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Journal articles on the topic 'Ultrafilters'

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Published: 4 June 2021

Last updated: 25 April 2022

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1

Benedikt, Michael. "Ultrafilters which extend measures." Journal of Symbolic Logic 63, no. 2 (June 1998): 638–62. http://dx.doi.org/10.2307/2586856.

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AbstractWe study classes of ultrafilters on ω defined by a natural property of the Loeb measure in the Nonstandard Universe corresponding to the ultrafilter. This class, the Property M ultrafilters, is shown to contain all ultrafilters built up by taking iterated products over collections of pairwise nonisomorphic selective ultrafilters. Results on Property M ultrafilters are applied to the construction of extensions of probability measures, and to the study of measurable reductions between ultrafilters.

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2

Baumgartner, James E. "Ultrafilters on ω." Journal of Symbolic Logic 60, no. 2 (June 1995): 624–39. http://dx.doi.org/10.2307/2275854.

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AbstractWe study the I-ultrafilters on ω, where I is a a collection of subsets of a set X, usually ℝ or ω1. The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.

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3

BLASS, ANDREAS, NATASHA DOBRINEN, and DILIP RAGHAVAN. "THE NEXT BEST THING TO A P-POINT." Journal of Symbolic Logic 80, no. 3 (July 22, 2015): 866–900. http://dx.doi.org/10.1017/jsl.2015.31.

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AbstractWe study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.

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4

Barney, Christopher. "Ultrafilters on the natural numbers." Journal of Symbolic Logic 68, no. 3 (September 2003): 764–84. http://dx.doi.org/10.2178/jsl/1058448437.

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AbstractWe study the problem of existence and generic existence of ultrafilters on ω. We prove a conjecture of Jörg Brendle's showing that there is an ultrafilter that is countably closed but is not an ordinal ultrafilter under CH. We also show that Canjar's previous partial characterization of the generic existence of Q-points is the best that can be done. More simply put, there is no normal cardinal invariant equality that fully characterizes the generic existence of Q-points. We then sharpen results on generic existence with the introduction of σ-compact ultrafilters. We show that the generic existence of said ultrafilters is equivalent to . This result, taken along with our result that there exists a Kσ, non-countably closed ultrafilter under CH, expands the size of the class of ultrafilters that were known to fit this description before. From the core of the proof, we get a new result on the cardinal invariants of the continuum, i.e., the cofinality of the sets with σ-compact closure is .

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5

Fremlin, D. H., and P. J. Nyikos. "Saturating ultrafilters on N." Journal of Symbolic Logic 54, no. 3 (September 1989): 708–18. http://dx.doi.org/10.2307/2274735.

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AbstractWe discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω, c)-saturating.

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6

Kanamori, Akihiro. "Finest partitions for ultrafilters." Journal of Symbolic Logic 51, no. 2 (June 1986): 327–32. http://dx.doi.org/10.1017/s0022481200031182.

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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 < 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<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) <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 > ω is indecomposable iff whenever ω < λ < κ, there is no V ≤RKU such that V is a (uniform) ultrafilter over λ.

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7

Palumbo, Justin. "Comparisons of Polychromatic and Monochromatic Ramsey Theory." Journal of Symbolic Logic 78, no. 3 (September 2013): 951–68. http://dx.doi.org/10.2178/jsl.7803130.

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AbstractWe compare the strength of polychromatic and monochromatic Ramsey theory in several set-theoretic domains. We show that the rainbow Ramsey theorem does not follow from ZF, nor does the rainbow Ramsey theorem imply Ramsey's theorem over ZF. Extending the classical result of Erdős and Rado we show that the axiom of choice precludes the natural infinite exponent partition relations for polychromatic Ramsey theory. We introduce rainbow Ramsey ultrafilters, a polychromatic analogue of the usual Ramsey ultrafilters. We investigate the relationship of rainbow Ramsey ultrafilters with various special classes of ultrafilters, showing for example that every rainbow Ramsey ultrafilter is nowhere dense but rainbow Ramsey ultrafilters need not be rapid. This entails comparison of the polychromatic and monochromatic Ramsey theorems as combinatorial principles on ω.

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8

FERNÁNDEZ-BRETÓN, DAVID JOSÉ. "STABLE ORDERED UNION ULTRAFILTERS AND cov." Journal of Symbolic Logic 84, no. 3 (April 3, 2019): 1176–93. http://dx.doi.org/10.1017/jsl.2019.20.

