Almost-free finite covers

Let W be a first-order structure and ρ be an Aut(W )-congruence on W . In this paper 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 almostfree 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 bindings groups equal at any point to a simple non-abelian regular permutation group G. Then we analyze almost-free finite covers of Ω, the set of ordered n-tuples of distinct elements from a countable set Ω, regarded as a structure with Aut(Ω) = Sym(Ω) and we show a result of biinterpretability. The material here presented addresses a problem which arises in the context of classification of totally categorical structures.