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Ultrafilter spaces on the semilattice of partitions

Lorenz Halbeisen, Benedikt Löwe
2001 Topology and its Applications  
Replacing the subsets of ω by partitions of ω in the construction of the ultrafilter space gives non-homeomorphic spaces of partition ultrafilters corresponding to βω.  ...  The Stone-Čech compactification of the natural numbers βω (or equivalently, the space of ultrafilters on the subsets of ω) is a well-studied space with interesting properties.  ...  of spaces of partition ultrafilters.  ... 
doi:10.1016/s0166-8641(00)00068-7 fatcat:dbx4p4xr6zbqrozgpvqws6mmby

Ultrafilter spaces on the semilattice of partitions [article]

Lorenz Halbeisen, Benedikt Loewe
2001 arXiv   pre-print
If one replaces the subsets of omega by partitions of omega, one can define corresponding, non-homeomorphic spaces of partition ultrafilters.  ...  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.  ...  We feel that we have illustrated how little is known about the fascinating field of ultrafilters on semilattices, and we would like to see more results along these lines.  ... 
arXiv:math/0109176v1 fatcat:iripgf2r25bhbg2ccvjc4tvs24

Coherent ultrafilters and nonhomogeneity

Jan Starý
2015 Commentationes Mathematicae Universitatis Carolinae  
We use these ultrafilters in a topological application: a coherent P-ultrafilter on an algebra B is an untouchable point in the Stone space of B, witnessing its nonhomogeneity.  ...  We introduce the notion of a coherent P-ultrafilter on a complete ccc Boolean algebra, strenghtening the notion of a P-point on ω, and show that these ultrafilters exist generically under c = d.  ...  For an ultrafilter U on B and P a partition of B, let U P = U ∩ B P , which is clearly an ultrafilter on B P . As B P is isomorphic to P (ω), the ultrafilter U P can be viewed as an ultrafilter on ω.  ... 
doi:10.14712/1213-7243.2015.123 fatcat:kbomgaxskbhcxa4zknnd2b3lfy

Selective and Ramsey Ultrafilters on $G$ -spaces

Oleksandr Petrenko, Igor Protasov
2017 Notre Dame Journal of Formal Logic  
A free ultrafilter on X is called G-selective if, for any G-invariant partition of X, either one cell of is a member of , or there is a member of which meets each cell of in at most one point.  ...  Theorems 5 and 6 give us a plenty of Z-selective ultrafilters on Z (as a regular Z-space) but not Z-Ramsey. We conjecture that each Z-Ramsey ultrafilter is selective.  ...  A free ultrafilter U on an infinite set X is said to be selective if, for any partition P of X, either one cell of P is a member of U, or some member of U meets each cell of P in at most one point.  ... 
doi:10.1215/00294527-3839090 fatcat:psb54u3xovhc3pp2ipkyelenna

Page 2591 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
Must the range of every ordinal-valued function on the set of positive integers be “small” on some set in the ultrafilter?  ...  The author considers the following sorts of questions: Must the range of every real-valued function on the set of positive integers be “small” on some set in the ultrafilter?  ... 

Ultrafilters on G-spaces [article]

Oleksandr Petrenko, Igor Protasov
2015 arXiv   pre-print
For a discrete group G and a discrete G-space X, we identify the Stone-Čech compactifications β G and β X with the sets of all ultrafilters on G and X, and apply the natural action of β G on β X to characterize  ...  We use G-invariant partitions and colorings to define G-selective and G-Ramsey ultrafilters on X.  ...  We note that Theorem 7.4 plays the key part in the proof of Theorem 6.1. For ultrafilters on metric spaces and balleans we address the reader to [12, 20, 24] .  ... 
arXiv:1507.00145v1 fatcat:xptr45bixzcvzcnnyqqmw5cho4

A Categorical Construction of Ultrafilters [article]

Daniel Litt, Zachary Abel, Scott D. Kominers
2009 arXiv   pre-print
of the profiniteness of Stone spaces.  ...  In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partitions; this is an elementary and intuitive presentation of a consequence  ...  They would also like to acknowledge Brett Harrison for his excellent suggestions on earlier drafts of this paper.  ... 
arXiv:0710.2497v2 fatcat:jtzky4fp4rdsrptlpmf3ksxpga

