On automorphisms groups of structures of countable cofinality
Ioannis Souldatos

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 (cf. theorem 1.2 too): (I)There is an uncountable model of the Scott sentence of . (II) There exists some , where is the closure of under the product topology in . (III) There is an - elementary embedding from to itself such that . We generalize his theorem to all cardinals of of cofinality (cf. theorem 4.2). The following are equivalent: (I) There is a model of the Scott sentence of of size . (II) For all , there exist functions in , such that for , \begin{equation}(*) j_{\gamma,\beta}\circ…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra · Advanced Topology and Set Theory
