Existence of many \(L_{\infty,\lambda}\)-equivalent, non-isomorphic models of T of power \(\lambda\) (Q1092040)

Let L, \(L_ 1\) be languages, \(L\subseteq L_ 1\), T, \(T_ 1\) be first order theories in L, \(L_ 1\) respectively, \(T\subseteq T_ 1\), T is complete in L and (a) T is not superstable or (b) \(T=T_ 1\), T has the dop or (c) \(T=T_ 1\), T is countable and T has the otop. For \(\lambda >| T_ 1|\) a regular cardinal or a strong limit cardinal of uncountable cofinality, the author constructs \(2^{\lambda}\) non- isomorphic, pairwise \(L_{\infty,\lambda}\)-equivalent models of T of power \(\lambda\) which are reducts of models of \(T_ 1\). The theorem is obtained by using generalized EM-models, and the proof applies even to appropriate non-elementary classes, too. For countable \(T_ 1=T\) it is the best possible result: for countable superstable T without dop or otop and \(\lambda >2^{\omega}\), any two \(L_{\infty,\lambda}\)-equivalent models of T of power \(\lambda\) are isomorphic. A corollary is that there are \(2^{\lambda}\) \(L_{\infty,\lambda}\)-equivalent non-isomorphic separable abelian p-groups of power \(\lambda\).