5 research outputs found
Homomorphisms on infinite direct products of groups, rings and monoids
We study properties of a group, abelian group, ring, or monoid which (a)
guarantee that every homomorphism from an infinite direct product
of objects of the same sort onto factors through the direct product of
finitely many ultraproducts of the (possibly after composition with the
natural map or some variant), and/or (b) guarantee that when a
map does so factor (and the index set has reasonable cardinality), the
ultrafilters involved must be principal.
A number of open questions, and topics for further investigation, are noted.Comment: 26 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. Version 2 has minor revisions in
wording etc. from version
Recommended from our members
Families of ultrafilters, and homomorphisms on infinite direct product algebras
Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many Îș-complete ultrafilters for a given uncountable cardinal Îș. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the ĆoĆâEda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the ErdËosâKaplansky theorem on dimensions of vector spaces DI (D a division ring) is extended to reduced products DI /F, and an application is noted
Families of ultrafilters, and homomorphisms on infinite direct product algebras
Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many Îș-complete ultrafilters for a given uncountable cardinal Îș. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the ĆoĆâEda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras. We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra. In a final section, the ErdËosâKaplansky theorem on dimensions of vector spaces DI (D a division ring) is extended to reduced products DI /F, and an application is noted
FAMILIES OF ULTRAFILTERS, AND HOMOMORPHISMS ON INFINITE DIRECT PRODUCT ALGEBRAS
Criteria are obtained for a filter F of subsets of a set I to be an
intersection of finitely many ultrafilters, respectively, finitely many
\kappa-complete ultrafilters for a given uncountable cardinal \kappa. From
these, general results are deduced concerning homomorphisms on infinite direct
product groups, which yield quick proofs of some results in the literature: the
{\L}o\'{s}-Eda theorem (characterizing homomorphisms from a
not-necessarily-countable direct product of modules to a slender module), and
some results of N. Nahlus and the author on homomorphisms on infinite direct
products of not-necessarily-associative k-algebras. The same tools allow other
results of Nahlus and the author to be nontrivially strengthened, and yield an
analog to one of their results, with nonabelian groups taking the place of
k-algebras.
We briefly examine the question of how the common technique used in applying
the general results of this note to k-algebras on the one hand, and to
nonabelian groups on the other, might be extended to more general varieties of
algebras in the sense of universal algebra.
In a final section, the Erd\H{o}s-Kaplansky Theorem on dimensions of vector
spaces D^I (D a division ring) is extended to reduced products D^I/F, and an
application is noted.Comment: 12 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv cop