论文标题
LászlóFuchs问题的模型理论解决方案
A model theoretic solution to a problem of László Fuchs
论文作者
论文摘要
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 $U_λ$.在本文中，我们使用抽象小学理论中的思想来显示： $ \ textbf {theorem。} $让$ p $为质数。如果$λ^{\ aleph_0} =λ$或$ \forallμ<λ（μ^{\ aleph_0} <λ）$，则有一个通用的Abelian $ P $ - 基数纯度$λ$。此外，对于$ n \ geq 2 $，如果$ 2^{\ aleph_0} \ leq \ aleph_n $，则有一个通用的Abelian $ p $ -p $ -p $ group，用于基数$ \ aleph_n $。 由于抽象基础类别的理论几乎没有用于解决代数问题，因此从代数的角度介绍了这一理论的努力。
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.
