LF groups, aec amalgamation, few automorphisms

TL;DR

This paper explores amalgamation bases, definability, indecomposability, and universality in locally finite groups, establishing conditions for extensions, automorphisms, and canonical models across various cardinalities.

Contribution

It introduces new criteria for amalgamation bases, analyzes automorphism groups, and constructs universal and canonical models in locally finite groups.

Findings

Many models are amalgamation bases under certain conditions.

Existence of universal models in strong limit cardinals with cofinality .

Every locally finite group of size  can be extended to a full group with desired properties.

Abstract

In S. 1 we deal with amalgamation bases, e.g., we define when an a.e.c. $k$ has $(\lambda,\kappa)$-amalgamation which means "many" M in $K^k_\lambda$ are amalgamation bases. We then consider what happens for the class of lf groups. In S. 2 we deal with weak definability of $a \in N\setminus M$ over $M$, for $ K_{exlf}$. In S. 3 we deal with indecomposable members of $K_{exlf}$ and with the existence of universal members of $K^k_\mu$, for $\mu$ strong limit of cofinality $\aleph_0$. Most noteworthy: if $K_{lf}$ has a universal model in $\mu$ then it has a canonical one similar to the special models, (the parallel to saturated ones in this cardinality). In S. 4 we prove "every $G \in K^{lf}_<\lambda$ can be extended to a complete $(\lambda,\theta)$-full G" for many cardinals. In a continuation we may consider "all the cardinals" or at least "almost all the cardinals"; also, we may consider a priori fixing the outer automorphism group.