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## Filters

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Ultrafilter spaces on the semilattice of partitions

2001
*
Topology and its Applications
*

Replacing the subsets

doi:10.1016/s0166-8641(00)00068-7
fatcat:dbx4p4xr6zbqrozgpvqws6mmby
*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*. ...##
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Ultrafilter spaces on the semilattice of partitions
[article]

2001
*
arXiv
*
pre-print

If

arXiv:math/0109176v1
fatcat:iripgf2r25bhbg2ccvjc4tvs24
*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. ...##
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Coherent ultrafilters and nonhomogeneity

2015
*
Commentationes Mathematicae Universitatis Carolinae
*

We use these

doi:10.14712/1213-7243.2015.123
fatcat:kbomgaxskbhcxa4zknnd2b3lfy
*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. ...##
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Selective and Ramsey Ultrafilters on $G$ -spaces

2017
*
Notre Dame Journal of Formal Logic
*

A free

doi:10.1215/00294527-3839090
fatcat:psb54u3xovhc3pp2ipkyelenna
*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. ...##
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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*? ...##
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Ultrafilters on G-spaces
[article]

2015
*
arXiv
*
pre-print

For a discrete group G and a discrete G-

arXiv:1507.00145v1
fatcat:xptr45bixzcvzcnnyqqmw5cho4
*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] . ...##
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A Categorical Construction of Ultrafilters
[article]

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. ...

##
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Ultrafilters (with dense elements) over closure spaces

2012
*
Reports on Mathematical Logic
*

" and the existence

dblp:journals/rml/Hinnion12
fatcat:zevhy3cbrjhdfbdvuzaxs7nmw4
*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. ...##
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Formulating Szemerédi's Theorem in Terms of Ultrafilters
[article]

2012
*
arXiv
*
pre-print

to a counting measure

arXiv:1206.0967v1
fatcat:zln2alfuyzfatmswrqr7d7s3w4
*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. ...##
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Topological Ramsey Spaces Dense in Forcings
[article]

2018
*
arXiv
*
pre-print

interested in using topological Ramsey

arXiv:1702.02668v2
fatcat:64gtar3mt5g5ppcpoivl5dlhk4
*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. ...##
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Ramseyan ultrafilters

2001
*
Fundamenta Mathematicae
*

Further we introduce an ordering

doi:10.4064/fm169-3-3
fatcat:anfano3j6fdb7jixmxpgvq6p7u
*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*. ...##
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Metrically Ramsey ultrafilters
[article]

2017
*
arXiv
*
pre-print

A free

arXiv:1704.07824v1
fatcat:w6l2z7eg45gmbgikttsi6gxtau
*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 . ...##
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Selective but not Ramsey
[article]

2013
*
arXiv
*
pre-print

We give the first example

arXiv:1312.5411v1
fatcat:y3jmrax3rrd25flnn7l6syb7hq
*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. ...##
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Selective but not Ramsey

2016
*
Topology and its Applications
*

We give the first example

doi:10.1016/j.topol.2015.12.088
fatcat:photasirynbfrdzff57nypdcy4
*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*. ...##
###
Metrically Ramsey ultrafilters

2018
*
Matematychni Studii
*

A free

doi:10.15330/ms.49.2.115-121
fatcat:xlsootvjozaqjkrksnsjchn4i4
*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 . ...
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