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On regular reduced products*

Published online by Cambridge University Press:  12 March 2014

Juliette Kennedy
Department of Mathematics, University of Helsinki, Helsinki, Finland, E-mail:
Saharon Shelah
Institute of Mathematics, Hebrew University, Jerusalem, Israel, E-mail:


Assume (ℵ0, ℵ1) → (λ, λ+). Assume M is a model of a first order theory T of cardinality at most λ+ in a language of cardinality ≤ λ. Let N be a model with the same language. Let Δ be a set of first order formulas in and let D be a regular filter on λ. Then M is Δ-embeddable into the reduced power Nλ/D, provided that every Δ-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, Mλ/D is λ++-universal. Our second result is as follows: For i < μ let Mi, and Ni, be elementarily equivalent models of a language which has cardinality ≤ λ. Suppose D is a regular filter on λ and (ℵ0, ℵ1) → (λ, λ+) holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ+ on ΠiMi/D and ΠiNi/D. This yields the following corollary: Assume GCH and λ regular (or just (ℵ0, ℵ1) → (λ, λ+) and 2λ = λ+. For L, Mi and Ni be as above, if D is a regular filter on λ, then ΠiMi/D ≅ ΠiNi/D.

Research Article
Copyright © Association for Symbolic Logic 2002

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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.



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