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AbstractA union ultrafilter is an ultrafilter over the finite subsets of ω that has a base of sets of the form ${\text{FU}}\left( X \right)$, where X is an infinite pairwise disjoint family and ${\text{FU}}(X) = \left\{ {\bigcup {F|F} \in [X]^{ < \omega } \setminus \{ \emptyset \} } \right\}$. The existence of these ultrafilters is not provable from the $ZFC$ axioms, but is known to follow from the assumption that ${\text{cov}}\left( \mathcal{M} \right) = \mathfrak{c}$. In this article we obtain various models of $ZFC$ that satisfy the existence of union ultrafilters while at the same time ${\text{cov}}\left( \mathcal{M} \right) = \mathfrak{c}$.

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9

Baldwin, Stewart. "The ⊲-ordering on normal ultrafilters." Journal of Symbolic Logic 50, no. 4 (December 1985): 936–52. http://dx.doi.org/10.2307/2273982.

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If κ is a measurable cardinal, then it is a well-known fact that there is at least one normal ultrafilter over κ. In [K-1], Kunen showed that one cannot say more without further assumptions, for if U is a normal ultrafilter over κ, then L[U] is an inner model of ZFC in which κ has exactly one normal measure. On the other hand, Kunen and Paris showed [K-P] that if κ is measurable in the ground model, then there is a forcing extension in which κ has normal ultrafilters, so it is consistent that κ has the maximum possible number of normal ultrafilters. Starting with assumptions stronger than measurability, Mitchell [Mi-1] filled in the gap by constructing models of ZFC + GCH satisfying “there are exactly λ normal ultrafilters over κ”, where λ could be κ+ or κ++ (measured in the model), or anything ≤ κ. Whether or not Mitchell's results can be obtained by starting only with a measurable cardinal in the ground model and defining a forcing extension is unknown.There are substantial differences between the Mitchell models and the Kunen-Paris models. In the Kunen-Paris models κ can be the only measurable cardinal. However, in the Mitchell model in which κ has exactly 2 normal ultrafilters, one of them contains the set {α < κ: α is measurable} while the other does not. Thus it is natural to ask if it is possible to get a model M of ZFC in which κ is the only measurable cardinal and κ has exactly 2 normal ultrafilters. In this paper we will show that, using appropriate large cardinal assumptions, the answer is yes.

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10

Barbanel, Julius B. "On the relationship between the partition property and the weak partition property for normal ultrafilters on Pκλ." Journal of Symbolic Logic 58, no. 1 (March 1993): 119–27. http://dx.doi.org/10.2307/2275328.

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AbstractSuppose κ is a supercompact cardinal and λ > κ. We study the relationship between the partition properly and the weak partition properly for normal ultrafilters on Pκλ. On the one hand, we show that the following statement is consistent, given an appropriate large cardinal assumption: The partition property and the weak partition properly are equivalent, there are many normal ultrafilters that satisfy these properties, and there are many normal ultrafilters that do not satisfy these properties. On the other hand, we consider the assumption that, for some λ > κ, there exists a normal ultrafilter U on Pκλ such that U satisfies the weak partition property but does not satisfy the partition property. We show that this assumption is implied by the assertion that there exists a cardinal γ > κ such that γ is γ+-supercompact, and, assuming the GCH, it implies the assertion that there exists a cardinal γ > κ such that γ is a measurable cardinal with a normal ultrafilter concentrating on measurable cardinals.

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11

Chentsov, A. G. "Some properties of ultrafilters of widely understood measurable spaces." Доклады Академии наук 486, no. 1 (May 10, 2019): 24–29. http://dx.doi.org/10.31857/s0869-5652486124-29.

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Filters and ultrafilters (maximal filters) on the π-system with “zero” and “unit” are considered (here, π-system is a nonempty family of sets closed with respect to finite intersections); so, our π-system contains including and empty sets. Characteristic properties of ultrafilters obtained from special representations for bases of two typical topologies connected with construction of Wallman extension and Stone compactums are investigated. The topology of Wallman type on the ultrafilters set of arbitrary π-system with “zero” and “unit” is defined. In addition, initial set is transformed in a compact T1-space with points in the form of ultrafilters of above-mentioned π-system. Under equipment of the resulting ultrafilter set with two topologies (by sense, Stone and Wallman topologies), bitopological space with comparable topologies is obtained; for this space, the degeneracy (in the sense of coincidence for above-mentioned topologies) and nondegeneracy conditions are indicated. The initial set is immersed in above-mentioned bitopological space as everywhere dense subset. Resulting construction is oriented on application in extensions of abstract attainability problems with constraints of asymptotic character (we keep in mind the possible application of ultrafilters as generalized elements).

