不可数情况下的非交换Specker现象
The non-commutative Specker phenomenon in the uncountable case
摘要：
Higman提出并研究了自由积概念的无限形式。设G_i（i在I中）是群，ast_{i in X} G_i是G_i（i在X中）对X子集I的自由积，p_{XY}：ast_{i in Y} G_{i} -> ast_{i in X} G_{i}是X子集q Y子集I的标准同态.（X子集I表示X是I的有限子集。）那么，无限制自由积是逆极限lim（ast_{i in X} G_i，p_{XY}：X子集Y子集I）。我们记为ast_{i in emptyset} G_i= {e}。定理：设F是自由群。然后，对每个同态h：limast G_i-> F，存在可数完全超滤子u_0，. u_m在I上，使得h = h。p_{U_0 cup ... cup U_m}对于u_0，.中的每个U_0 um_m in um_m.若指标集I的基数小于最小可测基数，则存在I的有限子集X_0和同态上交{h}：ast_{i in X_0} G_i-> F使得h=上交{h} .其中p_{X_0}：lim ast G_i-> ast_{i in X_0} G_i是正则投影。
英文摘要：
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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