A forcing construction of thin-tall Boolean algebras

Juan Martínez

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 2, page 99-113
  • ISSN: 0016-2736

Abstract

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It was proved by Juhász and Weiss that for every ordinal α with 0 < α < ω 2 there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that κ < κ = κ and α is an ordinal such that 0 < α < κ + + , then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all α < κ + + , we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height α and width κ for every α < κ + + . Consistency for specific κ, like ω 1 , then follows as a corollary.

How to cite

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Martínez, Juan. "A forcing construction of thin-tall Boolean algebras." Fundamenta Mathematicae 159.2 (1999): 99-113. <http://eudml.org/doc/212328>.

@article{Martínez1999,
abstract = {It was proved by Juhász and Weiss that for every ordinal α with $\{0 < α < ω_2\}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that $κ^\{< κ\} = κ$ and α is an ordinal such that $0 < α < κ^\{++\}$, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $α < κ^\{++\}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height α and width κ for every $α < κ^\{++\}$. Consistency for specific κ, like $ω_1$, then follows as a corollary.},
author = {Martínez, Juan},
journal = {Fundamenta Mathematicae},
keywords = {forcing; cardinal-preserving partial order; superatomic Boolean algebra; generic extension; consistency},
language = {eng},
number = {2},
pages = {99-113},
title = {A forcing construction of thin-tall Boolean algebras},
url = {http://eudml.org/doc/212328},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Martínez, Juan
TI - A forcing construction of thin-tall Boolean algebras
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 2
SP - 99
EP - 113
AB - It was proved by Juhász and Weiss that for every ordinal α with ${0 < α < ω_2}$ there is a superatomic Boolean algebra of height α and width ω. We prove that if κ is an infinite cardinal such that $κ^{< κ} = κ$ and α is an ordinal such that $0 < α < κ^{++}$, then there is a cardinal-preserving partial order that forces the existence of a superatomic Boolean algebra of height α and width κ. Furthermore, iterating this forcing through all $α < κ^{++}$, we obtain a notion of forcing that preserves cardinals and such that in the corresponding generic extension there is a superatomic Boolean algebra of height α and width κ for every $α < κ^{++}$. Consistency for specific κ, like $ω_1$, then follows as a corollary.
LA - eng
KW - forcing; cardinal-preserving partial order; superatomic Boolean algebra; generic extension; consistency
UR - http://eudml.org/doc/212328
ER -

References

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  1. [1] J. E. Baumgartner, Almost-disjoint sets, the dense set problem and the partition calculus, Ann. Math. Logic 10 (1976), 401-439. Zbl0339.04003
  2. [2] J. E. Baumgartner, Iterated forcing, in: Surveys in Set Theory, A. R. D. Mathias (ed.), Cambridge Univ. Press, 1983, 1-59. 
  3. [3] J. E. Baumgartner and S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic 33 (1987), 109-129. Zbl0643.03038
  4. [4] T. Jech, Set Theory, Academic Press, New York, 1978. 
  5. [5] I. Juhász and W. Weiss, On thin-tall scattered spaces, Colloq. Math. 40 (1978), 63-68. Zbl0416.54038
  6. [6] W. Just, Two consistency results concerning thin-tall Boolean algebras, Algebra Universalis 20 (1985), 135-142. Zbl0571.03022
  7. [7] P. Koepke and J. C. Martínez, Superatomic Boolean algebras constructed from morasses, J. Symbolic Logic 60 (1995), 940-951. Zbl0854.06018
  8. [8] S. Koppelberg, Handbook of Boolean Algebras, Vol. 1, J. D. Monk and R. Bonnet (eds.), North-Holland, Amsterdam, 1989. Zbl0676.06019
  9. [9] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980. 
  10. [10] J. C. Martínez, A consistency result on thin-tall superatomic Boolean algebras, Proc. Amer. Math. Soc. 115 (1992), 473-477. Zbl0767.03026
  11. [11] J. Roitman, Height and width of superatomic Boolean algebras, ibid. 94 (1985), 9-14. Zbl0534.06004
  12. [12] J. Roitman, Superatomic Boolean algebras, in: Handbook of Boolean Algebras, Vol. 3, J. D. Monk and R. Bonnet (eds.), North-Holland, Amsterdam, 1989, 719-740. 

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