论文标题
有界的超敏化独立性及其总莫利序列
Bounded ultraimaginary independence and its total Morley sequences
论文作者
论文摘要
我们研究以下模型理论独立关系：$ \ def \ indbu {{\ rlap {\ hspace11.9mu \ vert} \ lower7.5mu \ smile}^{\！\！\！\！ $ \ mathrm {bdd}^u（ab）\ cap \ mathrm {bdd}^u（ac）= \ mathrm {bdd}^u（a）$，其中$ \ mathrm {bdd}^u（x）^U（x）$是所有ultermaimaimaginaries of All ultraimaimaginaries by Bonded bounded bounded bonded by $ x $。特别是，我们提高了瓦格纳的结果，以表明$ b \ indbu_a \ hspace3mu c $ if，仅当$ \ langle \ langle \ mathrm {autf}（\ Mathbb {m}/ab）\ cup cup \ cup \ cup \ cup \ mathrm {autf}（autb}（autb}（autbb}（\ mathbb}）） \ mathrm {autf}（\ Mathbb {m}/a）$，我们建立了超级杂质参数的充分存在（即，对于任何一组Hyperimaginies $ a $ a $ a $ a $ a $ b $ b $ and $ b $和$ c $，有a $ b'\ equiv_ b'\ equiv_a b $ b $ b'$ b'\ indbe__ \ hsspace 3m.然后，扩展是直接推论的。 我们还研究总计$ \ hspace-5mu \ indbu $ -morley序列（即，$ a $ a $ a $ a-in-Indiscernible序列$ i $满足$ j \ indbu_a \ indbu_a \ hspace3mu k $ for $ j $ for AIN $ j $和$ j $ and $ j $ and $ k $ and $ j + k + k \ equiv equiv^equiv^{\ equiv^{ $ a $ - 印象的序列$ i $是总计$ \ hspace-5mu \ indbu $ -morley -morley序列$ a $时，并且仅当$ a $ a $时，并且只有当$ i $和$ i $ and $ i $'和$ i $和$ i'$具有相同的lascar type a $ a $，$ i $，$ i $和$ i'$与频繁的，符合性的关系相关的$ j $ j $ - 这也等同于Shelah在简单的不稳定理论的早期研究中定义的$ i $在$ a $上的“基于” $ a $。 最后，我们证明，如果有任何理论$ t $的任何$ a $和$ b $ <κ（α）$，那么总共有一个$ \ hspace-5mu \ indbu $ -morley序列$（b_i）_ {i <ω} $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ b_0 = b $。
We investigate the following model-theoretic independence relation: $\def\indbu{{\rlap{\hspace11.9mu\vert}\lower7.5mu\smile}^{\!\mathrm{bu}}} b \indbu_A\hspace3mu c$ if and only if $\mathrm{bdd}^u(Ab)\cap \mathrm{bdd}^u(Ac) = \mathrm{bdd}^u(A)$, where $\mathrm{bdd}^u(X)$ is the class of all ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner to show that $b \indbu_A\hspace3mu c$ if and only if $\langle \mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle = \mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b' \equiv_A b$ such that $b' \indbu_A\hspace3mu c$). Extension then follows as an immediate corollary. We also study total $\hspace-5mu\indbu$-Morley sequences (i.e., $A$-indiscernible sequences $I$ satisfying $J \indbu_A\hspace3mu K$ for any $J$ and $K$ with $J + K \equiv^{\mathrm{EM}}_A I$), and we prove that an $A$-indiscernible sequence $I$ is a total $\hspace-5mu\indbu$-Morley sequence over $A$ if and only if whenever $I$ and $I'$ have the same Lascar strong type over $A$, $I$ and $I'$ are related by the transitive, symmetric closure of the relation '$J+K$ is $A$-indiscernible.' This is also equivalent to $I$ being 'based on' $A$ in a sense defined by Shelah in his early study of simple unstable theories. Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an Erdös cardinal $κ(α)$ with $|Ab|+|T| < κ(α)$, then there is a total $\hspace-5mu\indbu$-Morley sequence $(b_i)_{i<ω}$ over $A$ with $b_0 = b$.
