Previous |  Up |  Next

# Article

Full entry | PDF   (0.2 MB)
Keywords:
quasi-balanced; almost balanced; Kravchenko classes
Summary:
An exact sequence $0\to A\to B\to C\to 0$ of torsion-free abelian groups is quasi-balanced if the induced sequence $$0\to \bold Q\otimes\operatorname{Hom}(X,A)\to\bold Q\otimes\operatorname{Hom}(X,B) \to\bold Q\otimes\operatorname{Hom}(X,C)\to 0$$ is exact for all rank-1 torsion-free abelian groups $X$. This paper sets forth the basic theory of quasi-balanced sequences, with particular attention given to the case in which $C$ is a Butler group. The special case where $B$ is almost completely decomposable gives rise to a descending chain of classes of Butler groups. This chain is a generalization of the chain of Kravchenko classes that arise from balanced sequences. As an application of our results concerning quasi-balanced sequences, the relationship between the two chains in the quasi-category of torsion-free abelian groups is illuminated.
References:
[A1] Arnold D.: Pure subgroups of finite rank completely decomposable groups. Abelian Group Theory Lecture Notes in Math. 874 Springer-Verlag New York (1982), 1-31. MR 0645913
[A2] Arnold D.: Finite Rank Torsion-Free Abelian Groups and Rings. Lecture Notes in Math. 931 Springer-Verlag New York (1982). MR 0665251 | Zbl 0493.20034
[AV] Arnold D., Vinsonhaler C.: Pure subgroups of finite rank completely decomposable groups $anII$. Abelian Group Theory Lecture Notes in Math. 1006 Springer-Verlag New York (1983), 97-143. MR 0722614
[B] Butler M.C.R.: A class of torsion-free abelian groups of finite rank. Proc. London Math. Soc. 15 (1965), 680-698. MR 0218446 | Zbl 0131.02501
[F] Fuchs L.: Infinite Abelian Groups. II Academic Press New York (1973). MR 0349869 | Zbl 0257.20035
[K] Kravchenko A.A.: Balanced and cobalanced Butler groups. Math. Notes Acad. Sci. USSR 45 (1989), 369-373. MR 1005459 | Zbl 0695.20032
[NV1] Nongxa L.G., Vinsonhaler C.: Balanced Butler groups. J. Algebra, to appear. MR 1378545 | Zbl 0846.20060
[NV2] Nongxa L.G., Vinsonhaler C.: Balanced representations of partially ordered sets. to appear.
[V] C. Vinsonhaler: A survey of balanced Butler groups and representations. Abelian Groups and Modules Lecture Notes in Pure and Applied Math. 182 Marcel Dekker (1996), 113-122. MR 1415625 | Zbl 0865.20040

Partner of