Abstract In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

中文翻译：

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度量阿贝尔群中集合的凸性

摘要 在本文中，我们引入了一个新的凸性概念，它是由阿贝尔群的自同态族产生的。在配备平移不变度量的阿贝尔群中，我们定义了有界、范数、注入模数和自同态的谱半径。除了研究它们的性质之外，我们的第一个主要目标是扩展著名的 Rådström 取消定理。另一个结果推广了诺依曼可逆定理。接下来，我们相对于自同态族定义集合的凸性，并描述此类集合的集合理论和代数结构。给定一个子集，我们还考虑使该子集凸出的自同态族，并建立该族的基本属性。我们的第一个主要结果建立了暗示中点凸性的条件。下一个主要结果，使用我们对 Rådström 对消定理的扩展，展示了使子集凸出的自同态族的进一步结构特性。