Notions of amalgamation for AECs and categoricity
Abstract
Motivated by the free products of groups, the direct sums of modules, and Shelah's $(\lambda,2)$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 $M_0\leq M_1, M_2$, and we show that if $K$ designates a unique strong amalgam to every triple $M_0\leq M_1, M_2$, then $K$ satisfies categoricity transfer at cardinals $\geq\theta(K)+2^{\text{LS}(K)}$, where $\theta(K)$ 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.13867
 arXiv:
 arXiv:2104.13867
 Bibcode:
 2021arXiv210413867C
 Keywords:

 Mathematics  Logic;
 03C48;
 03C45