Bounded ultraimaginary independence and its total Morley sequences

Abstract

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 $⟨\mathrm{Autf}(\mathbb{M}/Ab)\cup \mathrm{Autf}(\mathbb{M}/Ac)⟩=\mathrm{Autf}(\mathbb{M}/A)$, and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries $A$ and ultraimaginaries $b$ and $c$, there is a $b^{\prime }{\equiv }_{A}b$ 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 $J+K{\equiv }_A^{\mathrm{EM}}I$), 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 $I^{\prime }$ 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 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 $\kappa (\alpha )$ with $|Ab|+|T|<\kappa (\alpha )$, then there is a total $\hspace-5mu\indbu$-Morley sequence $(b_i{)}_{i<\omega }$ over $A$ with $b_0=b$.

22 pages

References in corpus (1)

• Tree indiscernibilities, revisited