# A forcing construction of thin-tall Boolean algebras

Fundamenta Mathematicae (1999)

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

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topMartínez, 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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- [9] K. Kunen, Set Theory, North-Holland, Amsterdam, 1980.
- [10] J. C. Martínez, A consistency result on thin-tall superatomic Boolean algebras, Proc. Amer. Math. Soc. 115 (1992), 473-477. Zbl0767.03026
- [11] J. Roitman, Height and width of superatomic Boolean algebras, ibid. 94 (1985), 9-14. Zbl0534.06004
- [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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