Uniform Ultrafilters Research Articles

Ultrafilters generated by a closed set of functions

AbstractLet κ and λ be infinite cardinals, a filter on κ and a set of functions from κ to κ. The filter is generated by if consists of those subsets of κ which contain the range of some element of . The set is <λ-closed if it is closed in the <λ-topology on κκ. (In general, the <λ-topology on IA has basic open sets all such that, for all i ∈ I, Ui ⊆ A and ∣{i ∈ I: Ui ≠ A} ∣<λ.) The primary question considered in this paper asks “Is there a uniform ultrafilter on κ which is generated by a closed set of functions?” (Closed means <ω-closed.) We also establish the independence of two related questions. One is due to Carlson: “Does there exist a regular cardinal κ and a subtree T of <κκ such that the set of branches of T generates a uniform ultrafilter on κ?”; and the other is due to Pouzet: “For all regular cardinals κ, is it true that no uniform ultrafilter on κ is it true that no uniform ultrafilter on κ analytic?”We show that if κ is a singular, strong limit cardinal, then there is a uniform ultrafilter on κ which is generated by a closed set of increasing functions. Also, from the consistency of an almost huge cardinal, we get the consistency of CH + “There is a uniform ultrafilter on ℵ1 which is generated by a closed set of increasing functions”. In contrast with the above results, we show that if Κ is any cardinal, λ is a regular cardinal less than or equal to κ and ℙ is the forcing notion for adding at least (κ<λ)+ generic subsets of λ, then in VP, no uniform ultrafilter on κ is generated by a closed set of functions.

Baire number of the spaces of uniform ultrafilters

The Baire number is defined for a topological space without isolated points as the minimal size of the family of nowhere dense sets covering the space in question. We prove that in the case ofU(κ), the space of uniform ultrafilters over uncountable κ, the Baire number equals eitherω 1 orω 2, depending on the cofinality of κ. The results are connected to the collapsing of cardinals when using the quotient algebraP(κ) mod[κ]<κ as the notion of forcing.

Non-(ω, ω 1)-regular ultrafilters and perfect κ-normality of product spaces

In this paper we give a topological characterization of non-(ω, ω 1)-regular ultrafilters by means of perfect κ-normality of product spaces. For a countably incomplete uniform ultrafilter p on λ > ω, we prove that p is not (ω, ω 1)-regular iff Y p × D τ is perfectly κ-normal for each cardinal τ, where D = {0, 1} and Y p is the subspace consisting of the point p and all isolated points in β D(λ). As corollaries, (1) Y p × D ω 1 is not perfectly κ-normal for each uniform ultrafilter p on ω 1, if we assume V = L or MA ω 1 , (2) Woodin's model contains a uniform ultrafilter p on ω 1 such that Y p × D τ is perfectly κ-normal for each cardinal τ.

Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters

For every cardinal κ ⩾ ω, the following is consistent: each x in U( κ) (the space of uniform ultrafilters on κ) is a nonnormality point of U( κ), i.e., U( κ)⧹{ x} is not normal, because the space of nonuniform ultrafilters on cf(2 κ) embeds as a closed subspace in U( κ)⧹{ x}.

Regularity of ultrafilters and the core model

We show that in the core model every uniform ultrafilter is regular. In addition, we prove that the existence of a nonregular uniform ultrafilter on a singular cardinal implies the existence of an inner model with a measurable cardinal.

Ultrafilters, independent collections, and applications to topology

The following are consequences of the main results in this paper: 1. (1) The number of countably compact, completely regular spaces of density κ is 2 2 2 κ . 2. (2) There are 2 2 κ points in U(κ) (= space of uniform ultrafilters on κ), each of which has tightness 2 κ in U(κ) and is a limit point of a countable subset of U(κ). 3. (3) There are 2 2 κ points in U(κ), each of which has tightness 2 κ and is a weak P-point of κ ∗. 4. (4) For each λ ⩽ κ there are at least 2 2 λ · κ points in βκ, each of which has tightness 2 λ in β κ and is a weak P-point of κ ∗. Moreover, under GCH there are at least 2 2 λ · κ λ such points.

Partition properties of M-ultrafilters and ideals

As is well known, the following are equivalent for any uniform ultrafilter U on an uncountable cardinal:(i) U is selective;(ii) U → ;(iii) U → .In §1 of this paper, we consider this result in terms of M-ultrafilters (Definition 1.1), where M is a transitive model of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). We define the partition properties and for M-ultrafilters (Definition 1.3), and characterize those M-ultrafilters that possess these properties (Theorem 1.5) so that the result mentioned at the beginning is subsumed as the special case that M is V, the universe of all sets. It turns out that the two properties have to be handled separately, and that, besides selectivity, we need to formulate additional conditions (Definition 1.4). The extra conditions become superfluous when M = V because they are then trivially satisfied. One of them is nothing new; it is none other than Kunen's iterability-of-ultrapowers condition.In §2, we obtain characterizations of the partition properties I+ → and I+ → (Definition 2.3) of uniform ideals I on an infinite cardinal κ (Theorem 2.6). This is done by applying the main results of §1 to the canonical -ultrafilter in the Boolean-valued model constructed from the completion of the quotient algebra P(κ)/I. They are related to certain known characterizations of weakly compact and of Ramsey cardinals.Our basic set theory is ZFC. In §1, it has to be supplemented by a new unary predicate symbol M and new nonlogical axioms that make M look like a transitive model of ZFC.

