Abstract
We prove in ZFC that the Baer-Specker group Zω has 2ℵ 1 non-free pure subgroups of cardinality ℵ1 which are almost disjoint: there is no non-free subgroup embeddable in any pair.
We thank the referee for constructive comments.
This research was partially supported by the German-Israel Foundation project No. G-545-173.06/97; publication number 683.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Baer, Abelian groups without elements of finite order, Duke Math. J. 3 (1937), 68–122.
A. Blass, Cardinal characteristics and the product of countably many infinite cyclic groups, J. Algebra 169 (1994), 512–540.
A. Blass and R. Göbel, Subgroups of the Baer-Specker group with few endomorphisms but large dual, Fund. Math. 149 (1996), 19–29.
W. W. Comfort and S. Negrepontis, The Theory of Ultrafilters, Berlin: Springer-Verlag (1974).
A. L. S. Corner and R. Göbel, Essentially rigid subgroups of the Baer-Specker group, Manuscripta math. 94 (1997), 319–326.
A. L. S. Corner and B. Goldsmith, On endomorphisms and automorphisms of some pure subgroups of the Baer-Specker group, in Abelian Group Theory and Related Topics (R. Göbel, P. Hill and W. Liebert, eds.), Contemp. Math. 171 (1994), 69–78.
M. Dugas and J. Irwin, On pure subgroups of cartesian products of integers, Results in Math. 15 (1989), 35–52.
K. Eda, A note on subgroups of Z N, in Abelian Group Theory (R. Göbel et al, eds.), Springer Lecture Notes in Mathematics 1006 (1983), 371–374.
P. C. Eklof and A. H. Mekler, Infinitary stationary logic and abelian groups, Fund. Math. 112 (1981), 1–15.
P. C. Eklof and A. H. Mekler, Almost Free Modules. Set-theoretic Methods, Amsterdam: North-Holland (1990).
P. C. Eklof, A. H. Mekler, and S. Shelah, Almost disjoint abelian groups, Israel J. Math. 49 (1984), 34–54.
P. C. Eklof and S. Shelah, A combinatorial principle equivalent to the existence of non-free Whitehead groups, in Abelian Group Theory and Related Topics (R. Göbel, P. Hill and W. Liebert, eds.), Contemp. Math. 171 (1994), 79–98.
R. Engelking and M. Karlowicz, Some theorems of set theory and their topological consequences, Fund. Math. 57 (1965), 275–285.
L. Fuchs, Infinite Abelian Groups, Vols. I and II, New York: Academic Press (1970, 1973).
L. Heindorf and L. B. Shapiro, Nearly Projective Boolean Algebras, Springer Lecture Notes in Mathematics 1596 (1994).
S. Shelah, Classification Theory and the Number of Non-Isomorphic Models, Revised edition, Studies in Logic and the Foundations of Mathematics, Vol. 92, Amsterdam: North-Holland (1990).
S. Shelah, Cardinal Arithmetic, Oxford: OUP (1994).
E. Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Math. 9 (1950), 131–140.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this chapter
Cite this chapter
Kolman, O., Shelah, S. (1999). Almost disjoint pure subgroups of the Baer-Specker group. In: Eklof, P.C., Göbel, R. (eds) Abelian Groups and Modules. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7591-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-0348-7591-2_17
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-7593-6
Online ISBN: 978-3-0348-7591-2
eBook Packages: Springer Book Archive