On \(\kappa\)-complete reduced products
Keisler's theorem states that (under GCH) elementarily equivalent structures have isomorphic ultrapowers. The author generalizes (the proof of) this result as follows: (1) For any two structures \(\mathfrak A\), \(\mathfrak B\), let \(F_ \alpha({\mathfrak A},{\mathfrak B})\) be the corresponding Ehrenfeucht-Fraissé game of length \(\alpha\). If --- under suitable cardinal assumptions --- for each \(\alpha<\kappa\) player \(E\) has a winning strategy for \(F_ \alpha({\mathfrak A},{\mathfrak B})\) and \(F\) is a \(\kappa\)-semigood filter over \(\lambda\), \(F'\) a \(\kappa\)-descendingly incomplete filter over \(\xi\), \(\xi\leq\lambda\), then the ultrapowers (of \(\mathfrak A\) and \(\mathfrak B\) resp.) modulo any filter \(D\) over \(\xi\times\lambda\) are isomorphic, provided \(F'\times F\subseteq D\). (2) Filters \(F\) as required in (1) exist. (3) The filter \(D\) in (1) can be \(\kappa\)-complete. (4) There are applications to a generalized notion of universality (w.r.t. infinitary languages).
scientific article; zbMATH DE number 55778