Normal elementary maps (Q1267139)

Let \(T\) be a countable complete theory, \(M\) be a model of \(T\). A partial elementary map \(f\) on \(M\) is called normal if \(f\) extends to an elementary map on \(M\) whose domain or image just equals \(M\). In the paper under review, normal elementary maps are investigated when \(T\) is \(\omega\)-stable. It is proved that some ``special'' triples of models of \(T\) play a key role in this setting. In detail, a triple \((M_0, M_1, N)\) of models of \(T\) is called weakly special if, for \(i =0,1\), (a) \(N <M_i\) and \(N \not=M_i\), (b) there is \(a_i \in M_i- N\) such that no element in \(M_{1-i}\) satisfies the same type as \(a_i\) over \(N\). When the latter condition is replaced by the stronger assumption that for every \(a_0 \in M_0 -N\) and \(a_1 \in M_1 -N\), \(\text{tp} (a_0 /N) \not= \text{tp} (a_1/N)\), the triple is called almost special. It is shown that, for \(T\) \(\omega\)-stable, the following propositions are equivalent: 1. for every \(M \models T\), all elementary maps on \(M\) are normal, 2. \(T\) has no weakly special triple of models, 3. \(T\) has no almost special triple of models. It is also proved that, when \(T\) admits an almost special triple satisfying some further assumptions (\(N\) is \(\omega\)-saturated, \(M_0\) and \(M_1\) are isomorphic, \(N\) and \((M_0 -N) \cup (M_1 -N)\) are independent over some finite subset of \(N\)), there are \(M \models T\) and an elementary map \(f\) on \(M\) such that \(f\) is not normal, both \(M- \text{Dom }f\) and \(M- \text{Im }f\) are countable, but \(| \text{Dom }f|=| \text{Im }f|=\aleph_1\).