Abstract This paper deals with the extension of Pontryagin reflexivity of the topological groups to the class of convergence groups. First, we give an example of a non-topological convergence on a group which is compatible with the algebraic structure of the group. Then we introduce the notion of local quasi-convexity for the class of convergence groups and present the results obtained while investigating the class of locally quasi-convex convergence abelian groups. We obtain that in contrast to the topological case, locally compact abelian convergence groups do not lie in the class of locally quasi-convex convergence groups. Finally, we prove that a compact Hausdorff convergence abelian group which is embedded is reflexive.

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局部拟凸收敛群

摘要 本文讨论了将拓扑群的庞特里亚金自反性扩展到收敛群类的问题。首先，我们给出一个与群的代数结构兼容的群上的非拓扑收敛的例子。然后我们引入了收敛群类的局部拟凸性的概念，并给出了在研究局部拟凸收敛阿贝尔群类时获得的结果。我们得到，与拓扑情况相反，局部紧阿贝尔收敛群不属于局部拟凸收敛群。最后，我们证明了嵌入的紧Hausdorff收敛阿贝尔群是自反的。