Forcing minimal extensions of Boolean algebras

Author:
Piotr Koszmider

Journal:
Trans. Amer. Math. Soc. **351** (1999), 3073-3117

MSC (1991):
Primary 03E35, 03E50, 03E99, 06E15, 06E99, 54A25, 54A35, 54B35, 54G12, 54G99, 54H10

DOI:
https://doi.org/10.1090/S0002-9947-99-02145-5

Published electronically:
April 8, 1999

MathSciNet review:
1467471

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications:

1) We make Fedorchuk's method more flexible, obtaining, for every cardinal of uncountable cofinality, a consistent example of a Boolean algebra whose every infinite homomorphic image is of cardinality and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra can be strictly less than the minimal size of a homomorphic image of , answering a question of J. D. Monk.

2) We prove that for every cardinal of uncountable cofinality it is consistent that and both and exist.

3) By combining these algebras we obtain many examples that answer questions of J.D. Monk.

4) We prove the consistency of MA + CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangelskii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.

5) We prove that for any cardinal of uncountable cofinality it is consistent that there is a countably tight Boolean algebra with a distinguished ultrafilter such that for every the algebra is countable and has hereditary character .

**[BSV]**B.Balcar, P.Simon, P.Vojtas;*Refinement properties and extensions of filters in Boolean algebras*; Trans. AMS. 267, 1981, pp. 265-283. MR**82k:06014****[Bal]**Z.Balogh;*On Compact Hausdorff Spaces of Countable Tightness*; Proc. Amer. Math. Soc., 105, 1989, pp. 755-764. MR**89h:03088****[BMR]**J.Baumgartner, J.Malitz, W.Reinhard;*Embedding trees in the rationals*; Proc. Nat. Acad. Sci. U.S.A. 67. (1970), pp. 3073-3117.j MR**47:3172****[Ba1]**J.Baumgartner;*Iterated Forcing*; in A. Mathias, (ed);*Surveys in Set Theory*; Lecture Note Series 87, London Mathematical Society 1983, pp. 1-59. MR**87c:03099****[Ba 2]**J.Baumgartner;*Applications of Proper Forcing*; in [KV], pp. 913-959. MR**86g:03084****[BK]**J. Baumgartner, P. Komjath;*Boolean algebras in which every chain and every antichain is countable*; Fund.Math 111, 1981. pp. 125-133. MR**82j:06023****[BMS]**J.Baumgartner, D.Martin, S.Shelah (eds);*Axiomatic Set Theory*; Contemporary Mathematics 31, 1984. MR**85g:03004****[BS]**J.Baumgartner, S.Shelah;*Remarks on Superatomic Boolean Algebras*; Ann. Pure. Appl. Logic 33. 1987 pp. 109-130. MR**88d:03100****[De]**K.Devlin;*-trees*; Annals of Mathematical Logic, 13. 1978 pp. 267-330. MR**80c:03053****[vDMR]**E.vanDouwen, J.D.Monk, M.Rubin;*Some Questions About Boolean Algebras*; Algebra Universalis 11, 1988, pp. 220-243. MR**82a:06024****[vD]**E.vanDouwen;*Cardinal Functions on Boolean Spaces*; in [MB], pp. 417-467. CMP**21:10****[Do]**A.Dow*Applications of Elementary submodels to Topology*; Topology Proceedings Vol. 13, No 1, 1988, pp. 17-72. MR**91a:54003****[Fe1]**V. Fedorchuk;*On the cardinality of hereditarily separable compact Hausdorff spaces*; Soviet Math. Dokl. 16, 1975 pp. 651-655.**[Fe2]**V. Fedorchuk;*A compact space having the cardinality of the continuum with no convergent sequences*; Math. Proc. Camb. Phil. Soc. 81, 1977, pp. 177-181. MR**54:13827****[G]**S. Grigorieff;*Combinatorics on Ideals and Forcing*; Annals of Mathematical Logic 3, 1971, pp. 363-393. MR**45:6614****[H]**R. Hodel;*Cardinal Functions I*; in [KV]. MR**86j:54007****[Ju1]**I.Juhasz;*Cardinal Functions in Topology*; Math. Centre Tracts 34, 1971 Amsterdam. MR**49:4778****[Ju2]**I. Juhasz;*Cardinal Functions in Topology - Ten Years Later*; Math. Centre Tracts 123, 1981 Amsterdam. MR**82a:54002****[Ju3]**I. Juhasz;*Cardinal Functions II*; in [KV]. MR**86j:54008****[Ju4]**I. Juhasz;*On the minimal character of points in compact spaces*; Colloq. Math. Soc., János Bolyai 55, North-Holland, Amsterdam, 1993. MR**94k:54004****[Ju5]**I. Juhasz;*A weakening of , with applications to topology*; CMUC 29, 1988, pp. 767-773. MR**90d:54005****[JSz]**I. Juhasz, Z.Szentmiklossy;*Convergent Sequences in Compact Spaces*; Proc. AMS, 116, 1992, pp. 1153-1160. MR**93b:54024****[Just]**W. Just;*Remarks on the altitude of Boolean algebras*; Algebra Universalis, 25 (1988) pp. 283-289. MR**90d:03101****[Kop1]**S. Koppelberg;*Minimally Generated