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12

Henle, J. M. "Concerning ultrafilters on ultrapowers." Journal of Symbolic Logic 52, no. 1 (March 1987): 149–51. http://dx.doi.org/10.2307/2273868.

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This paper concerns ultrafilters on a cardinal γ extending the filter of λ-closed, unbounded sets, λ < γ. The history of these ultrafilters is closely connected with that of the axiom of determinacy (AD). Solovay noticed first that, under AD, there was such an ultrafilter for γ = ℵ1; λ = ω. Later, Kleinberg found that the existence of such ultrafilters followed from the partition relation γ → (γ)λ+λ. Specific instances of this and more powerful relations on cardinals were then proved from AD by Martin, Kunen, Paris, and others. The axiom of determinacy was recently shown consistent with ZF relative to something less than a supercompact cardinal by Martin and Steel. Solovay's and Kleinberg's results were actually stronger, and we discuss this at the end of the paper. Good references for these results include [K2] and [KM].We are interested here in the case where γ is the ultrapower of a strong partition cardinal κ (a cardinal satisfying for all α < κ). Such cardinals exist in great abundance assuming AD, and in fact, if sufficiently many cardinals are strong, then AD holds in L[R] [KKMW].

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13

DOBRINEN, NATASHA. "HIGH DIMENSIONAL ELLENTUCK SPACES AND INITIAL CHAINS IN THE TUKEY STRUCTURE OF NON-P-POINTS." Journal of Symbolic Logic 81, no. 1 (March 2016): 237–63. http://dx.doi.org/10.1017/jsl.2015.10.

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AbstractThe generic ultrafilter${\cal G}_2 $forced by${\cal P}\left( {\omega \times \omega } \right)/\left( {{\rm{Fin}} \otimes {\rm{Fin}}} \right)$was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters ([1]), but it was left open where exactly in the Tukey order it lies. We prove${\cal G}_2 $that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter${\cal G}_k $forced by${\cal P}\left( {\omega ^k } \right)/{\rm{Fin}}^{ \otimes k} $forms a chain of lengthk. Essential to the proof is the extraction of a dense subsetεkfrom (Fin⊗k)+which we prove to be a topological Ramsey space. The spacesεk,k≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εkare proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below${\cal G}_k $.

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14

MALLIARIS, M., and S. SHELAH. "MODEL-THEORETIC PROPERTIES OF ULTRAFILTERS BUILT BY INDEPENDENT FAMILIES OF FUNCTIONS." Journal of Symbolic Logic 79, no. 01 (March 2014): 103–34. http://dx.doi.org/10.1017/jsl.2013.28.

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Abstract Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of ${\cal P}\left( \lambda \right)/{\cal D}$ have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter ${\cal D}$ on λ so that any ultrafilter extending ${\cal D}$ fails to ${\lambda ^ + }$ -saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on $\lambda > \kappa$ which is λ-flexible but not ${\kappa ^{ + + }}$ -good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that ${\rm{lcf}}\left( {{\aleph _0}} \right)$ may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.

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15

Takahashi, Joji. "Partition properties of M-ultrafilters and ideals." Journal of Symbolic Logic 52, no. 4 (December 1987): 897–907. http://dx.doi.org/10.2307/2273824.

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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.

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16

MARKS, ANDREW. "The universality of polynomial time Turing equivalence." Mathematical Structures in Computer Science 28, no. 3 (July 13, 2016): 448–56. http://dx.doi.org/10.1017/s0960129516000232.

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We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

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17

Laflamme, Claude, and Jian-Ping Zhu. "The Rudin-Blass ordering of ultrafilters." Journal of Symbolic Logic 63, no. 2 (June 1998): 584–92. http://dx.doi.org/10.2307/2586852.

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AbstractWe discuss the finite-to-one Rudin-Keisler ordering of ultrafilters on the natural numbers, which we baptize the Rudin-Blass ordering in honour of Professor Andreas Blass who worked extensively in the area.We develop and summarize many of its properties in relation to its bounding and dominating numbers, directedness, and provide applications to continuum theory. In particular, we prove in ZFC alone that there exists an ultrafilter with no Q-point below in the Rudin-Blass ordering.

