Unbounded Monotone Subgroups of the Baer–Specker Group

Abstract

We consider special subgroups of the Baer–Specker group $\mathbb{Z}^{\omega }$ of all integer valued functions on ω, which L. Fuchs called monotone groups (Fuchs, Infinite abelian groups. Academic, Boston, MA, 1973, Specker, Portugaliae Math 9:131–140, 1950). Together with R. Göbel the author defined an equivalence relation between monotone groups which corresponds to a behavior of homomorphisms from a monotone group into an abelian group (Göbel and Wald, Symp Math 23:201–239, 1979). The group $\mathbb{Z}^{\omega }$ and the subgroup B of all bounded functions form two equivalence classes with just a single member. A third class is build by all bounded monotone groups, which are monotone groups where the growth of all elements is bounded by the growth of some given function b. An unbounded monotone group different from $\mathbb{Z}^{\omega }$ can be constructed by an ultrafilter of ω. So the number of equivalence classes of monotone groups is at least 4. In Göbel and Wald (Math Z 172:107–121, 1980) it is proved that the number is $2^{2^{\aleph _{0}} }$ if the Continuum Hypothesis, CH, or alternatively Martin’s Axiom is assumed. Later A. Blass and C. Laflamme showed that it is relatively consistent with ZFC, that the number of equivalence classes is 4. In this case all unbounded monotone groups different from $\mathbb{Z}^{\omega }$ are equivalent (Blass and Laflamme, J Symb Log 54:54–56, 1989). Further investigations on monotone groups by O. Kolman and the author led to a special technical assumption on monotone groups (Kolman and Wald, Isr J Math 217, to appear). In the present paper we call these monotone groups comfortable and show that the existence of a monotone group that is not comfortable, is independent of ZFC.