Boolean Algebras*; Order 5, 1989, pp. 393-406. MR**90g:06022****[Kop2]**S.Koppelberg;*Counterexamples in minimally generated Boolean algebras*; Acta Univ. Carolin. Math. Phys. 29, 1988, pp 37-46. MR**90a:06014****[Kop3]**S. Koppelberg;*General Theory of Boolean Algebras*; in [MB], Vol I. MR**90k:06002****[Kosz1]**P. Koszmider;*The Consistency of*; Algebra Universalis vol.27, 1990, pp. 80-87. MR**91a:03101****[Kosz2]**P. Koszmider;*On The Complete Invariance Property in Some Uncountable Products*; Canadian Mathematical Bulletin, Vol 35 (2), 1992, pp. 221-229. MR**93g:54057****[Kosz3]**P. Koszmider;*Semimorasses and Nonreflection at Singular Cardinals*; Annals of Pure and Applied Logic 72 (1995), 1-23. MR**96i:03043****[K]**K.Kunen;*Set Theory. An Introduction to Independence Proofs*; North Holland, 1980. MR**82f:03001****[KV]**K. Kunen, J.E.Vaughan, (eds);*Handbook of Set-Theoretic Topology*; North Holland, 1984. MR**85k:54001****[Mal]**V.Malykhin;*A Frechet-Urysohn compactum without points of countable character*; Math. Notes, 41, pp. 210-216, original in Mat. Zametki 41, 1987, pp. 365-376. MR**88g:54645****[MB]**J.D. Monk, R.Bennet; (eds)*Handbook of Boolean Algebras*; North-Holland, 1989. MR**90k:06004****[M1]**J.D. Monk;*Cardinal Functions on Boolean Algebras*; Lectures in Mathematics, ETH Zurich, Birkhäuser-Verlag, 1990. MR**92d:06033****[M2]**J.D. Monk;*Cardinal Functions on Boolean Algebras*a revised version of [M1], 17 September 1992, circulated notes.**[N1]**P. Nyikos;*Forcing compact non-sequential spaces of countable tightness*; Preprint.**[N2]**P. Nyikos;*Dichotomies in Compact spaces and -spaces; in New Classic problems;*Topology Proceedings, vol 15, pp. 201-220.**[Ra]**M. Rabus;*An -minimal Boolean algebra*; Trans. Amer. Math. Soc. 348, 1996, pp. 3235-3244. MR**96j:03070**1994.**[R]**J. Roitman;*Superatomic Boolean Algebras*; in [MB] pp. 719-740. CMP**21:10****[Si]**R. Sikorski;*Boolean Algebras*; Springer-Verlag 1964. MR**31:2178****[Shah]**D.Shahmatov;*Compact spaces and their generalizations*; in M. Husek, J. vanMill*Recent progress in General Topology*; NHPC 1992, pp. 571- 640. MR**95g:54004****[Shap]**B.Shapirovskii;*On -character and -weight of compact Hausdorff spaces*; Soviet math. Dokl. 16, 1975, pp. 999-1003.**[Sh1]**S.Shelah;*On uncountable Boolean algebras with no uncountable pairwise comparable or incomparable sets of elements*; Notre Dame Journal of Formal Logic 22, 1981, pp. 301-308. MR**83d:03060****[Sh2]**S. Shelah;*Constructions of many complicated uncountable structures and Boolean algebras*; Israel J. Math. Vol 45, No 2-3, 1983, pp. 100-146. MR**86k:06010****[Sh3]**S. Shelah;*Around Classification Theory of Models*; Lecture Notes in Mathematics 1182, Springer-Verlag, 1986. MR**90a:03037****[Sz]**Z.Szentmiklossy;*S-spaces and L-spaces under Martin's axiom*; Coll. Math. Soc. Janos Bolyai 23, Budapest, 1978, 1139-1145. MR**81k:54032****[Ve]**D.Velleman;*Souslin Trees Constructed from Morasses*; in [BMS], pp. 219-241. MR**86b:03066****[W]**W. Weiss;*Versions of Martin's Axiom*; in [KV], pp. 827-886. MR**86d:03088****[Z]**W.Zwicker;*Combinatorics I: Stationary coding sets rationalize the club filter*; in [BMS] , pp. 243-259. MR**86e:03046**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
03E35,
03E50,
03E99,
06E15,
06E99,
54A25,
54A35,
54B35,
54G12,
54G99,
54H10

Retrieve articles in all journals with MSC (1991): 03E35, 03E50, 03E99, 06E15, 06E99, 54A25, 54A35, 54B35, 54G12, 54G99, 54H10

Additional Information

**Piotr Koszmider**

Affiliation:
Departmento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, SP, CEP: 05315-970, Brasil

Email:
piotr@ime.usp.br

DOI:
https://doi.org/10.1090/S0002-9947-99-02145-5

Received by editor(s):
June 6, 1994

Received by editor(s) in revised form:
January 20, 1997

Published electronically:
April 8, 1999

Additional Notes:
Some results presented in this paper were obtained when the author was a Ph.D. student at the University of Toronto, under the supervision of Professors F.D. Tall and W. Weiss. Other results of this paper were obtained when the author was a National Science and Engineering Research Council of Canada postdoctoral fellow at York University, under the supervision of Professor J. Steprans.

Article copyright:
© Copyright 1999
American Mathematical Society