### On regular reduced products

by Kennedy and Shelah. [KeSh:769]

J Symbolic Logic, 2002

Assume < aleph_0, aleph_1>-> <
lambda, lambda^+> . Assume M is a model of a first order
theory T of cardinality at most lambda^+ in a vocabulary
L (T) of cardinality <= lambda . Let N be a model
with the same vocabulary. Let Delta be a set of first order
formulas in 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
<= lambda . Suppose D is a regular filter on mu and <
aleph_0, aleph_1>-> < lambda, lambda^+ >
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 < aleph_0, aleph_1
>-> < lambda, lambda^+ > 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 .

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