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Chains of strongly non-reflexive dual groups
of integer-valued continuous functions

Author: Haruto Ohta
Journal: Proc. Amer. Math. Soc. 124 (1996), 961-967
MSC (1991): Primary 54C40, 20K20, 20K45
MathSciNet review: 1327034
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Abstract: Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of $\omega _{i}$ consisting of ordinals of cofinality $\omega $ for each $1 < i < \omega $, then there exist abelian groups $A_{n} (n \in {\mathbb{Z}})$ such that $A_{n}^{\ast } \cong A_{n+1}$ and $A_{n} \not \cong A_{n+2}$ for each $n \in {\mathbb{Z}}$. (2) There exist abelian groups $A_{n} (n \in {\mathbb{Z}})$ such that $A_{n}^{\ast } \cong A_{n+1}$ for each $n \in {\mathbb{Z}}$ and $A_{n} \not \cong A_{n+2}$ for each $n < 0$. The groups $A_{n}$ are the groups of $\mathbb{Z}$-valued continuous functions on a topological space and their dual groups.

References [Enhancements On Off] (What's this?)

  • 1. K. Eda and H. Ohta, On abelian groups of integer-valued continuous functions, their $\mathbb{Z}$-duals and $\mathbb{Z}$-reflexivity, Abelian Group Theory (R. Göbel and E. Walker, eds.), Gordon and Breach, London, 1985, pp. 241--257. MR 90f:20081
  • 2. K. Eda, T. Kiyosawa and H. Ohta, $N$-compactness and its applications, Topics in General Topology (K. Morita and J. Nagata, eds.), North-Holland, Amsterdam, 1989, pp. 459--521. MR 95m:54018
  • 3. K. Eda, S. Kamo and H. Ohta, Abelian groups of continuous functions and their duals, Topology and its Appl. 53 (1993), 131--151. MR 94m:20108
  • 4. P. C. Eklof and A. H. Mekler, Almost Free Modules, Set-theoretic Methods, North-Holland, Amsterdam, 1990. MR 92e:20001
  • 5. P. C. Eklof, A. H. Mekler and S. Shelah, On strongly-non-reflexive groups, Israel J. Math. 59 (1987), 283--298. MR 89c:20080
  • 6. G. Schlitt, Sheaves of abelian groups and the quotients $A^{\ast \ast }/A$, J. Algebra 158 (1993), 50-60. MR 94e:20072

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Additional Information

Haruto Ohta
Affiliation: Faculty of Education, Shizuoka University, Ohya, Shizuoka, 422 Japan

Keywords: Abelian group, continuous function, dual group, reflexivity, strong non-reflexivity, $\mathbb{Z}$-compact
Received by editor(s): July 6, 1994
Additional Notes: Research partially supported by Grant-in-Aid for Scientific Research No. 06640125, Ministry of Education, Science and Culture.
Dedicated: Dedicated to Professor Shōzō Sasada on his $60$th birthday
Communicated by: Franklin D. Tall
Article copyright: © Copyright 1996 American Mathematical Society

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