论文标题
晶格有序的阿贝尔群体的维度和模块的模型理论在prüfer域上
Dimensions on Lattice Ordered Abelian Groups and Model Theory of Modules over Prüfer Domains
论文作者
论文摘要
我们证明了一个转移定理，当与Jaffard-Kaplansky-Ohm定理结合使用时，可以通过其价值组将模型的模型理论转换为Prüfer域上的结果。扩展了Puninski和Toffalori的工作，我们表明，Prüfer域的价值组的扩展正锥具有M维时，并且仅当它的PP-$ 1 $ -Formulae晶格具有宽度（等效宽度）并且这些维度相等。此外，我们表明，这些维度的存在等同于具有M-二极二的PP- $ 1 $形式的晶格（因此，具有Cantor-Bendixson等级的Ziegler Spectrum和Superdecosable-Bendixson等级的Ziegler Spectrum。最后，我们在PP-$ 1 $ 1 $ lattice的m维度上为Prüfer域的m维数提供了最佳限制，从其价值组的扩展正锥的m维数表示，并表明所有未列出的级别+λ$ 1 $ for $λ$ for $λ$ for A limitiminal的序列的限制均为PP-$ 1 $ 1 $ 1 $ 1 $ 1-lattice的m-级别的限制。
We prove a transfer theorem which, when combined with the Jaffard-Kaplansky-Ohm Theorem, allows results in model theory of modules over Bézout domains to be translated into results over Prüfer domains via their value groups. Extending work of Puninski and Toffalori, we show that the extended positive cone of the value group of a Prüfer domain has m-dimension if and only if its lattice of pp-$1$-formulae has breadth (equivalently width) and that these dimensions are equal. Further, we show that the existence of these dimensions is equivalent to the lattice of pp-$1$-formulae having m-dimension (and hence to its Ziegler spectrum having Cantor-Bendixson rank) and the non-existence of superdecomposable pure-injective modules. Finally, we give a best possible upper bound for the m-dimension of the pp-$1$-lattice of a Prüfer domain in terms of the m-dimension of the extended positive cone of its value group and show that all ordinals which are not of the form $λ+1$ for $λ$ a limit ordinal occur as the m-dimension of the pp-$1$-lattice of a Bézout domain.
