Interpreting groups and fields in simple, finitary AECs

Categoricity transfer in simple finitary abstract elementary classes

We continue to study nitary abstract elementary classes, de ned in [7]. We introduce a concept of weak κ -categoricity and an f-primary model in a א0 -stable simple nitary AEC with the extension property, and gain the following theorem: Let (K,4K) be a simple nitary AEC, weakly categorical in some uncountable κ . Then (K,4K) is weakly categorical in each λ ≥ min{κ,i(2א0 )+} . We have that if th...

Lawlessness and Rank Restrictions in Certain Finitary Groups

We give two applications of the recent classification of locally finite simple finitary skew linear groups. We show that certain irreducible finitary skew linear groups of infinite dimension, generate the variety of all groups and have infinite Prüfer rank.

Lascar Types and Lascar Automorphisms in Abstract Elementary Classes

We study Lascar strong types and Galois types and especially their relation to notions of type which have finite character. We define a notion of a strong type with finite character, so called Lascar type. We show that this notion is stronger than Galois type over countable sets in simple and superstable finitary AECs. Furthermore we give and example where the Galois type itself does not have f...

Finitary Permutation Groups

A finitary permutation group is a natural generalization of a finite permutation group. The structure of a transitive finitary permutation group is surprisingly simple when its degree is infinite. Here we study primitivity, following P. M. Neumann’s work in the 1970s. We also study generalized solubility conditions on these groups. These notes arose from lectures aimed at an audience who had se...

Interpreting a Finitary Pi-calculus in Differential Interaction Nets