一个LászlóFuchs问题的模型论解法
A model theoretic solution to a problem of László Fuchs
摘要：
[Fuc 15]第181页中的问题5.1要求找到基数$λ$，使得存在用于基数纯度$λ$的通用阿贝尔$p$ -群，即，交换$p$ -基数$λ$的群$U_λ$，使得每个交换$p$ -基数$\leq λ$的群纯嵌入$U_λ$中。在本文中，我们使用的想法从理论的抽象初等类显示：$\textbf{Theorem.}$让$p$是一个素数。如果是$λ^{\aleph_0}=λ$或$\forall μ< λ( μ^{\aleph_0} < λ)$，则有一个泛阿贝尔$p$ -群，用于基数纯度$λ$。此外，对于$n\geq 2$，有一个通用的交换$p$ -群，用于基数$\aleph_n$的纯度当且仅当$2^{\aleph_0} \leq \aleph_n$。由于抽象初等类的理论几乎没有被用来解决代数问题，因此人们努力从代数的角度介绍这一理论。
英文摘要：
Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $λ$ such that there is a universal abelian $p$-group for purity of cardinality $λ$, i.e., an abelian $p$-group $U_λ$ of cardinality $λ$ such that every abelian $p$-group of cardinality $\leq λ$ purely embeds in $U_λ$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $λ^{\aleph_0}=λ$ or $\forall μ< λ( μ^{\aleph_0} < λ)$, then there is a universal abelian $p$-group for purity of cardinality $λ$. Moreover for $n\geq 2$, there is a universal abelian $p$-group for purity of cardinality $\aleph_n$ if and only if $2^{\aleph_0} \leq \aleph_n$. As the theory of abstract elementary classes has barely been used to tackle algebraic questions, an effort was made to introduce this theory from an algebraic perspective.
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