### The non-commutative Specker phenomenon in the uncountable case

by Eda and Shelah. [EdSh:751]

J Algebra, 2002

An infinitary version of the notion of free products has been
introduced and investigated by G.Higman. Let G_i (for i in I) be
groups and ast_{i in X} G_i the free product of G_i (i in X)
for X Subset I and p_{XY}: ast_{i in Y} G_{i}->
ast_{i in X} G_{i} the canonical homomorphism for X subseteq Y
Subset I. (X Subset I denotes that X is a finite subset of
I .) Then, the unrestricted free product is the inverse limit
lim (ast_{i in X} G_i, p_{XY}: X subseteq Y Subset I).
We remark ast_{i in emptyset} G_i= {e} . We prove:
Theorem: Let F be a free group. Then, for each homomorphism h:
lim ast G_i-> F there exist countably complete ultrafilters
u_0, ...,u_m on I such that h = h . p_{U_0 cup ... cup
U_m} for every U_0 in u_0, ...,U_m in u_m .
If the cardinality of the index set I is less than the least
measurable cardinal, then there exists a finite subset X_0 of I
and a homomorphism overline {h}: ast_{i in X_0}G_i-> F such that
h= overline {h} . p_{X_0}, where p_{X_0}: lim ast G_i->
ast_{i in X_0}G_i is the canonical projection.

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