The ideal structure of the space of $\kappa$-uniform ultrafilters on a discrete semigroup

The ideal structure of the space of $\kappa$-uniform ultrafilters on a discrete semigroup

Finest partitions for ultrafilters

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ−({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α &lt; κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω1 can be amplified to incorporate a finest partition.Henceforth, let us assume that all ultrafilters are uniform.There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If Ui is an ultrafilter over Iii for i = 1, 2, then U1 ≤RKU2 iff there is a projecting function Ψ:I2 → I1 such that U1 = Ψ*(U2) = {X ⊆, I1∣ Ψ−1(X) ϵ U2}· U1, =RKU2 iff U1, ≤RK and U2 and U2≤RKU1; and U1&lt;RKU2 iff U1≤RKU2 yet U1 ≠RKU2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ*(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g*(U) &lt;RKU, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g*(U) = h*(ƒ*(U)). Say now that an ultrafilter U over κ &gt; ω is indecomposable iff whenever ω &lt; λ &lt; κ, there is no V ≤RKU such that V is a (uniform) ultrafilter over λ.

Finest partitions for ultrafilters

If a uniform ultrafilter U over an uncountable cardinal κ is not outright countably complete, probably the next best thing is that it have a finest partition: a master function f:κ → ω with ƒ−({n}) ∉ U for each n ϵ ω such that for any g: κ → κ, either (a) it is one-to-one on a set in U, or (b) it factors through ƒ (mod U), i.e. for some function h, {α &lt; κ ∣ h(f(α)) = g(α)} ϵ U. In this paper, it is shown that recent contructions of irregular ultrafilters over ω1 can be amplified to incorporate a finest partition.Henceforth, let us assume that all ultrafilters are uniform.There has been an extensive study of substantial hypotheses, which are nonetheless weaker than countable completeness, on ultrafilters over uncountable cardinals. To survey some results and to establish a context, let us first recall the Rudin-Keisler (RK) ordering on ultrafilters: If Ui is an ultrafilter over Iii for i = 1, 2, then U1 ≤RKU2 iff there is a projecting function Ψ:I2 → I1 such that U1 = Ψ*(U2) = {X ⊆, I1∣ Ψ−1(X) ϵ U2}· U1, =RKU2 iff U1, ≤RK and U2 and U2≤RKU1; and U1&lt;RKU2 iff U1≤RKU2 yet U1 ≠RKU2. In terms of this ordering, if an ultrafilter U has a finest partition ƒ, then ƒ*(U) over ω is maximum amongst all RK predecessors of U: for any g:κ → κ, if g*(U) &lt;RKU, then g is not one-to-one on a set in U, so since g factors through ƒ with some h,g*(U) = h*(ƒ*(U)). Say now that an ultrafilter U over κ &gt; ω is indecomposable iff whenever ω &lt; λ &lt; κ, there is no V ≤RKU such that V is a (uniform) ultrafilter over λ.

Negative Partition Relations for Ultrafilters on Uncountable Cardinals

Assuming GCH, we prove that if $\kappa$ is a successor cardinal and $U$ is a uniform ultrafilter on $\kappa$, then $U \nrightarrow {(U,3)^2}$. The case $\kappa = {\omega _1}$ is an old result of Hajnal. Our proof makes use of several known results concerning nonregular, weakly normal and indecomposable ultrafilters, as well as some negative partition relations for uncountable ordinals.

Negative partition relations for ultrafilters on uncountable cardinals

Assuming GCH, we prove that if κ \kappa is a successor cardinal and U U is a uniform ultrafilter on κ \kappa , then U ↛ ( U , 3 ) 2 U \nrightarrow {(U,3)^2} . The case κ = ω 1 \kappa = {\omega _1} is an old result of Hajnal. Our proof makes use of several known results concerning nonregular, weakly normal and indecomposable ultrafilters, as well as some negative partition relations for uncountable ordinals.

Almost disjoint refinement of families of subsets of 𝑁

Without any set-theoretic assumptions, we prove that every uniform ultrafilter on the set N of all natural numbers has a Comfort system, that is, an almost disjoint refinement. Moreover, we describe one type of ideal such that the family of all subsets of N that are not contained in it has an almost disjoint refinement.

