Representing free Boolean algebras

Alan Dow; P. Nyikos

Fundamenta Mathematicae (1992)

  • Volume: 141, Issue: 1, page 21-30
  • ISSN: 0016-2736

Abstract

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Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with ℵ 1 generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra of any partitioner algebra.

How to cite

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Dow, Alan, and Nyikos, P.. "Representing free Boolean algebras." Fundamenta Mathematicae 141.1 (1992): 21-30. <http://eudml.org/doc/211949>.

@article{Dow1992,
abstract = {Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with $ℵ_1$ generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra of any partitioner algebra.},
author = {Dow, Alan, Nyikos, P.},
journal = {Fundamenta Mathematicae},
keywords = {free Boolean algebra; partitioner algebra},
language = {eng},
number = {1},
pages = {21-30},
title = {Representing free Boolean algebras},
url = {http://eudml.org/doc/211949},
volume = {141},
year = {1992},
}

TY - JOUR
AU - Dow, Alan
AU - Nyikos, P.
TI - Representing free Boolean algebras
JO - Fundamenta Mathematicae
PY - 1992
VL - 141
IS - 1
SP - 21
EP - 30
AB - Partitioner algebras are defined in [2] and are natural tools for studying the properties of maximal almost disjoint families of subsets of ω. In this paper we investigate which free algebras can be represented as partitioner algebras or as subalgebras of partitioner algebras. In so doing we answer a question raised in [2] by showing that the free algebra with $ℵ_1$ generators is represented. It was shown in [2] that it is consistent that the free Boolean algebra of size continuum is not a subalgebra of any partitioner algebra.
LA - eng
KW - free Boolean algebra; partitioner algebra
UR - http://eudml.org/doc/211949
ER -

References

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  1. [1] B. Balcar, J. Pelant, and P. Simon, The space of ultrafilters on N covered by nowhere dense sets, Fund. Math. 110 (1980), 11-24. Zbl0568.54004
  2. [2] J. E. Baumgartner and M. Weese, Partition algebras for almost-disjoint families, Trans. Amer. Math. Soc. 274 (1982), 619-630. Zbl0515.03032
  3. [3] D. Booth, A Boolean view of sequential compactness, Fund. Math. 85 (1974), 99-102. Zbl0297.54020
  4. [4] A. Dow and R. Frankiewicz, Remarks on partitioner algebras, Proc. Amer. Math. Soc. 113 (1991), 1067-1070. Zbl0789.03043
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  6. [6] S. Mr'owka, On completely regular spaces, Fund. Math. 41 (1954), 105-106. 
  7. [7] S. Mr'owka, Some set-theoretic constructions in topology, ibid. 94 (1977), 83-92. 
  8. [8] J. Roitman, Adding a random or a Cohen real: topological consequences and the effect on Martin's axiom, ibid. 103 (1979), 47-60; Correction, ibid. 129 (1988), 141. Zbl0442.03034
  9. [9] F. Rothberger, On some problems of Hausdorff and of Sierpiński, ibid. 35 (1948), 29-46. Zbl0032.33702
  10. [10] J. Teresawa, Spaces N ∪ ℛ and their dimensions, Topology Appl. 11 (1980), 93-102. 
  11. [11] E. K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan (eds.), North-Holland, 1984, 111-168. 
  12. [12] J. Vaughan, Small uncountable cardinals and topology, in: Open Problems in Topology, J. van Mill and G. M. Reed (eds.), North-Holland, 1990, 195-218. 

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