Submission date: 28 April 2021

Abstract:

Motivated by the free products of groups, the direct sums of modules, and Shelah's (λ,2)-goodness, we study strong amalgamation properties in Abstract Elementary Classes. Such a notion of amalgamation consists of a selection of certain amalgams for every triple M_0 ≤ M_1, M_2, and we show that if K designates a unique strong amalgam to every triple M_0 ≤ M_1, M_2, then K satisfies categoricity transfer at cardinals ≥ θ(K)+2^{LS(K)}, where θ(K) is a cardinal associated with the notion of amalgamation. We also show that if such a unique choice does not exist, then there is some model M in K having 2^{|M|} many extensions which cannot be embedded in each other over M. Thus, for AECs which admit a notion of amalgamation, the property of having unique amalgams is a dichotomy property in the sense of Shelah's classification theory.

Mathematics Subject Classification: 03C48, 03C45

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Full text arXiv 2104.13867: pdf, ps.