Abstract
Analogues of Nunke’s theorem are proved which characterize variants of slenderness. For a bounded monotone subgroup M of ℤω, a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G. It is consistent relative to ordinary set theory (ZFC) that if M ≠ ℤω is an unbounded monotone subgroup of ℤω, then a torsion-free reduced abelian group G is M-slender if, and only if, there is no monomorphism from M into G.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Bagaria and M. Magidor, Group radicals and strongly compact cardinals, Trans. Amer. Math. Soc. 366 (2014), 1857–1877.
G. M. Bergman, Families of ultrafilters, and homomorphisms on infinite direct product algebras, J. Symb. Log. 79 (2014), 223–239.
A. Blass and C. Laflamme, Consistency results about filters and the number of inequivalent growth types, J. Symbolic Logic 54 (1989), 50–56.
K. Eda, A Boolean power and a direct product of abelian groups, Tsukuba J. Math. 6 (1982), 187–193.
K. Eda, On a Boolean power of a torsion free abelian group, J. Algebra 82 (1983), 84–93.
P. C. Eklof and A. H. Mekler, Almost free modules, revised ed., North-Holland Mathematical Library, Vol. 65, North-Holland Publishing Co., Amsterdam, 2002, Set-theoretic methods.
L. Fuchs, Infinite abelian groups. Vol. II, Academic Press, New York-London, 1973, Pure and Applied Mathematics. Vol. 36-II.
R. Göbel and J. Trlifaj, Approximations and endomorphism algebras of modules, de Gruyter Expositions in Mathematics, Vol. 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2006.
R. Göbel and B. Wald, Wachstumstypen und schlanke Gruppen, in Symposia Mathematica, Vol. XXIII (Conf. Abelian Groups and their Relationship to the Theory of Modules, INDAM, Rome, 1977), Academic Press, London-New York, 1979, pp. 201–239.
R. Göbel and B. Wald, Martin’s axiom implies the existence of certain slender groups, Math. Z. 172 (1980), 107–121.
O. Kolman and S. Shelah, Almost disjoint pure subgroups of the Baer-Specker group, in Abelian groups and modules (Dublin, 1998), Trends Math., Birkhäuser, Basel, 1999, pp. 225–230.
M. Machura and B. Tsaban, The combinatorics of the Baer-Specker group, Israel J. Math. 168 (2008), 125–151.
G. Nöbeling, Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math. 6 (1968), 41–55.
R. J. Nunke, Slender groups, Acta Sci. Math. (Szeged) 23 (1962), 67–73.
E. Sąsiada, Proof that every countable and reduced torsion-free Abelian group is slender, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 7 (1959), 143–144 (unbound insert).
S. Shelah and O. Kolman, Infinitary axiomatizability of slender and cotorsion-free groups, Bull. Belg. Math. Soc. Simon Stevin 7 (2000), 623–629.
E. Specker, Additive Gruppen von Folgen ganzer Zahlen, Portugaliae Math. 9 (1950), 131–140.
B. Wald, Schlankheitsgrade kotorsionsfreier gruppen, doctoral dissertation, Universität Essen (1979).
Author information
Authors and Affiliations
Corresponding author
Additional information
In memoriam Professor Rüdiger Göbel
Oren Kolman died in 2015 after completing the article.
Rights and permissions
About this article
Cite this article
Kolman, O., Wald, B. M-slenderness. Isr. J. Math. 217, 303–312 (2017). https://doi.org/10.1007/s11856-017-1447-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-017-1447-5