相对ω-稳定性、相对范畴性与内部覆盖
Relative stability, relative categoricity and internal covers
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中文总结 AI 辅助
研究相对范畴理论\((T,P)\)中\(T\)是\(T^P\)内部覆盖的情况，给出其结构理论，证明\(T\)相对ω-稳定当且仅当对偶绑定群\(H\)在可定义子群上有降链条件。
中文摘要 AI 辅助
我们研究相对范畴理论\((T,P)\)的一种特殊情况，即\(T\)是\(T^P\)的内部覆盖。我们给出了相对范畴内部覆盖的结构理论。在进入\(T^{eq}\)并从\(P\)部分命名一个参数后，它们恰好是\(T^P\)的“纯挠子覆盖”，通过为\(T^P\)中一个空可定义群\(G\)的主齐性空间或挠子添加一个新类\(S\)以及一个作用符号得到。我们证明，如[4]中所定义，\(T\)是相对ω-稳定的（或在\(P\)上是ω-稳定的）当且仅当对偶绑定群\(H\)在可定义子群上具有降链条件。
英文摘要
Let $T$ be a countable complete theory with a distinguished unary predicate $P$, and let $T^{P}$ be the theory of the $P$-parts of models of $T$ with the induced structure. $T$ is said to be relatively categorical or categorical over $P$ if any isomorphism between the $P$-parts of two models of $T$ lifts to an isomorphism of the models in question. We study the special case of relative categoricity where $T$ is internal to $T^{P}$ (that is, every model $M$ of $T$ is in the definable closure of $P(M)$ together with additional parameters from $M$). We first give a structure theory for such $T$: after passing to $T^{eq}$ and naming a parameter, $T$ is the same thing as a "pure torsor cover" of $T^{P}$, namely simply adjoining to $T^{P}$ a new sort for a torsor $S$ for a $\emptyset$-definable group $G$ in $T^{P}$, with no additional structure. We discuss relative stability, or stability over $P$, and give a characterization of relative stability, superstability, and $ω$-stability of $T$ in terms of $H$ having the stable chain condition, superstable chain condition, and $ω$-stable chain condition, respectively.
