On regular reduced products
Abstract
Assume <aleph0,aleph1>-><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 Nlambda/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, Mlambda/D is lambda++-universal. Our second result is as follows: For i<mu let Mi and Ni be elementarily equivalent models of a vocabulary which has has cardinality <=lambda. Suppose D is a regular filter on mu and <aleph0,aleph1>-><lambda,lambda+> holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length lambda+ on prodi Mi/D and prodi Ni/D. This yields the following corollary: Assume GCH and lambda regular). For L, Mi and Ni as above, if D is a regular filter on lambda, then prodi Mi/D cong prodi Ni/D .
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