Almost-Free Finite Covers

Abstract

Let W be a first-order structure and ρ be an Aut(W)-congruence on W. In this article we define the almost-free finite covers of W with respect to ρ, and we show how to construct them. These are a generalization of free finite covers.

A consequence of a result of [5] 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 ρ 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 binding groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of Ω⁽ⁿ⁾, the set of ordered n-tuples of distinct elements from a countable set Ω, regarded as a structure with Aut(Ω⁽ⁿ⁾) = Sym(Ω) and we show a result about biinterpretability.

The material here presented addresses a problem which arises in the context of classification of totally categorical structures.

Key Words:

• Infinite permutation groups
• Model theory

2000 Mathematics Subject Classification:

• 20B27
• 20E22
• 03C30

ACKNOWLEDGMENTS

The author wishes to express her thanks to D. M. Evans for several stimulating conversations and hospitality at UEA. She thanks as well O. Puglisi for his helpful suggestions related to this article.

Notes

Communicated by D. Macpherson.