Bounded ultraimaginary independence and its total Morley sequences

TL;DR

This paper explores a specific model-theoretic independence relation involving ultraimaginaries, establishing its properties, existence of Morley sequences, and connections to Shelah's concepts, with implications for theories with large cardinals.

Contribution

It introduces and analyzes bounded ultraimaginary independence, proving full existence, characterizing total Morley sequences, and linking to Shelah's notion of being 'based on' in simple unstable theories.

Findings

Characterization of ultraimaginary independence in terms of automorphism groups

Existence of total Morley sequences over hyperimaginary parameters

Connection between Morley sequences and Shelah's 'based on' concept

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 $\langle \mathrm{Autf}(\mathbb{M}/Ab)\cup\mathrm{Autf}(\mathbb{M}/Ac) \rangle = \mathrm{Autf}(\mathbb{M}/A)$, 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 Erd\"os 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 $b_0 = b$.