Amalgamations preserving ℵ1-categoricity

1. F = Countably saturated strongly minimal structures with the DMP. In [3], Hrushovski showed that any two strongly minimal theories formulated in totally different languages have a common extension which is still strongly minimal and with the DMP. (DMP is the property that states that if a point b is sufficiently close to a, then Q(x, b) has the same rank and the same degree as Q(x, a).) His proof essentially shows that if Lo = 0 then any two countably saturated strongly minimal structures with the DMP have a strongly minimal amalgamation. Also he gave an example that shows the condition Lo = 0 is necessary 2. F = N -categorical countable structures. Let M1 be the structure (Q, +) and let M2 be the {E, F}-structure defined by: (i) E is an equivalence relation which divides the universe into two infinite classes A and B, (ii) F is a bijection between A and B. Clearly both structures are N l-categorical. But there is no amalgamation which is NI -categorical: Suppose that M were such an amalgamation. By (ii), the Morley degree of the structure is greater than one. So Q = MIL1 must have a subgroup G such that 1 1 then it has a subgroup of finite index. So the same example as in 2 works. I.e., there is a pair of co-stable structures whose amalgamation is always non-cw-stable. 4. F = No-categorical countable structures. In [6], Schmerl treated the case Lo = 0, and showed that if in M1 acl(A) = A holds for A c IM1 I then an cccategorical amalgamation of Ml and M2 exists. (In this paper IMI denotes the underlying set of a given structure M.)