Categoricity and closure of amalgamation bases under unions

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Let K⁡\operatorname{\mathcal{K}} be an abstract elementary class, let Hanf⁡(K⁡)\operatorname{Hanf}(\operatorname{\mathcal{K}}) be its Hanf number, and let LS(K⁡)LS(\operatorname{\mathcal{K}}) be its Löwenheim–Skolem number. For a cardinal μ\mu, let K⁡μam\operatorname{\mathcal{K}}^{am}_\mu be the class of amalgamation bases in K⁡\operatorname{\mathcal{K}} of cardinality μ\mu. Closure conjecture for amalgamation bases. If there exists a λ≥Hanf⁡(K⁡)\lambda\geq\operatorname{Hanf}(\operatorname{\mathcal{K}}) such that K⁡\operatorname{\mathcal{K}} is categorical in λ\lambda, then K⁡μam\operatorname{\mathcal{K}}^{am}_\mu is closed under unions of length <μ+<\mu^+ for all μ\mu with LS(K⁡)≤μ<λLS(\operatorname{\mathcal{K}})\leq\mu<\lambda. This is presented as a weakening of Grossberg's Intermediate Categoricity Conjecture; the source does not give a resolution.

References

Primary source

Monica VanDieren, “Categoricity in Abstract Elementary Classes with No Maximal Models”, arXiv:math/0510579 (2005).

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