具有纯嵌入的作用的抽象初等类
On the abstract elementary class of acts with pure embeddings
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中文总结 AI 辅助
该论文研究具有纯嵌入的作用的抽象初等类，用幺半群S的性质刻画其稳定性与超稳定性，在集合论假设下刻画稳定性谱，还应用得到纯内射作用的Baer型判据并给出相关结论的另一证明。
中文摘要 AI 辅助
我们研究具有纯嵌入的作用的抽象初等类。特别地，我们证明该类中的稳定性和超稳定性可分别通过幺半群S是LO（对任意s,t∈S，有s∈St或t∈Ss）和弱诺特（每个理想有限生成）来刻画。此外，在温和的集合论假设下，我们通过每个理想的生成集的最小基数来刻画稳定性谱。作为应用，当S是LO时，我们得到纯内射作用的类Baer型判据，并用其给出当S是LO时作用类具有足够多纯内射对象的另一证明。
英文摘要
We study the abstract elementary class of acts with pure embeddings. In particular, we show that stability and superstability in this class can be characterized in terms of the monoid $S$ being LO (for every $s,t \in S$, we have that $s \in St$ or $t \in Ss$) and weakly noetherian (every ideal is finitely generated), respectively. Moreover, under mild set-theoretic assumptions, we characterize the stability spectrum via the minimal cardinality of a generating set for every ideal. As an application, we obtain a Baer-like criterion for pure injective acts when $S$ is LO. We use this to provide an alternative proof that the class of acts has enough pure injectives when $S$ is LO.
