arXiv math/01091762001-09-23math.LO

Ultrafilter spaces on the semilattice of partitions

Lorenz Halbeisen, Benedikt Loewe

发表信息：: Topology and its Applications 115(3) (2001) 317-332

英文摘要：

The Stone-Cech compactification of the natural numbers bN, or equivalently, the space of ultrafilters on the subsets of omega, is a well-studied space with interesting properties. If one replaces the subsets of omega by partitions of omega, one can define corresponding, non-homeomorphic spaces of partition ultrafilters. It will be shown that these spaces still have some of the nice properties of bN, even though none is homeomorphic to bN. Further, in a particular space, the minimal height of a tree pi-base and P-points are investigated.

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

Strongly productive ultrafilters on semigroups

We prove that if S is a commutative semigroup with well founded universal semilattice or a solvable inverse semigroup with well founded semilattice of idempotents, then every strongly productive ultrafilter on S is idempotent. Moreover we show that any very strongly productive ultrafilter on the free semigroup with countably many generators is sparse, answering a question of Hindman and Legette Jones.

2015-08-06math.GRmath.LO

Ramseyan ultrafilters

We investigate families of partitions of omega which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions. The combinatorial properties of these partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to those of Ramsey ultrafilters. For example it will be shown that dual Mathias forcing restricted to a Ramseyan ultrafilter has the same features as Mathias forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on the set of partition-filters and consider the dual form of some cardina…

2001-09-16math.LO

Partitions and conservativity

We study the partition properties enjoyed by the "next best thing to a P-point'' ultrafilters introduced recently in joint work with Dobrinen and Raghavan. That work established some finite-exponent partition relations, and we now analyze the connections between these relations for different exponents and the notion of conservativity introduced much earlier by Phillips. In addition, we establish some infinite-exponent partition relations for these ultrafilters and also for sums of non-isomorphic selective ultrafilters indexed by selective ultrafilters.

2015-10-16math.LO

A note on the Canonical Ramsey Theorem and Ramsey ultrafilters

We give a characterizations of Ramsey ultrafilters on $\mathscr P(ω)$ in terms of functions $f:ω^n\toω$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[ω]^n$ there is a finite partition~$\mathcal Q$ of $[ω]^{2n}$ such that any set $X\subseteq ω$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.

2022-03-24math.LO

Partition theorems from creatures and idempotent ultrafilters

We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.

2011-03-23math.COmath.LO

大基数上的Galvin性质及其在划分演算中的应用

Galvin's property at large cardinals and an application to partition calculus

本文研究大基数上不具有Galvin性质的κ-完全超滤子的一致性，证明基模型超滤子既可扩张为非Galvin超滤子，也可扩张为满足Galvin性质的P-点超滤子，并由此在划分演算问题λ→(λ,ω+1)^2中获得一致的新实例。

2022-09-25math.LO

更多相关论文收起更多论文

Ultrafilters, monads and combinatorial properties

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterizing their combinatorial properties by means of their monads. This extends to arbitrary sets methods previously used to study partition regular Diophantine equations on \N. Several applications are described by means of multiple examples.

2017-12-17math.LO

Ultrafilters on $G$-spaces

For a discrete group $G$ and a discrete $G$-space $X$, we identify the Stone-Čech compactifications $βG$ and $βX$ with the sets of all ultrafilters on $G$ and $X$, and apply the natural action of $βG$ on $βX$ to characterize large, thick, thin, sparse and scattered subsets of $X$. We use $G$-invariant partitions and colorings to define $G$-selective and $G$-Ramsey ultrafilters on $X$. We show that, in contrast to the set-theoretical case, these two classes of ultrafilters are distinct. We consider also universally thin ultrafilters on $ω$, the $T$-points, and study interrelations between these…

2015-07-01math.GNmath.LO

Ramsey degrees of ultrafilters, pseudointersection numbers, and the tools of topological Ramsey spaces

This paper investigates properties of $σ$-closed forcings which generate ultrafilters satisfying weak partition relations. The Ramsey degree of an ultrafilter $\mathcal{U}$ for $n$-tuples, denoted $t(\mathcal{U},n)$, is the smallest number $t$ such that given any $l\ge 2$ and coloring $c:[ω]^n\rightarrow l$, there is a member $X\in\mathcal{U}$ such that the restriction of $c$ to $[X]^n$ has no more than $t$ colors. Many well-known $σ$-closed forcings are known to generate ultrafilters with finite Ramsey degrees, but finding the precise degrees can sometimes prove elusive or quite involved, at …

2019-11-21math.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

A new class of Ramsey-classification theorems and their applications in the Tukey theory of ultrafilters

Motivated by Tukey classification problems and building on work in \cite{Dobrinen/Todorcevic11}, we develop a new hierarchy of topological Ramsey spaces $\mathcal{R}_α$, $α<ω_1$. These spaces form a natural hierarchy of complexity, $\mathcal{R}_0$ being the Ellentuck space, and for each $α<ω_1$, $\mathcal{R}_{α+1}$ coming immediately after $\mathcal{R}_α$ in complexity. Associated with each $\mathcal{R}_α$ is an ultrafilter $\mathcal{U}_α$, which is Ramsey for $\mathcal{R}_α$, and in particular, is a rapid p-point satisfying certain partition properties. We prove Ramsey-classification theorems…

2012-05-26math.COmath.GNmath.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

Narrow coverings of omega-product spaces

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset in a small set. For example, if the set (omega \times omega) is partitioned into two pieces along the diagonal, then one piece meets every horizontal line in a finite set, and the other piece meets each vertical line in a finite set. Such partitions or coverings can exist only when the sets forming the product are of limited size. This paper considers such …

1996-02-20math.LO

Set-Theoretical Problems in Asymptology

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $ω$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $ω$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.

2021-11-09math.GNmath.LO

Ramsey properties of nonlinear Diophantine equations

We prove general sufficient and necessary conditions for the partition regularity of Diophantine equations, which extend the classic Rado's Theorem by covering large classes of nonlinear equations. Sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters betaN, grounding on combinatorial properties of positive density sets and IP sets. Necessary conditions are proved by a new technique in nonstandard analysis, based on the use of the relation of u-equivalence for the hypernatural numbers *N.

2016-06-07math.COmath.LOmath.NT

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