### More on regular reduced products

by Kennedy and Shelah. [KeSh:852]

J Symbolic Logic, 2004

The authors show, by means of a finitary version
square^{fin}_{lambda,D} of the combinatorial principle
square^{b^*}_{lambda}, the consistency of the failure, relative
to the consistency of supercompact cardinals, of the following: for
all regular filters D on a cardinal lambda, if M_i and N_i
are elementarily equivalent models of a language of size
<= lambda, then the second player has a winning strategy in the
Ehrenfeucht-Fra{i}sse game of length lambda^+ on prod_i
M_i/D and prod_i N_i/D . If in addition 2^{lambda}= lambda^+
and i< lambda implies |M_i|+|N_i| <= lambda^+ this means that
the ultrapowers are isomorphic.

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