Almost-free finite covers

TL;DR

This paper introduces the concept of almost-free finite covers, generalizing free finite covers, and explores their construction, properties, and applications in structures with rich automorphism groups, including symmetric groups.

Contribution

It defines almost-free finite covers relative to an automorphism congruence and characterizes their kernels, extending the theory of finite covers in model theory.

Findings

Almost-free finite covers generalize free finite covers.

Finite covers with simple non-abelian binding groups are almost-free.

Biinterpretability results for covers of ordered n-tuples from countable sets.

Abstract

Finite covers are a technique for building new structures from simpler ones. The original motivation to study finite covers is in the Ladder theorem of Zilber which describes how totally categorical structures are built from strictly minimal sets by a sequence of covers. Let W be a first-order structure and r be an Aut(W)-congruence on W. In this paper we define the almost-free finite covers of W with respect to r, and we show how to construct them. These are a generalization of free finite covers. A consequence of a result of Evans and Hrushovski in the paper "On the automorphism groups of finite covers" is that any finite cover of W with binding groups all equal to a simple non-abelian permutation group is almost-free with respect to some r on W. Our main result gives a description (up to isomorphism) in terms of the Aut(W)-congruences on W of the kernels of principal finite covers of W with bindings groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of the set of ordered n-tuples of distinct elements from a countable set Omega, regarded as a structure with automorphism group equal to the Sym(Omega) and we show a result of biinterpretability.