Ultrafilters (with dense elements) over closure spaces

Roland Hinnion
2012 Reports on Mathematical Logic  
" and the existence of special ultrafilters (namely ultrafilters which elements are dense) on such structures. .  ...  Several notions and results that are useful for directed sets (and their applications) can be extended to the more general context of closure spaces; inter alia the so-called "finite intersection property  ...  A partition F of an acyclic closure space E is "fatal" iff no element X of F is dense in E. Remark.  ... 
dblp:journals/rml/Hinnion12 fatcat:zevhy3cbrjhdfbdvuzaxs7nmw4

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

Heinrich-Gregor Zirnstein
2012 arXiv   pre-print
to a counting measure on βN, almost all ultrafilters have these properties.  ...  This diploma thesis gives an interpretation of Szemerédi's theorem in terms of ultrafilters as well.  ...  Chapter 4 introduces a family of counting measures on the space of ultrafilters βN.  ... 
arXiv:1206.0967v1 fatcat:zln2alfuyzfatmswrqr7d7s3w4

Topological Ramsey Spaces Dense in Forcings [article]

Natasha Dobrinen
2018 arXiv   pre-print
interested in using topological Ramsey space techniques to gain finer insight into ultrafilters satisfying weak partition relations.  ...  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.  ...  Ultrafilters of Laflamme with increasingly weak partition relations.  ... 
arXiv:1702.02668v2 fatcat:64gtar3mt5g5ppcpoivl5dlhk4

Ramseyan ultrafilters

Lorenz Halbeisen
2001 Fundamenta Mathematicae  
Further we introduce an ordering on the set of partition-filters and consider the dual form of some cardinal characteristics of the continuum.  ...  We investigate families of partitions of ω which are related to special coideals, so-called happy families, and give a dual form of Ramsey ultrafilters in terms of partitions.  ...  The proof is the same as the proof of Lemma 2.1.2, but restricted to the partition-ultrafilter U . 2.2. On the existence of Ramseyan ultrafilters.  ... 
doi:10.4064/fm169-3-3 fatcat:anfano3j6fdb7jixmxpgvq6p7u

Metrically Ramsey ultrafilters [article]

Igor Protasov, Ksenia Protasova
2017 arXiv   pre-print
A free ultrafilter U on an infinite metric space (X,d) is called metrically Ramsey if, for every isometric coloring χ of [X]^2, there is a member U∈U such that the set [U]^2 is χ-monochrome.  ...  We prove that every metrically Ramsey ultrafilter U on N has a member with no arithmetic progression of length 2, and if U has a thin member then there is a mapping f:N⟶ω such that f(U) is a Ramsey ultrafilter  ...  We say that a free ultrafilter on an infinite metric space (X, d) is metrically Ramsey if U is Ramsey with respect to all isometric colorings of [X] 2 .  ... 
arXiv:1704.07824v1 fatcat:w6l2z7eg45gmbgikttsi6gxtau

Selective but not Ramsey [article]

Timothy Trujillo
2013 arXiv   pre-print
We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space, and in fact a countable collection of such examples.  ...  If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a well-known result of Kunen the two are equivalent.  ...  In [1] , Mijares generalizes the notion of selective ultrafilter on ω to a notion of selective ultrafilter on an arbitrary topological Ramsey space R.  ... 
arXiv:1312.5411v1 fatcat:y3jmrax3rrd25flnn7l6syb7hq

Selective but not Ramsey

Timothy Trujillo
2016 Topology and its Applications  
We give the first example of an ultrafilter on a topological Ramsey space that is selective but not Ramsey for the space, and in fact a countable collection of such examples.  ...  If R is taken to be the Ellentuck space then the two concepts reduce to the familiar notions of Ramsey and selective ultrafilters on ω; so by a wellknown result of Kunen the two are equivalent.  ...  one part of the partition.  ... 
doi:10.1016/j.topol.2015.12.088 fatcat:photasirynbfrdzff57nypdcy4

Metrically Ramsey ultrafilters

I. V. Protasov, K. D. Protasova
2018 Matematychni Studii  
A free ultrafilter U on an infinite metric space (X, d) is called metrically Ramsey if, for every isometric coloring χ of [X] 2 , there is a member U ∈ U such that the set [U ] 2 is χ-monochrome.  ...  We prove that each infinite ultrametric space (X, d) has a countable subset Y such that each free ultrafilter U on X satisfying Y ∈ U is metrically Ramsey.  ...  We say that a free ultrafilter on an infinite metric space (X, d) is metrically Ramsey if U is Ramsey with respect to all isometric colorings of [X] 2 .  ... 
doi:10.15330/ms.49.2.115-121 fatcat:xlsootvjozaqjkrksnsjchn4i4
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