Simple homogeneous models

Abstract:

Geometrical stability theory is a powerful set of model-theoretic tools that can lead to structural results on models of a simple first-order theory. Typical results offer a characterization of the groups definable in a model of the theory. The work is carried out in a universal domain of the theory (a saturated model) in which the Stone space topology on ultrafilters of definable relations is compact. Here we operate in the more general setting of homogeneous models, which typically have noncompact Stone topologies. A structure $M$ equipped with a class of finitary relations $\mathcal {R}$ is strongly $\lambda$-homogeneous if orbits under automorphisms of $(M,\mathcal {R})$ have finite character in the following sense: Given $\alpha$ an ordinal $<\lambda \leq |M|$ and sequences $\bar {a}=\{ a_i:\:i<\alpha \}$, $\bar {b}=\{ b_i:\:i<\alpha \}$ from $M$, if $(a_{i_1},\dots ,a_{i_n})$ and $(b_{i_1},\dots ,b_{i_n})$ have the same orbit, for all $n$ and $i_1<\dots <i_n<\alpha$, then $f(\bar {a})=\bar {b}$ for some automorphism $f$ of $(M,\mathcal {R})$. In this paper strongly $\lambda$-homogeneous models $(M,\mathcal {R})$ in which the elements of $\mathcal {R}$ induce a symmetric and transitive notion of independence with bounded character are studied. This notion of independence, defined using a combinatorial condition called “dividing”, agrees with forking independence when $(M,\mathcal {R})$ is saturated. A concept central to the development of stability theory for saturated structures, namely parallelism, is also shown to be well-behaved in this setting. These results broaden the scope of the methods of geometrical stability theory.