Criteria are obtained for a filter
${\cal F}$
of subsets of a set I to be an intersection
of finitely many ultrafilters, respectively, finitely many
κ-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 Łoś–Eda
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 Erdős–Kaplansky theorem on
dimensions of vector spaces
${D^I}$
$(D$
a division ring) is extended to reduced products
${D^I}/{\cal F},$
and an application is noted.