On regular reduced products

Juliette Kennedy, Saharon Shelah

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

Assume 〈א0, א1〉 → 〈λ,λ+〉. Assume M is a model of a first order theory T of cardinality at most λ+ in a language ℒ (T) of cardinality ≤ λ. Let N be a model with the same language. Let δ be a set of first order formulas in ℒ(T) 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 ∏i Mi/D and ∏i Ni/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 ∏i Mi/D ≅ ∏i Ni/D.

Original languageEnglish
Pages (from-to)1169-1177
Number of pages9
JournalJournal of Symbolic Logic
Volume67
Issue number3
DOIs
StatePublished - 2002
Externally publishedYes

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