The non-commutative Specker phenomenon in the uncountable case
Abstract
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.
Keywords
Cite
@article{arxiv.math/0011231,
title = {The non-commutative Specker phenomenon in the uncountable case},
author = {Katsuya Eda and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0011231},
year = {2007}
}