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18

Ronco, C., G. Cappelli, M. Ballestri, E. Lusvarghi, P. Frisone, M. Milan, R. Dell'Aquila, et al. "On line filtration of dialysate: structural and functional features of an asymmetric polysulfone hollow fiber ultrafilter (Diaclean®)." International Journal of Artificial Organs 17, no. 10 (October 1994): 515–20. http://dx.doi.org/10.1177/039139889401701002.

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The endotoxin transfer across dialysis membranes has been investigated using specific in vitro circuits. Backdiffusion and backfiltration have been analyzed and most dialysis membranes have shown to be permeable to LAL positive substances. Synthetic membranes however display the better capacity of retention of these products despite their higher porosity and permeability. For such reason synthetic polysulfone ultrafilters are used as pyrogen filters to obtain ultrapure dialysate. We have investigated the characteristics of a polysulfone ultrafilter named Diaclean and manufactured by Amicon Ireland. The capacity of endotoxin retention has been investigated both in filtration and backfiltration modes on new and used ultrafilters. The capacity of endotoxin adsorption was investigated as well. Used ultrafilters appeared to maintain the retention capacity and the adsorption capacity up to 4 months of use. Only slight differences were noted from the baseline values (p = n.s.). The best adsorption capacity is always displayed by the outer layer of the membrane suggesting its best utilization in back filtration mode with tangential flow. No morphological changes were observed in the used membrane analyzed by scanning electron microscopy.

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19

Komjáth, Péter, and Vilmos Totik. "Ultrafilters." American Mathematical Monthly 115, no. 1 (January 2008): 33–44. http://dx.doi.org/10.1080/00029890.2008.11920493.

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20

KREUZER, ALEXANDER P. "ON IDEMPOTENT ULTRAFILTERS IN HIGHER-ORDER REVERSE MATHEMATICS." Journal of Symbolic Logic 80, no. 1 (March 2015): 179–93. http://dx.doi.org/10.1017/jsl.2014.58.

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AbstractWe analyze the strength of the existence of idempotent ultrafilters in higher-order reverse mathematics.Let $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ be the statement that an idempotent ultrafilter on ℕ exists. We show that over $ACA_0^\omega$, the higher-order extension of ACA0, the statement $\left( {{{\cal U}_{{\rm{idem}}}}} \right)$ implies the iterated Hindman’s theorem (IHT) and we show that $ACA_0^\omega + \left( {{{\cal U}_{{\rm{idem}}}}} \right)$ is ${\rm{\Pi }}_2^1$-conservative over $ACA_0^\omega + IHT$ and thus over $ACA_0^ +$.

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21

Blass, Andreas, and Neil Hindman. "On strongly summable ultrafilters and union ultrafilters." Transactions of the American Mathematical Society 304, no. 1 (January 1, 1987): 83. http://dx.doi.org/10.1090/s0002-9947-1987-0906807-4.

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22

Herrlich, Horst, Paul Howard, and Kyriakos Keremedis. "On extensions of countable filterbases to ultrafilters and ultrafilter compactness." Quaestiones Mathematicae 41, no. 2 (September 19, 2017): 213–25. http://dx.doi.org/10.2989/16073606.2017.1376229.

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23

Mouadi, Hassan, and Driss Karim. "Some topology on zero-dimensional subrings of product of rings." Filomat 34, no. 14 (2020): 4589–95. http://dx.doi.org/10.2298/fil2014589m.

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Let R be a ring and {Ri}i?I a family of zero-dimensional rings. We define the Zariski topology on Z(R,?Ri) and study their basic properties. Moreover, we define a topology on Z(R,?Ri) by using ultrafilters; it is called the ultrafilter topology and we demonstrate that this topology is finer than the Zariski topology. We show that the ultrafilter limit point of a collections of subrings of Z(R,?Ri) is a zero-dimensional ring. Its relationship with F-lim and the direct limit of a family of rings are studied.

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24

Halbeisen, Lorenz. "Ramseyan ultrafilters." Fundamenta Mathematicae 169, no. 3 (2001): 233–48. http://dx.doi.org/10.4064/fm169-3-3.

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Fernández-Bretón, David J., and Michael Hrušák. "Gruff ultrafilters." Topology and its Applications 210 (September 2016): 355–65. http://dx.doi.org/10.1016/j.topol.2016.08.012.

