Sh:769

• Kennedy, J. C., & Shelah, S. (2002). On regular reduced products. J. Symbolic Logic, 67(3), 1169–1177. arXiv: math/0105135 DOI: 10.2178/jsl/1190150156 MR: 1926605
• Abstract:
  Assume \langle\aleph_0,\aleph_1\rangle\rightarrow\langle \lambda,\lambda^+\rangle. Assume M is a model of a first order theory T of cardinality at most \lambda^+ in a vocabulary {\mathcal L}(T) of cardinality \leq\lambda. Let N be a model with the same vocabulary. Let \Delta be a set of first order formulas in {\mathcal L}(T) and let D be a regular filter on \lambda. Then M is \Delta-embeddable into the reduced power N^\lambda/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, M^\lambda/ D is \lambda^{++}-universal. Our second result is as follows: For i<\mu let M_i and N_i be elementarily equivalent models of a vocabulary which has has cardinality \le\lambda. Suppose D is a regular filter on \mu and \langle \aleph_0,\aleph_1\rangle\rightarrow\langle\lambda,\lambda^+ \rangle holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraisse game of length \lambda^+ on \prod_i M_i/D and \prod_i N_i/D. This yields the following corollary: Assume GCH and \lambda regular (or just \langle\aleph_0,\aleph_1 \rangle \rightarrow \langle \lambda,\lambda^+ \rangle and 2^\lambda=\lambda^+). For L, M_i and N_i as above, if D is a regular filter on \lambda, then \prod_i M_i/D\cong\prod_i N_i/D.
• Version 2001-05-08_11 (13p) published version (10p)

@article{Sh:769,
 author = {Kennedy, Juliette Cara and Shelah, Saharon},
 title = {{On regular reduced products}},
 journal = {J. Symbolic Logic},
 fjournal = {The Journal of Symbolic Logic},
 volume = {67},
 number = {3},
 year = {2002},
 pages = {1169--1177},
 issn = {0022-4812},
 mrnumber = {1926605},
 mrclass = {03C20 (03C55)},
 doi = {10.2178/jsl/1190150156},
 note = {\href{https://arxiv.org/abs/math/0105135}{arXiv: math/0105135}},
 arxiv_number = {math/0105135}
}