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 κ-complete ultrafilters for a given uncountable cardinal κ. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Loś-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 Erdős-Kaplansky Theorem on dimensions of vector spaces DI (D a division ring) is extended to reduced products DI/F , and an application is noted. 1. Results on filters and ultrafilters The definition below recalls some standard concepts. Readers not familiar with some of these might skim those parts of the definition now, and return to them as one or another concept is called on. (For a thorough development of ultrafilters and related topics, see works such as [7] or [8].) Definition 1. If I is a set, then a filter on I means a set F of subsets of I, such that (i) I ∈ F , (ii) if J ∈ F and J ⊆ K ⊆ I, then K ∈ F , and (iii) if J, K ∈ F , then J ∩K ∈ F . A filter F on I is proper if it is not the set of all subsets of I, equivalently, if ∅ / ∈ F . A filter F on I is κ-complete, for κ an infinite cardinal, if F is closed under intersections of families of < κ elements. (Thus, every filter is א0-complete.) A filter which is א1-complete, i.e., closed under countable intersections, is called countably complete. A maximal proper filter on I is called an ultrafilter. An ultrafilter of the form {J ⊆ I | i0 ∈ J} for some i0 ∈ I is called principal; all other ultrafilters are called nonprincipal. An infinite cardinal κ is called measurable if there exists a nonprincipal κ-complete ultrafilter on κ. The use to which filters will be put in this note arises from the following observation. 2010 Mathematics Subject Classification. Primary: 03C20, 03E75, 17A01, Secondary: 03E55, 08B25, 16S60, 20A15.