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26

Bergman, George M., and Ehud Hrushovski. "Linear ultrafilters." Communications in Algebra 26, no. 12 (January 1998): 4079–113. http://dx.doi.org/10.1080/00927879808826396.

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Petrenko, O., and I. V. Protasov. "Thin Ultrafilters." Notre Dame Journal of Formal Logic 53, no. 1 (2012): 79–88. http://dx.doi.org/10.1215/00294527-1626536.

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Di Nasso, Mauro, and Marco Forti. "Hausdorff ultrafilters." Proceedings of the American Mathematical Society 134, no. 6 (January 4, 2006): 1809–18. http://dx.doi.org/10.1090/s0002-9939-06-08433-4.

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29

Gitik, Moti. "Strange ultrafilters." Archive for Mathematical Logic 58, no. 1-2 (March 1, 2018): 35–52. http://dx.doi.org/10.1007/s00153-018-0620-9.

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30

Barbanel, Julius B. "Supercompact cardinals, trees of normal ultrafilters, and the partition property." Journal of Symbolic Logic 51, no. 3 (September 1986): 701–8. http://dx.doi.org/10.2307/2274023.

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AbstractSuppose κ is a supercompact cardinal. It is known that for every λ ≥ κ, many normal ultrafilters on Pκ(λ) have the partition property. It is also known that certain large cardinal assumptions imply the existence of normal ultrafilters without the partition property. In [1], we introduced the tree T of normal ultrafilters associated with κ. We investigate the distribution throughout T of normal ultrafilters with and normal ultrafilters without the partition property.

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31

Machura, Michał, and Andrzej Starosolski. "Thin ultrafilters and the P-hierarchy of ultrafilters." Topology and its Applications 281 (August 2020): 107205. http://dx.doi.org/10.1016/j.topol.2020.107205.

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32

Tembrowski, Bronis?aw. "Q-ultrafilters and normal ultrafilters in B-algebras." Studia Logica 45, no. 2 (June 1986): 167–79. http://dx.doi.org/10.1007/bf00373272.

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Malliaris, M. E. "Hypergraph sequences as a tool for saturation of ultrapowers." Journal of Symbolic Logic 77, no. 1 (March 2012): 195–223. http://dx.doi.org/10.2178/jsl/1327068699.

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AbstractLet T1, T2 be countable first-order theories, Mi ⊨ Ti and any regular ultrafilter on λ ≥ ℵ0. A longstanding open problem of Keisler asks when T2 is more complex than T1, as measured by the fact that for any such λ, , if the ultrapower realizes all types over sets of size ≤ λ, then so must the ultrapower . In this paper, building on the author's prior work [12] [13] [14], we show that the relative complexity of first-order theories in Keisler's sense is reflected in the relative graph-theoretic complexity of sequences of hypergraphs associated to formulas of the theory. After reviewing prior work on Keisler's order, we present the new construction in the context of ultrapowers, give various applications to the open question of the unstable classification, and investigate the interaction between theories and regularizing sets. We show that there is a minimum unstable theory, a minimum TP2 theory, and that maximality is implied by the density of certain graph edges (between components arising from Szemerédi-regular decompositions) remaining bounded away from 0, 1. We also introduce and discuss flexible ultrafilters, a relevant class of regular ultrafilters which reflect the sensitivity of certain unstable (non low) theories to the sizes of regularizing sets, and prove that any ultrafilter which saturates the minimal TP2 theory is flexible.

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Bešlagić, Amer, and Eric K. van Douwen. "Spaces of nonuniform ultrafilters in spaces of uniform ultrafilters." Topology and its Applications 35, no. 2-3 (June 1990): 253–60. http://dx.doi.org/10.1016/0166-8641(90)90110-n.

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35

MONTALBÁN, ANTONIO, and RICHARD A. SHORE. "CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC." Journal of Symbolic Logic 83, no. 2 (June 2018): 740–65. http://dx.doi.org/10.1017/jsl.2017.76.

