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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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on the weak freese nation property of Complete Boolean Algebras
Annals of Pure and Applied Logic, 2001Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.
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on the weak freese nation property of Complete Boolean Algebras
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)
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on absolutely divergent series
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(omega)/fin. This complements Vojtas' result, that under cf(c)=p the two Algebras are isomorphic
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on absolutely divergent series
Fundamenta Mathematicae, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the @2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(!)=fln. This complements Vojtas' result, that under cf(c) = p the two Algebras are isomorphic (15).
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a Complete Boolean Algebra that has no proper atomless Complete subAlgebra
arXiv: Logic, 1995Co-Authors: Thomas Jech, Saharon ShelahAbstract:There exists a Complete atomless Boolean Algebra that has no proper atomless Complete subAlgebra.
Sakae Fuchino - One of the best experts on this subject based on the ideXlab platform.
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on the weak freese nation property of Complete Boolean Algebras
Annals of Pure and Applied Logic, 2001Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.
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on the weak freese nation property of Complete Boolean Algebras
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)
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on absolutely divergent series
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(omega)/fin. This complements Vojtas' result, that under cf(c)=p the two Algebras are isomorphic
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on absolutely divergent series
Fundamenta Mathematicae, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the @2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(!)=fln. This complements Vojtas' result, that under cf(c) = p the two Algebras are isomorphic (15).
Francesco Ciraulo - One of the best experts on this subject based on the ideXlab platform.
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Overlap Algebras: a Constructive Look at Complete Boolean Algebras
arXiv: Logic in Computer Science, 2019Co-Authors: Francesco Ciraulo, Michele ContenteAbstract:The notion of a Complete Boolean Algebra, although Completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap Algebra, although classically equivalent to that of a Complete Boolean Algebra, has powersets and other natural structures as instances. In this paper we study the category of overlap Algebras as an extension of the category of sets and relations, and we establish some basic facts about mono-epi-isomorphisms and (co)limits; here a morphism is a symmetrizable function (with classical logic this is just a function which preserves joins). Then we specialize to the case of morphisms which preserve also finite meets: classically, this is the usual category of Complete Boolean Algebras. Finally, we connect overlap Algebras with locales, and their morphisms with open maps between locales, thus obtaining constructive versions of some results about Boolean locales.
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Constructive version of Boolean Algebra
2016Co-Authors: Francesco Ciraulo, Maria Emilia Maietti, Paola TotoAbstract:The notion of overlap Algebra introduced by G. Sambin provides a constructive version of Complete Boolean Algebra. Here we show that his notion of overlap morphism corresponds classically to that of map preserving arbitrary joins. Moreover we prove that the power collection of a set is the free overlap Algebra join-generated from the set. Then, we generalize the concept of overlap Algebra and overlap mor-phism in various ways to provide constructive versions of the category of Boolean Algebras with maps preserving arbitrary existing joins.
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regular opens in constructive topology and a representation theorem for overlap Algebras
Annals of Pure and Applied Logic, 2013Co-Authors: Francesco CirauloAbstract:Abstract Giovanni Sambin has recently introduced the notion of an overlap Algebra in order to give a constructive counterpart to a Complete Boolean Algebra. We propose a new notion of regular open subset within the framework of intuitionistic, predicative topology and we use it to give a representation theorem for (set-based) overlap Algebras. In particular we show that there exists a duality between the category of set-based overlap Algebras and a particular category of topologies in which all open subsets are regular.
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constructive version of Boolean Algebra
arXiv: Logic, 2012Co-Authors: Francesco Ciraulo, Maria Emilia Maietti, Paola TotoAbstract:The notion of overlap Algebra introduced by G. Sambin provides a constructive version of Complete Boolean Algebra. Here we first show some properties concerning overlap Algebras: we prove that the notion of overlap morphism corresponds classically to that of map preserving arbitrary joins; we provide a description of atomic set-based overlap Algebras in the language of formal topology, thus giving a predicative characterization of discrete locales; we show that the power-collection of a set is the free overlap Algebra join-generated from the set. Then, we generalize the concept of overlap Algebra and overlap morphism in various ways to provide constructive versions of the category of Boolean Algebras with maps preserving arbitrary existing joins.
Lajos Soukup - One of the best experts on this subject based on the ideXlab platform.
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on the weak freese nation property of Complete Boolean Algebras
Annals of Pure and Applied Logic, 2001Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:Abstract The following results are proved: (a) In a model obtained by adding ℵ 2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property is consistent with GCH. (c) If a weak form of □μ and cof ([μ] ℵ 0 ,⊆)=μ + hold for each μ>cf(μ)=ω, then the weak Freese-Nation property of 〈 P (ω),⊆〉 is equivalent to the weak Freese-Nation property of any of C (κ) or R (κ) for uncountable κ. (d) Modulo the consistency of (ℵ ω+1 ,ℵ ω )↠(ℵ 1 ,ℵ 0 ) , it is consistent with GCH that C (ℵ ω ) does not have the weak Freese-Nation property and hence the assertion in (c) does not hold, and also that adding ℵ ω Cohen reals destroys the weak Freese-Nation property of 〈 P (ω), ⊆ 〉 . These results solve all of the problems except Problem 1 in S. Fuchino, L. Soukup, Fundament. Math. 154 (1997) 159–176, and some other problems posed by Geschke.
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on the weak freese nation property of Complete Boolean Algebras
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Stefan Geschke, Lajos SoukupAbstract:The following results are proved: (a) In a model obtained by adding aleph_2 Cohen reals, there is always a c.c.c. Complete Boolean Algebra without the weak Freese-Nation property. (b) Modulo the consistency strength of a supercompact cardinal, the existence of a c.c.c. Complete Boolean Algebras without the weak Freese-Nation property consistent with GCH. (c) Under some consequences of the negation of 0^#, the weak Freese-Nation property of (P(omega),subseteq) is equivalent to the weak Freese-Nation property of any of C(kappa) or R(kappa) for uncountable kappa. (d) Modulo consistency of (aleph_{omega+1},aleph_omega)-->(aleph_1,aleph_0), it is consistent with GCH that the assertion in (c) does not hold and also that adding aleph_omega Cohen reals destroys the weak Freese-Nation property of (P(omega),subseteq)
Peter Vojtas - One of the best experts on this subject based on the ideXlab platform.
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on absolutely divergent series
arXiv: Logic, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(omega)/fin. This complements Vojtas' result, that under cf(c)=p the two Algebras are isomorphic
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on absolutely divergent series
Fundamenta Mathematicae, 1999Co-Authors: Sakae Fuchino, Saharon Shelah, Heike Mildenberger, Peter VojtasAbstract:We show that in the @2-stage countable support iteration of Mathias forcing over a model of CH the Complete Boolean Algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(!)=fln. This complements Vojtas' result, that under cf(c) = p the two Algebras are isomorphic (15).