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On union ultrafilters [article]

Peter Krautzberger
2011 arXiv   pre-print
We present some new results on union ultrafilters. We characterize stability for union ultrafilters and, as the main result, we construct a new kind of unordered union ultrafilter.  ...  of Berlin.  ...  Acknowledgments This work evolved out of the author's Ph.D. thesis [Kra09] written under the supervision of Sabine Koppelberg at the Freie Universität Berlin and supported by the NaFöG grant of the state  ... 
arXiv:1012.4532v2 fatcat:twd4cf6alnhohmfqdltqvaffem

Slow ultrafilters and asymptotic cones of proper metric spaces [article]

Lars Scheele
2010 arXiv   pre-print
Furthermore I answer a question of Drutu and Sapir concerning slow ultrafilters.  ...  In this paper I present an elementary construction to prove that any proper metric space can arise as the asymptotic cone of another proper metric space.  ...  Let α be the sequence of scaling factors obtained by ordering the elements of A in the natural order and β the sequence obtained from the set B.  ... 
arXiv:1010.1699v1 fatcat:3h7ujapxijbpvcxivhryfth2ui

Selective ultrafilters and $\omega \rightarrow (\omega )^\omega $

Todd Eisworth
1999 Proceedings of the American Mathematical Society  
Logic. 12 (1977) , 59-111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in L(R) is U-Ramsey with respect to every selective ultrafilter  ...  Now we turn to the construction. We define for each α < ω 1 the sequence Z α = Z ξ : ξ < ω · 2α (3.4) (so Z α+1 is obtained from Z α by appending a sequence of order-type ω + ω).  ...  Such ultrafilters do not necessarily exist -in [3] Kunen constructs a model of ZFC in which there are no selective ultrafilters.  ... 
doi:10.1090/s0002-9939-99-04835-2 fatcat:af7xc5hm7ng6zn3o2mq3esgg5y

On Milliken-Taylor Ultrafilters

Heike Mildenberger
2011 Notre Dame Journal of Formal Logic  
We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model.  ...  We show that k-coloured Milliken-Taylor ultrafilters have at least k + 1 near coherence classes of ultrafilters in its projection to ω.  ...  I thank Jordi-López Abad for his question about the number of near coherence classes in the core of a Milliken-Taylor ultrafilter at this special semester.  ... 
doi:10.1215/00294527-1499345 fatcat:pncmsldelfabjd73js3vfcjezm

Minimal idempotent ultrafilters and the Auslander-Ellis theorem [article]

Alexander P. Kreuzer
2015 arXiv   pre-print
We obtain that the existence of minimal idempotent ultrafilters restricted to countable algebras of sets is equivalent to the Auslander-Ellis theorem (AET) and that the existence of minimal idempotent  ...  ultrafilters as higher-order objects is Π^1_2-conservative over a refinement of AET.  ...  To construct this object from the sequence (n i ) one in general needs µ. The construction in the proof of Lemma 10 can be summed up in the following way.  ... 
arXiv:1305.6530v2 fatcat:t7jrssot4re3rcevejv274tn6e

A parametrized diamond principle and union ultrafilters

David Fernández-Bretón, Michael Hrušák
2018 Colloquium Mathematicum  
As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.  ...  We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters.  ...  Acknowledgments The first author was partially supported by Postdoctoral Fellowship number 275049 from the Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico.  ... 
doi:10.4064/cm7364-10-2017 fatcat:buf4qoxlhbey7pledm5pckddsy

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

Roberto E. Madariaga-Garcia, Artur Hideyuki Tomita
2007 Topology and its Applications  
In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MA countable .  ...  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.  ...  The authors thank the referee for a careful reading of the manuscript and for making several suggestions to improve the presentation of the article.  ... 
doi:10.1016/j.topol.2006.03.028 fatcat:ueqte3swjfcdxge5asrdlugtpy

MODEL-THEORETIC PROPERTIES OF ULTRAFILTERS BUILT BY INDEPENDENT FAMILIES OF FUNCTIONS

M. MALLIARIS, S. SHELAH
2014 Journal of Symbolic Logic (JSL)  
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.  ...  less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory.  ...  a large set U of κ so that any two < ζ from U have the same associated pattern.  ... 
doi:10.1017/jsl.2013.28 fatcat:zbxjxmg7irbwponfpxpuqhpube