On the existence of nonregular ultrafilters and the cardinality of ultrapowers

Assuming the consistency of huge cardinals, we prove that ω 3 {\omega _3} can carry an ultrafilter D such that ω 1 ω 3 / D {\omega _1}^{{\omega _3}}/D has cardinality ω 3 {\omega _3} . (Hence D is not ( ω 3 , ω 1 ) ({\omega _3},\,{\omega _1}) regular.) Similarly ω 2 {\omega _2} can carry an ultrafilter D such that ω ω 2 / D {\omega ^{{\omega _2}}}/D has cardinality ω 2 {\omega _2} . (Hence D is not ( ω 2 , ω ) ({\omega _2},\,\omega ) regular.)

Some consequences of an infinite-exponent partition relation

Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).Our basic theorem will be the following: If k &gt; ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

Weakly Normal Filters and Irregular Ultrafilters

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal $\kappa$. (a) If $\kappa = {\lambda ^ + }$, then U is not $(\lambda ,{\lambda ^ + })$-regular iff U has a least function f such that $\{ \xi < {\lambda ^ + }|{\text {cf}}(f(\xi )) = \lambda \} \in U$. (b) If $\omega \leqslant \mu < \kappa$ and U is not $(\omega ,\mu )$-regular, then U has a least function.

Weakly normal filters and irregular ultrafilters

For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafi'lters, are then proved: Let U be a uniform ultrafilter over a regular cardinal K. (a) If K = X+, then U is not (X, A +)-regular iff U has a least function f such that {t < X+1 cf(f(0)) = X} E U. (b) If w S A < K and U is not (w, ,)-regular, then U has a least function. In this paper is considered a relatively new class of filters, those which satisfy a property abstracted from normal ultrafilters over a measurable cardinal. The first section discusses these filters in a general context, and the second shows their relevance to the study of the regularity of ultrafilters. The set theoretical notation and terminology is standard, in particular a, j, y, . . . are variables for ordinals while K, X, t, .... are reserved for cardinals. In fact, K will denote an arbitrary but fixed regular cardinal throughout the discussion. It is always assumed that a filter over K is proper and contains the sets {tl x < t < K} for every a < K, SO that ultrafilters are always uniform. This material forms part of the third chapter of the author's doctoral dissertation [41, but the results (except 2.5) were obtained some time ago in 1974. 1. Weakly normal filters. A series of easy definitions culminate in the main concept; recall that a set X has positive measure with respect to a filter F iff X meets every element of F. 1.1. DEFINITIONS. Let F be a filter over K. (i) f E KK is unbounded (mod F) iff{t < KIt <f(Q)} has positive measure for every a < K. (ii) f E KK is almost 1-1 iff for every a < K, If ({a})I < K. f E KK iS almost 1-1 (mod F) iff there is a set X of positive measure so that fIX is almost l-1,i.e. for every a< K, if-I({a}) fnXI < K. (iii) F is a p-point filter iff every function unbounded (mod F) is almost 1-1 (mod F). (iv) f E KK is a least function (mod F) iff f is unbounded (mod F) yet Received by the editors May 7, 1975 and, in revised form, June 2, 1975. AMS (MOS) subject classifications (1970). Primary 04A10, 02K35.

Regularity of ultrafilters

If there arek ++ eventually functions fromk + intok or if there arek ++ eventually different functions fromk + then uniform ultrafilters onk + are (k, k +)-regular.

Ultrafilters and almost disjoint sets

Let μκ denote the space of uniform ultrafilters on κ, and let λ be a cardinal. Uϵμκ is said to be a λ-point of μκ if U is a boundary point of λ pairwise disjoint open subsets of μκ. We prove that if κ is a successor cardinal, 2 κ= κ +, and Kurepa's hypothesis for κ holds, then each U ϵ μκ is a 2 κ-point of μκ.

A result concerning cardinalities of ultraproducts

The cardinality problem for ultraproducts is as follows: Given an ultrafilter over a set I and cardinals αi, i ∈ I, what is the cardinality of the ultraproduct ? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter over a set I of power κ (uniform means that all elements of have power κ). It is open whether every countably incomplete has the property that, for all infinite α, the ultra-power has power ακ. However, it is shown in [4] that certain countably incomplete , namely the κ-regular , have this property.This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the κ-good ultrafilters, do have this property. The κ-good ultrafilters are important in model theory because they are exactly the ultrafilters such that every ultraproduct modulo is κ-saturated (see [5]).Let be an ultrafilter on a set I. Let αi, n, i ∈ I, n ∈ ω, be cardinals and αi, n, ≥ αi, m if n &lt; m. Let.Then ρn are nonincreasing and therefore there is some m and ρ such that ρn = ρ if n ≥ m. We call ρ the eventual value (abbreviated ev val) of ρn.