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AbstractWe extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example, Hindman’s theorem and its generalizations such as the Galvin--Glazer theorem and iterated versions of these theorems (IHT and IGG). We prove that adding these axioms to IHT produce conservative extensions of ACA0+IHT,${\rm{ACA}}_{\rm{0}}^ +$, ATR0,${\rm{\Pi }}_2^1$-CA0, and${\rm{\Pi }}_2^1$-CA0for all sentences of second order arithmetic and for full Z2for the class of${\rm{\Pi }}_4^1$sentences. We also generalize and strengthen a metamathematical result of Wang (1984) to show, for example, that any${\rm{\Pi }}_2^1$theorem ∀X∃YΘ(X,Y) provable in ACA0or${\rm{ACA}}_{\rm{0}}^ +$there aree,k∈ ℕ such that ACA0or${\rm{ACA}}_{\rm{0}}^ +$proves that ∀X(Θ(X, Φe(J(k)(X))) where Φeis theeth Turing reduction andJ(k)is thekth iterate of the Turing or Arithmetic jump, respectively. (A similar result is derived for${\rm{\Pi }}_3^1$theorems of${\rm{\Pi }}_1^1$-CA0and the hyperjump.)

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36

Goldberg, Gabriel. "The linearity of the Mitchell order." Journal of Mathematical Logic 18, no. 01 (June 2018): 1850005. http://dx.doi.org/10.1142/s0219061318500058.

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We show from an abstract comparison principle (the Ultrapower Axiom) that the Mitchell order is linear on sufficiently strong ultrafilters: normal ultrafilters, Dodd solid ultrafilters, and assuming GCH, generalized normal ultrafilters. This gives a conditional answer to the well-known question of whether a [Formula: see text]-supercompact cardinal [Formula: see text] must carry more than one normal measure of order 0. Conditioned on a very plausible iteration hypothesis, the answer is no, since the Ultrapower Axiom holds in the canonical inner models at the finite levels of supercompactness.

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37

Barbanel, Julius B. "A note on a result of Kunen and Pelletier." Journal of Symbolic Logic 57, no. 2 (June 1992): 461–65. http://dx.doi.org/10.2307/2275281.

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AbstractSuppose that U and U′ are normal ultrafilters associated with some supercompact cardinal. How may we compare U and U′? In what ways are they similar, and in what ways are they different? Partial answers are given in [1], [2], [3], [5], [6], and [7]. In this paper, we continue this study.In [6], Menas introduced a combinatorial principle χ(U) of normal ultrafilters U associated with supercompact cardinals, and showed that normal ultrafilters satisfying this property also satisfy a partition property. In [5], Kunen and Pelletier showed that this partition property for U does not imply χ(U). Using results from [3], we present a different method of finding such normal ultrafilters which satisfy the partition property but do not satisfy χ(U). Our method yields a large collection of such normal ultrafilters.

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38

Huberich, Markus. "Non-regular ultrafilters." Israel Journal of Mathematics 87, no. 1-3 (February 1994): 275–88. http://dx.doi.org/10.1007/bf02772999.

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39

Rosłanowski, Andrzej, and Saharon Shelah. "Reasonable Ultrafilters, Again." Notre Dame Journal of Formal Logic 52, no. 2 (April 2011): 113–47. http://dx.doi.org/10.1215/00294527-1306154.

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40

Coplakova, Eva, and Klaas Pieter Hart. "Crowded rational ultrafilters." Topology and its Applications 97, no. 1-2 (October 1999): 79–84. http://dx.doi.org/10.1016/s0166-8641(98)00069-8.

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41

Protasov, I. V., and S. V. Slobodianiuk. "Ultrafilters on Balleans." Ukrainian Mathematical Journal 67, no. 12 (May 2016): 1922–31. http://dx.doi.org/10.1007/s11253-016-1200-y.

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42

Šobot, Boris. "|˜-divisibility of ultrafilters." Annals of Pure and Applied Logic 172, no. 1 (January 2021): 102857. http://dx.doi.org/10.1016/j.apal.2020.102857.

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43

Krautzberger, Peter. "On Union Ultrafilters." Order 29, no. 2 (July 16, 2011): 317–43. http://dx.doi.org/10.1007/s11083-011-9223-3.

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44

Börger, Reinhard. "Coproducts and ultrafilters." Journal of Pure and Applied Algebra 46, no. 1 (1987): 35–47. http://dx.doi.org/10.1016/0022-4049(87)90041-7.

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45

Fernández Bretón, David J. "Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property." Canadian Journal of Mathematics 68, no. 1 (February 1, 2016): 44–66. http://dx.doi.org/10.4153/cjm-2015-023-9.

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AbstractWe answer two questions of Hindman, Steprāns, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., , a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.

