On automorphisms groups of structures of countable cofinality

TL;DR

This paper generalizes Gao's characterization of automorphism groups for countable models to models of uncountable cofinality, establishing equivalences involving models' Scott sentences, automorphism closures, and elementary embeddings.

Contribution

It extends Gao's theorem from countable models to all cardinals of cofinality , providing new equivalences involving automorphism groups and elementary embeddings.

Findings

Generalization of Gao's theorem to uncountable cofinality cardinals

Equivalence between models' Scott sentences and automorphism group properties

Existence of  automorphisms implying large automorphism groups

Abstract

In [2] Su Gao proves that the following are equivalent for a countable $M$ (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of $M$. (II) There exists some $j\in \overline{Aut(M)}\setminus Aut(M)$, where $\overline{Aut(M)}$ is the closure of $Aut(M)$ under the product topology in $\omega^\omega$. (III) There is an $L_{\omega_1,\omega}$- elementary embedding $j$ from $M$ to itself such that $range(j)\subset M$. We generalize his theorem to all cardinals $\kappa$ of of cofinality $\omega$ (cf. theorem 4.2). The following are equivalent: (I$^*$) There is a model of the Scott sentence of $M$ of size $\kappa^+$. (II$^*$) For all $\alpha<\beta<\kappa^+$, there exist functions $j_{\beta,\alpha}$ in $\overline{Aut(M)}^{T}\setminus Aut(M)$, such that for $\alpha< \beta<\gamma<\kappa^+$, \begin{equation}(*) j_{\gamma,\beta}\circ j_{\beta,\alpha}=j_{\gamma,\alpha},\end{equation} where $\overline{Aut(M)}^{T}$ is the closure of $Aut(M)$ under the product topology in $\kappa^\kappa$. (III$^*$) For every $\beta<\kappa^+$, there exist $L_{\infty,\kappa}^{fin}$- elementary embeddings (cf. definition 2.5) $(j_\alpha)_{\alpha<\beta}$ from $M$ to itself such that $\alpha_1<\alpha_2\Rightarrow range(j_{\alpha_1})\subset range(j_{\alpha_2})$. Theorem 4.2 holds both for countable and uncountable $\kappa$. Condition (*) in (II$^*$), which does not appear in the countable case, can not be removed when $\kappa$ is uncountable (cf. theorem 4.5). Condition (II$^*$) imply the existence of at least $\kappa^\omega$ automorphisms of $M$ (cf. corollary 4.6). It is unknown to the author whether a purely topological proof of corollary 4.6 exists.