We study generalized amalgamation properties in simple theories. We formulate a notion of generalized amalgamation in such a way so that the properties are preserved when we pass from T to Teq or Theq; we provide several equivalent ways of formulating the notion of generalized amalgamation.We define two distinct hierarchies of simple theories characterized by their amalgamation properties; examples are given to show the difference between the hierarchies.

In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n ⩾ 2, there exists a stable theory having -existence and k-uniqueness, for every k ⩽ n, but has neither -existence nor -uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3. © 2011 WILEY-VCH Verlag GmbH & Co. (...) KGaA, Weinheim. (shrink)

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the (...) failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split. (shrink)

Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.