Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω₁]^<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.     

Filters and semigroup properties

The existence of non-fixed, almost translation invariant ultrafilters on any infinite semigroupS satisfying some algebraic properties is established using an ultrafilter approach. The structure of the Stone-?ech compactification of any discrete semigroup is investigated using filters and closed subsets ofßS.     

Nice ℵ1 generated non-P-points,Part I

We define a family of non-principal ultrafilters on $\mathbb{N}$ which are, in a sense, very far from P-points. We prove the existence of such ultrafilters under reasonable conditions. In subsequent articles, we intend to prove that such ultrafilters may exist while no P-point exists. Though our primary motivations came from forcing and independence results, the family of ultrafilters introduced here should be interesting from combinatorial point of view too.     

Between p-points and nowhere dense ultrafilters

We discuss the problem of existence and that of generic existence for various classes of ultrafilters onω. For example, we prove it is consistent that there are nowhere dense ultrafilters while there are no measure zero ultrafilters, and that there are measure zero ultrafilters while there are no P-points. We also prove that every filter base of size <c can be extended to a nowhere dense ultrafilter iff cof ( % MathType!End!2!1!)=c, and that every filter base of size <c can be extended to an ordinal ultrafilter iff ∂=c. Along the way we get a few new results on cardinal invariants of the continuum.     

Countable ultraproducts without CH

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω, using ultrafilters on a countable index set, which we also assume to be ω. Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors.     

P-points and Q-points over a measurable cardinal

This paper presents constructions of κ-ultrafilters over a measurable cardinal κ, specifically p-points and q-points, in the continuation of works of Gitik, Kanamori and Ketonen. On the assumption of supercompactness, Kunen has shown the existence of a chain of p-points, in the Rudin–Keisler ordering, of maximal length. We shall prove a similar result for q-points. Results of Mitchell [8,9] establish connections between Rudin—Keisler chains of κ-ultrafilters and inner models of “?ν(o(ν)= ν⁺⁺)”. This shows the necessity of some strong large cardinal hypothesis. The second part of the paper is devoted to a separation property of κ- ultrafilters (cf. Kanamori and Taylor). To answer a question of Taylor concerning the existence of a non-separating p-point, we use a combination of Silver's Forcing and iterated ultrapowers; the proof itself may be of some interest.     

Selective ultrafilters and homogeneity

We develop a game-theoretic approach to partition theorems, like those of Mathias, Taylor, and Louveau, involving ultrafilters. Using this approach, we extend these theorems to contexts involving several ultrafilters. We also develop an analog of Mathias forcing for such contexts and use it to show that the proposition (considered by Laver and Prikry) “every non-trivial c.c.c. forcing adjoins Cohen-generic reals or random reals” implies the non-existence of P-points. We show that, in the model obtained by Lévy collapsing to ω all cardinals below a Mahlo cardinal ;, any countably many selective ultrafilters are mutually generic over the Solovay (Lebesgue measure) submodel. Finally, we show that a certain natural group of self-homeomorphisms of βω-ω, chosen so as to preserve selectivity of ultrafilters, in fact preserves isomorphism types.     

On the maximal resolvability of monotonically normal spaces

We continue the work started in [6] and show that all monotonically normal (in short: MN) spaces are maximally resolvable if and only if all uniform ultrafilters are maximally decomposable. As a consequence we get that the existence of an MN space which is not maximally resolvable is equi-consistent with the existence of a measurable cardinal. We also show that it is consistent (modulo the consistency of a measurable cardinal) that there is an MN space X with |X| = Δ(X) = ? _ω which is not ω ₁-resolvable. It follows from the results of [6] that this is best possible.     

Open Ultrafilters and Separability with the Use of the Operation of Closure

We study ultrafilters of topologies as well as sets of ultrafilters that each time dominate the open neighborhood filter of some fixed point in a topological space. The sets of ultrafilters are considered as “enlarged points” of the original space. We study conditions that provide the distinguishability of (enlarged) “points” of this type. We use nontraditional separability axioms and study their connection with the known axioms T₀, T₁, and T₂.     

Countably compact topological group topologies on free Abelian groups from selective ultrafilters

Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size c² using a forcing model in which CH holds.Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MAcountable.In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom.     

Some constructions of ultrafilters over a measurable cardinal

Some non-normal κ-complete ultrafilters over a measurable κ with special properties are constructed. Questions by A. Kanamori [4] about infinite Rudin-Frolik sequences, discreteness and products are answered.     

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 ultraproducts of (finite) cardinals modulo D which are infinite.     

Adding ultrafilters by definable quotients

Forcing notions of the type $\mathcal{P}(\omega)/\mathcal{I}$ which do not add reals naturally add ultrafilters on ω. We investigate what classes of ultrafilters can be added in this way when $\mathcal{I}$ is a definable ideal. In particular, we show that if $\mathcal{I}$ is an F _σ P-ideal the generic ultrafilter will be a P-point without rapid RK-predecessors which is not a strong P-point. This provides an answer to long standing open questions of Canjar and Laflamme.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Ultrafilters on ω and atoms in the lattice of uniformities I

This is the first part of a paper devoted to atoms in the lattice of uniformities (on a given set). It is shown that there are two types of atoms: proximally discrete and proximally non-discrete. Both are related to ultrafilters. Proximally non-discrete atoms are completely described; they correspond to ultrafilters in a one-to-one way. This correspondence is used to classify ultrafilters on a countable set according as the corresponding atoms are, or are not, proximally fine.     

On rapid idempotent ultrafilters

This short note contains the proofs of two small but somewhat surprising results about ultrafilters on \(\mathbb {N}\) : (1) strongly summable ultrafilters are rapid, (2) every rapid ultrafilter induces a closed left ideal of rapid ultrafilters. As a consequence, there will be rapid minimal idempotents in all models of set theory with rapid ultrafilters. The history of this result has been published as an experiment in mathematical writing on the author’s website (Krautzberger, One Day in Colorado or Strongly Summable Ultrafilters are Rapid, 2012) and (Krautzberger, Rapid Idempotent Ultrafilters, 2012) where you can can also find additional remarks by Blass and Hindman, offering a form of peer-review.     

A dynamical systems approach to the Kadison-Singer problem

In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends uniquely to the von Neumann algebra of the group. We prove that if any state arising in the Kadison-Singer problem has a unique extension, then the injective envelope of a C*-crossed product algebra associated with the state necessarily contains the full von Neumann algebra of the group. We prove that this latter property holds for states arising from rare ultrafilters and δ-stable ultrafilters, independent, of the group action and also for states corresponding to non-recurrent points in the corona of the group.     

Triquotient maps via ultrafilter convergence

In this paper we characterize triquotient maps as those that are surjective on chains of convergent ultrafilters, extending the result known for triquotient maps between finite topological spaces.

     

Spectra of Tukey types of ultrafilters on Boolean algebras

Extending recent investigations on the structure of Tukey types of ultrafilters on \({\mathcal{P}(\omega)}\) to Boolean algebras in general, we classify the spectra of Tukey types of ultrafilters for several classes of Boolean algebras, including interval algebras, tree algebras, and pseudo-tree algebras.     

Ideals and free Pairs in the semigroup β ℤ

We prove that the equations ξ+x=mξ+y, x+ξ=y+mξ have no solutions in the semigroup β ? for every free ultrafilter ξ and every integer m∈0, 1. We study semigroups generated by the ultrafilters ξ, mξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ?.