Higher Amalgamation and Finite Covers

Higher amalgamation is a model theoretic property. It was also studied under the name generalised independence theorem. This property is defined in stable, or more generally simple or rosy theories. In this thesis we study how higher amalgamation behaves under expansion by finite covers and algebraic covers. We first show that finite and algebraic covers are mild expansions, in the sense that they preserve many model theoretic properties and behave well when imaginaries are added to them. Then we show that in pregeometric theories higher amalgamation over ; implies higher amalgamation over parameters. We also show that in general this is not true. In fact, for any stable theory with an algebraic closed set which is not a model we construct a finite cover which fails 4-Amalgamation. With some additional assumption we can also preserve higher amalgamation over the empty set. We apply this result to abelian groups and show that (Z/4Z)ω satisfies these assumptions. Then we take the opposite direction: rather then investigating covers which have malicious properties towards amalgamation, we construct covers which will make higher amalgamation become true. First we give a new proof for the fact that there exists an algebraic cover of any stable Teq acleq(Ø) with higher amalgamation over Ø. A proof sketch of this was given by Hrushovski and a full proof appeared in an unpublished work by D. Evans. The new proof uses the notion of symmetric witness which was introduced by Goodrick, Kim and Kolesnikov. We also show with a similar approach that there exists an algebraic cover of any stable, omega-categorical theory with higher amalgamation over parameters