On regular reduced products

by Kennedy and Shelah. [KeSh:769]
J Symbolic Logic, 2002
Assume < aleph_0, aleph_1>-> < lambda, lambda^+> . Assume M is a model of a first order theory T of cardinality at most lambda^+ in a vocabulary L (T) of cardinality <= lambda . Let N be a model with the same vocabulary. Let Delta be a set of first order formulas in L (T) and let D be a regular filter on lambda . Then M is Delta-embeddable into the reduced power N^lambda /D, provided that every Delta-existential formula true in M is true also in N . We obtain the following corollary: for M as above and D a regular ultrafilter over lambda, M^lambda / D is lambda^{++}-universal. Our second result is as follows: For i< mu let M_i and N_i be elementarily equivalent models of a vocabulary which has has cardinality <= lambda . Suppose D is a regular filter on mu and < aleph_0, aleph_1>-> < lambda, lambda^+ > holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda^+ on prod_i M_i/D and prod_i N_i/D . This yields the following corollary: Assume GCH and lambda regular (or just < aleph_0, aleph_1 >-> < lambda, lambda^+ > and 2^lambda = lambda^+). For L, M_i and N_i as above, if D is a regular filter on lambda, then prod_i M_i/D cong prod_i N_i/D .

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