On Regular Reduced Products

Assume 〈א0,א1〉 → 〈λ, λ +〉. Assume M is a model of a first order theory T of cardinality at most λ+ in a vocabulary L(T ) of cardinality ≤ λ. Let N be a model with the same vocabulary. Let ∆ be a set of first order formulas in L(T ) and let D be a regular filter on λ. Then M is ∆-embeddable into the reduced power N /D, provided that every ∆-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 λ, M/D is λ++-universal. Our second result is as follows: For i < μ letMi and Ni be elementarily equivalent models of a vocabulary which has has cardinality ≤ λ. Suppose D is a regular filter on μ and 〈א0,א1〉 → 〈λ, λ +〉 holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length λ+ on ∏ iMi/D and ∏ iNi/D. This yields the following corollary: Assume GCH and λ regular (or just 〈א0,א1〉 → 〈λ, λ +〉 and 2 = λ+). For L, Mi and Ni be as above, if D is a regular filter on λ, then ∏ iMi/D ∼= ∏ iNi/D. This paper was written while the authors were guests of the Mittag-Leffler Institute, Djursholm, Sweden. The authors are grateful to the Institute for its support. Research partially supported by grant 1011049 of the Academy of Finland. Research partially supported by the Binational Science Foundation. Publication number 769.