Abstract

We investigate the following model-theoretic independence relation: $b{|⌣}_A^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}c$ if and only if ${\mathrm{bdd}<!--FUNCTION\; APPLICATION-->}^{u<!--FUNCTION\; APPLICATION-->}(Ab)\cap {\mathrm{bdd}<!--FUNCTION\; APPLICATION-->}^{u<!--FUNCTION\; APPLICATION-->}(Ac)={\mathrm{bdd}<!--FUNCTION\; APPLICATION-->}^{u<!--FUNCTION\; APPLICATION-->}(A)$, where ${\mathrm{bdd}<!--FUNCTION\; APPLICATION-->}^{u<!--FUNCTION\; APPLICATION-->}(X)$ is the class of all ultraimaginaries bounded over $X$. In particular, we sharpen a result of Wagner to show that $b{|⌣}_A^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}c$ if and only if $⟨\mathrm{Autf}<!--FUNCTION\; APPLICATION-->(\mathbb{𝕄}∕Ab)\cup \mathrm{Autf}<!--FUNCTION\; APPLICATION-->(\mathbb{𝕄}∕Ac)⟩=\mathrm{Autf}<!--FUNCTION\; APPLICATION-->(\mathbb{𝕄}∕A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b^{\prime }{\equiv }_{A}b$ such that $b^{\prime }{|⌣}_A^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}c$). Extension then follows as an immediate corollary.

We also study total ${|⌣}^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}$-Morley sequences (i.e., $A$-indiscernible sequences $I$ satisfying $J{|⌣}_A^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}K$ for any $J$ and $K$ with $J+K{\equiv }_A^{\mathrm{EM}<!--FUNCTION\; APPLICATION-->}I$), and we prove that an $A$-indiscernible sequence $I$ is a total ${|⌣}^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}$-Morley sequence over $A$ if and only if whenever $I$ and $I^{\prime }$ have the same Lascar strong type over $A$, $I$ and $I^{\prime }$ 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 (1980) in his study of simple unstable theories.

Finally, we show that for any $A$ and $b$ in any theory $T$, if there is an Erdős cardinal $\kappa (\alpha )$ with $\left|Ab\right|+\left|T\right|<\kappa (\alpha )$, then there is a total ${|⌣}^{\mathrm{bu}<!--FUNCTION\; APPLICATION-->}$-Morley sequence ${(b_i)}_{i<\omega }$ over $A$ with $b_0=b$.

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