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Preprint Number 903

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903. Monica M. VanDieren and Sebastien Vasey
Transferring symmetry downward and applications
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Submission date: 13 August 2015.

Abstract:

For K an abstract elementary class satisfying the amalgamation property, we prove a downward transfer of the symmetry property for splitting (previously isolated by the first author). This allows us to deduce uniqueness of limit models from categoricity in a cardinal of high-enough cofinality, improving on a 16-year-old result of Shelah:

Theorem
Suppose λ and μ are cardinals so that cf(λ)>μ ≥ LS(K) and assume that K has no maximal models and is categorical in λ. If M_0,M_1,M_2 in K_μ are such that both M_1 and M_2 are limit models over M_0, we have that M_1\cong_{M_0}M_2.

Another application of the symmetry transfer utilizes tameness (a locality property for types) and improves on the work of Will Boney and the second author:

Theorem
Let μ ≥ LS(K). If K is μ-superstable and μ-tame, then:
* If M_0, M_1, M_2 in K_μ are such that both M_1 and M_2 are limit models over M_0, then M_1 \cong_{M_0} M_2.
* For any λ > μ, the union of an increasing chain of λ-saturated models is λ-saturated.
* There exists a unique type-full good μ^+-frame with underlying class the saturated models in K_{μ^+}.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1508.03252: pdf, ps.


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