arXiv math/05041992005-04-10math.LO

Ultrafilters and partial products of infinite cyclic groups

Andreas Blass, Saharon Shelah

发表信息：: Comm. Algebra 33 No. 6 (2005) 1997--2007

英文摘要：

We consider, for infinite cardinals kappa and alpha <= kappa^+, the group Pi(kappa,< alpha) of sequences of integers, of length kappa, with non-zero entries in fewer than alpha positions. Our main result tells when Pi(kappa,< alpha) can be embedded in Pi(lambda,< beta). The proof involves some set-theoretic results, one about familes of finite sets and one about families of ultrafilters.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Differentiating Infinite Voting Populations using Ultrafilters

In this short paper, we use the Rudin-Frolik order to shed light on the differing structures of invisible dictatorships given by Arrow-type social welfare functions over a countably infinite number of voters.

2016-07-29cs.GTmath.LO

The spectrum of ultraproducts of finite cardinals for an ultrafilter

We complete the characterization of the possible spectrum of regular ultrafilters D on a set I, where the spectrum is the set of infinite cardinals which are ultraproducts of finite cardinals modulo D.

2021-09-02math.LO

Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. IV

We extend to singular cardinals the model-theoretical relation $λ\stackrelκ{\Rightarrow} μ$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione Matematica Italiana ser. VII, {\bf 4-B} 875--903 (1990). We extend some results obtained in Part II, finding equivalent conditions involving uniformity of ultrafilters and the existence of certain infinite matrices. Our present definition suggests a new compactness property for abstract logics.

2008-05-11math.LO

Combinatorial and model-theoretical principles related to regularity of ultrafilters and compactness of topological spaces. I

We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.

2008-03-25math.GNmath.LO

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 Łoś-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-…

2013-01-27math.LOmath.RA

$κ$-勉强独立族与超滤子的 Tukey 类型

$κ$-barely independent families and Tukey types of ultrafilters

本文引入 $λ$ 上的 $κ$-勉强独立族，结合广义收割数给出其存在条件，并据此刻画一致超滤子的 Tukey 顶性质与极大特征。

2025-07-23math.LO

更多相关论文收起更多论文

Every strongly summable ultrafilter on $\bigoplus\mathbb Z_2$ is sparse

We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Steprāns and Strauss.

2013-03-25math.GNmath.LO

Fréchet cardinals

An infinite cardinal $λ$ is called Fréchet if the Fréchet filter on $λ$ extends to a countably complete ultrafilter. We investigate the relationship between Fréchet cardinals and strongly compact cardinals under a hypothesis called the Ultrapower Axiom.

2018-10-09math.LO

Topological Ramsey Spaces Dense in Forcings

Topological Ramsey spaces are spaces which support infinite dimensional Ramsey theory similarly to the Ellentuck space. Each topological Ramsey space is endowed with a partial ordering which can be modified to a $σ$-closed `almost reduction' relation analogously to the partial ordering of `mod finite' on $[ω]^ω$. Such forcings add new ultrafilters satisfying weak partition relations and have complete combinatorics. In cases where a forcing turned out to be equivalent to a topological Ramsey space, the strong Ramsey-theoretic techniques have aided in a fine-tuned analysis of the Rudin-Keisler a…

2018-05-21math.COmath.LO

Factoring a minimal ultrafilter into a thick part and a syndetic part

Let $S$ be an infinite discrete semigroup. The operation on $S$ extends uniquely to the Stone-Čech compactification $βS$ making $βS$ a compact right topological semigroup with $S$ contained in its topological center. As such, $βS$ has a smallest two sided ideal, $K(βS)$. An ultrafilter $p$ on $S$ is \emph{minimal} if and only if $p \in K(βS)$. We show that any minimal ultrafilter $p$ factors into a thick part and a syndetic part. That is, there exist filters $\mathcal F$ and $\mathcal G$ such that $\mathcal F$ consists only of thick sets, $\mathcal G$ consists only of syndetic sets, and $p$ is…

2018-05-18math.GNmath.LOmath.RA

Cardinal Characteristics and the Product of Countably Many Infinite Cyclic Groups

Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard unit vectors. Specifically, we relate the smallest possible size of such a subgroup to several of the standard cardinal characteristics of the continuum. We also study some related properties and cardinals, both group-theoretic and set-theoretic. One of the set-theoretic properties and the associated cardinal are combinatorially natural, independently of any …

1992-09-08math.LO

ZF + DC下有限集合超积的基数

Cardinalities of Ultraproducts of Finite Sets in ZF + DC

该研究在ZF + DC体系中证明，ω上特定超滤子U下有限集合的超积基数相等的充要条件，且该结论在Solovay模型的通用扩张中成立。

2026-09-03math.LO

SPM Bulletin 27

Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An in…

2009-04-22math.GNmath.LO

Homomorphisms on infinite direct products of groups, rings and monoids

We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ 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.

2014-10-02math.GRmath.LOmath.RA