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46

Simmons, Otto D., Mark D. Sobsey, Christopher D. Heaney, Frank W. Schaefer, and Donna S. Francy. "Concentration and Detection of Cryptosporidium Oocysts in Surface Water Samples by Method 1622 Using Ultrafiltration and Capsule Filtration." Applied and Environmental Microbiology 67, no. 3 (March 1, 2001): 1123–27. http://dx.doi.org/10.1128/aem.67.3.1123-1127.2001.

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ABSTRACT The protozoan parasite Cryptosporidium parvumis known to occur widely in both source and drinking water and has caused waterborne outbreaks of gastroenteritis. To improve monitoring, the U.S. Environmental Protection Agency developed method 1622 for isolation and detection of Cryptosporidium oocysts in water. Method 1622 is performance based and involves filtration, concentration, immunomagnetic separation, fluorescent-antibody staining and 4′,6-diamidino-2-phenylindole (DAPI) counterstaining, and microscopic evaluation. The capsule filter system currently recommended for method 1622 was compared to a hollow-fiber ultrafilter system for primary concentration of C. parvum oocysts in seeded reagent water and untreated surface waters. Samples were otherwise processed according to method 1622. Rates of C. parvumoocyst recovery from seeded 10-liter volumes of reagent water in precision and recovery experiments with filter pairs were 42% (standard deviation [SD], 24%) and 46% (SD, 18%) for hollow-fiber ultrafilters and capsule filters, respectively. Mean oocyst recovery rates in experiments testing both filters on seeded surface water samples were 42% (SD, 27%) and 15% (SD, 12%) for hollow-fiber ultrafilters and capsule filters, respectively. Although C. parvum oocysts were recovered from surface waters by using the approved filter of method 1622, the recovery rates were significantly lower and more variable than those from reagent grade water. In contrast, the disposable hollow-fiber ultrafilter system was compatible with subsequent method 1622 processing steps, and it recovered C. parvum oocysts from seeded surface waters with significantly greater efficiency and reliability than the filter suggested for use in the version of method 1622 tested.

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47

Chentsov, Aleksandr Georgievich. "MAXIMAL LINKED SYSTEMS AND ULTRAFILTERS OF WIDELY UNDERSTOOD MEASURABLE SPACES." Tambov University Reports. Series: Natural and Technical Sciences, no. 124 (2018): 846–60. http://dx.doi.org/10.20310/1810-0198-2018-23-124-846-860.

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Two types of set families (ultrafilters or maximal filters and maximal linked systems) for widely understood measurable space are considered. The resulting sets of ultrafilters and maximal linked systems are equipped with the pair of comparable topologies (within the meaning of «Wallman» and «Stone»). As a result, two bitopological spaces are realized; one of them turns out a subspace of another. More precisely, ultrafilters are maximal linked systems and the totality of the latter forms a cumulative bitopological space. With employment of topological constructions some characteristic properties of ultrafilters and (in smaller power) maximal linked systems are obtained (the question is necessary and sufficient conditions of maximality of filters and linked systems).

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48

Kennedy, Juliette, Saharon Shelah, and Jouko Väänänen. "Regular ultrafilters and finite square principles." Journal of Symbolic Logic 73, no. 3 (September 2008): 817–23. http://dx.doi.org/10.2178/jsl/1230396748.

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AbstractWe show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle introduced in [3]. For such ultrafilters D and cardinals λ there are models of size λ for which Mλ/D is not λ++-universal and elementarily equivalent models M and N of size λ for which Mλ/D and Nλ/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [1].

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49

Starosolski, Andrzej. "P-hierarchy on βω." Journal of Symbolic Logic 73, no. 4 (December 2008): 1202–14. http://dx.doi.org/10.2178/jsl/1230396914.

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AbstractWe classify ultrafilters on ω with respect to sequential contours (see [4]. [5]) of different ranks. In this way we obtain an ω1 sequence of disjoint classes. We prove that non-emptiness of for successor α ≥ 2 is equivalent to the existence of P-point. We investigate relations between P-hierarchy and ordinal ultrafilters (introduced by J. E. Baumgartner in [1]). we prove that it is relatively consistent with ZFC that the successor classes (for α ≥ 2) of P-hierarchy and ordinal ultrafilters intersect but are not the same.

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50

Kanamori, Akihiro. "Finest Partitions for Ultrafilters." Journal of Symbolic Logic 51, no. 2 (June 1986): 327. http://dx.doi.org/10.2307/2274055.

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