TY - JOUR
T1 - On regular reduced products
AU - Kennedy, Juliette
AU - Shelah, Saharon
PY - 2002
Y1 - 2002
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0036744462&partnerID=8YFLogxK
U2 - 10.2178/jsl/1190150156
DO - 10.2178/jsl/1190150156
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AN - SCOPUS:0036744462
SN - 0022-4812
VL - 67
SP - 1169
EP - 1177
JO - Journal of Symbolic Logic
JF - Journal of Symbolic Logic
IS - 3
ER -