Abstract In this paper, we will explain the reason that filter topological characterization of 2-divisible abelian lattice ordered groups cannot be easily translated into MV-algebras. Then we prove that a 2-divisible MV-algebra A is semisimple if and only if it is Hausdorff with respect to the filter topology induced by the lattice filter of the form F = ⋃ n ∈ N A a , n for any a ∈ A which exceeds zero. We also show that an MV-algebra is simple if and only if it is Hausdorff with respect to any filter topology. Furthermore, an MV-algebra A is semisimple if and only if for any lattice filter F, which satisfies F ∩ I = ∅ for any proper MV-algebra ideal I, A is Hausdorff with respect to F.

中文翻译：

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半简单MV-代数的拓扑表征

摘要 在本文中，我们将解释2-可分阿贝尔格有序群的滤波器拓扑特征不能轻易转化为MV-代数的原因。然后我们证明一个 2 可分的 MV-代数 A 是半简单的当且仅当它是 Hausdorff 相对于由形式 F = ⋃ n ∈ NA a 的格子滤波器引入的滤波器拓扑结构，n 对于任何 a ∈ A超过零。我们还表明，当且仅当它对于任何滤波器拓扑都是 Hausdorff 时，MV 代数是简单的。此外，MV-代数A是半简单的当且仅当对于任何格滤波器F，对于任何适当的MV-代数理想I满足F ∩ I = ∅，A是关于F的Hausdorff。