Gruff ultrafilters

David J. Fernández-Bretón, Michael Hrušák
2016 Topology and its Applications  
We investigate the question of whether Q carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen).  ...  at the end of the paper.]  ...  The authors would like to thank Andreas Blass for some very fruitful discussions, particularly concerning Cohen's paper [5] , as well as for greatly simplifying the proof of Lemma 1.2.  ... 
doi:10.1016/j.topol.2016.08.012 fatcat:ggrem5nugrafdoj66m2kbzity4

Social choice with infinite populations: construction of a rule and impossibility results

Graciela Chichilnisky, Geoffrey Heal
1997 Social Choice and Welfare  
We provide a simple construction of social choice rules for economies with infinite populations .  ...  The rules are continuous, Pareto and nondictatorial ; they are constructed as limits of individual preferences when the limit exists, and otherwise as adequate generalizations .  ...  Hence we can construct a sequence zi , converging on y from above, everyone satisfying zi p x.  ... 
doi:10.1007/s003550050067 fatcat:br32u5uh5beofkzxjv4fmhnivy

Social choice with infinite populations: construction of a rule and impossibility results [chapter]

Graciela Chichilnisky, Geoffrey Heal
1997 Topological Social Choice  
We provide a simple construction of social choice rules for economies with infinite populations .  ...  The rules are continuous, Pareto and nondictatorial ; they are constructed as limits of individual preferences when the limit exists, and otherwise as adequate generalizations .  ...  Hence we can construct a sequence zi , converging on y from above, everyone satisfying zi p x.  ... 
doi:10.1007/978-3-642-60891-9_10 fatcat:bxxuasmszfethpngtwsqqt6nta

Iterated asymptotic cones [article]

Lars Scheele, Alessandro Sisto
2011 arXiv   pre-print
In this paper we define a class of metric spaces which display a wide range of behaviors with respect to iterated asymptotic cones, and we use those to construct examples within the class of groups.  ...  Iterated asymptotic cones have been used by Druţu and Sapir to construct a group with uncountably many pairwise non-homeomorphic asymptotic cones.  ...  Take X as above from Theorem 2.7, using the sequence α to define variable and fixed parts. Let G be a group as in Theorem 3.8. Fix an ultrafilter µ and set C := Cone µ (G, e, α) .  ... 
arXiv:1108.1146v1 fatcat:7hlkzoh3tzhjfbmy2pdiewlzhq

Formulating Szemerédi's Theorem in Terms of Ultrafilters [article]

Heinrich-Gregor Zirnstein
2012 arXiv   pre-print
This diploma thesis gives an interpretation of Szemerédi's theorem in terms of ultrafilters as well.  ...  Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that  ...  Back to ultrafilter construction. In a sense, condition 4 from the ultrafilter definition is also a condition of maximality.  ... 
arXiv:1206.0967v1 fatcat:zln2alfuyzfatmswrqr7d7s3w4

Finite and countable families of algebras of sets

L. S. Grinblat
2010 Mathematical Research Letters  
In the formulation of these conditions we use ω-saturated algebras and finite sequences of ultrafilters on X. 1. The formulation of results 1.1.  ...  As usual, P(M ) denotes the set of all subsets of the set M . When it is clear from the context, we will not state explicitly that a set belongs to P(X).  ...  sequence of ω-cycles.  ... 
doi:10.4310/mrl.2010.v17.n4.a3 fatcat:spues2soirc7piyopxegy6xn7y

$G_\delta $ and co-meager semifilters

William R. Brian, Jonathan L. Verner
2016 Fundamenta Mathematicae  
The ultrafilters on the partial order ([ω]^ω,⊆^*) are the free ultrafilters on ω, which constitute the space ω^*, the Stone-Cech remainder of ω.  ...  Most notably, we will prove that (ω^*,+) contains minimal left ideals that are also weak P-sets.  ...  Acknowledgement The second author would like to thank the members of the Prague Set Theory seminar for fruitful discussions of the topic.  ... 
doi:10.4064/fm182-2-2016 fatcat:gl2gjv5iqzcxtgrk2ch2hx57k4
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