church Algebras. We prove that the set of all semicentral elements in a right Church algebra forms a right-handed skew Boolean algebra for the properly defined operations. The main result of this paper states that the variety of all semicentral right Church Algebras of type \({\tau}\) is term equivalent to the variety of right-handed skew Boolean Algebras with additional regular operations. As a corollary to this result, we show that a pointed variety is a discriminator variety if and only if it is a 0-regular variety of right-handed skew Boolean Algebras.
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stone duality for skew Boolean Algebras with intersections
Houston Journal of Mathematics, 2013Co-Authors: Andrej Bauer, Karin CvetkovahAbstract:We extend Stone duality between generalized Boolean Algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean Algebras with intersections is dual to the cate
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stone duality for skew Boolean Algebras with intersections
arXiv: Rings and Algebras, 2011Co-Authors: Andrej Bauer, Karin CvetkovahAbstract:We extend Stone duality between generalized Boolean Algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean Algebras with intersections is dual to the category of surjective etale maps between Boolean spaces. We then extend the duality to skew Boolean Algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.
Jindřich Zapletal - One of the best experts on this subject based on the ideXlab platform.
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Semi-Cohen Boolean Algebras☆
Annals of Pure and Applied Logic, 1997Co-Authors: Bohuslav Balcar, Thomas Jech, Jindřich ZapletalAbstract:Abstract We investigate classes of Boolean Algebras related to the notion of forcing that adds Cohen reals. A Cohen algebra is a Boolean algebra that is dense in the completion of a free Boolean algebra. We introduce and study generalizations of Cohen Algebras: semi-Cohen Algebras, pseudo-Cohen Algebras and potentially Cohen Algebras. These classes of Boolean Algebras are closed under completion.
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Semi-Cohen Boolean Algebras
arXiv: Logic, 1995Co-Authors: Bohuslav Balcar, Thomas Jech, Jindřich ZapletalAbstract:We investigate classes of Boolean Algebras related to the notion of forcing that adds Cohen reals. A >>Cohen algebra
