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Analytic ideals and cofinal types

Alain Louveau, Boban Velickovi
1999 Annals of Pure and Applied Logic  
We also study ideals associated to classical Banach spaces and ideals of compacts sets in a Polish space.  ...  We study the class of analytic ideals on the set of natural numbers ordered under Tukey reducibility.  ...  Is .F the maximum cofinal type among analytic a-ideals of compact sets?  ... 
doi:10.1016/s0168-0072(98)00065-7 fatcat:64v66i4wm5foxiupdym6xnjzua

Cofinal types of topological directed orders

SŁawomir Solecki, Stevo Todorcevic
2004 Annales de l'Institut Fourier  
We investigate the structure of the Tukey ordering among directed orders arising naturally in topology and measure theory.  ...  The relation D ≡ T E is equivalent to the existence of a directed partial order into which both D and E embed as cofinal subsets. We say then that D and E realize the same cofinal type [14] .  ...  We establish a result about definability of cofinal subsets of analytic ideals.  ... 
doi:10.5802/aif.2070 fatcat:pwlg5ruhkjbv3j267awzgpx3sq

Tukey Order, Calibres and the Rationals [article]

Paul Gartside, Ana Mamatelashvili
2016 arXiv   pre-print
Particular emphasis is placed on the position of K(M) when M is: completely metrizable, the rationals Q, co-analytic or analytic.  ...  One partially ordered set, Q, is a Tukey quotient of another, P, denoted P ≥_T Q, if there is a map ϕ : P → Q carrying cofinal sets of P to cofinal sets of Q.  ...  Note also that the Tukey type and cofinality of a K(B) where B is totally imperfect and has size ≥ ℵ ω is dependent on what happens at ℵ ω . 3.3. General Calibres.  ... 
arXiv:1606.06493v2 fatcat:6ik5numlnrdq5bvx7baegeol6m

Near coherence of filters. II. Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups

Andreas Blass
1987 Transactions of the American Mathematical Society  
Thus, all these statements are also consistent and independent.  ...  We show that NCF is equivalent to the following statements, among others: (1) The ideal of compact operators on Hilbert space is not the sum of two smaller ideals. (2) The Stone-Cech remainder of a half-line  ...  Thus, a growth type is an ideal in w ~ w that is closed under doubling. Of the ideals discussed in the last section, B and w ~ w are growth types.  ... 
doi:10.1090/s0002-9947-1987-0876466-8 fatcat:ta6dzmf6ojh5zngewj2uvzndvy

Page 32 of Mathematical Reviews Vol. , Issue 93a [page]

1993 Mathematical Reviews  
In the paper under review, the authors show that in the Cohen model for 2%° = N>, the property S — (cofinal subset)3 (and also S — (cofinal subset);, for r,k € w) holds if and only if S has finite character  ...  (J-NAGOS) Mansfield and Solovay type results on covering plane sets by lines. Nagoya Math. J. 124 (1991), 145-155. Let 2 be a family of subsets of the plane R? and let x be a cardinal number.  ... 

Dimension zero vs measure zero

Ondřej Zindulka
1999 Proceedings of the American Mathematical Society  
Similar questions concerning perfectly meager sets and other types of small sets are also discussed.  ...  It is shown that in a model of set theory it is so for separable metric spaces and that under the Martin's Axiom there are separable metric spaces of positive dimension yet of universal measure zero.  ...  If X is a space and J an ideal on X, call J proper if it contains all singletons and X / ∈ J .  ... 
doi:10.1090/s0002-9939-99-05225-9 fatcat:spufqea4mbh67lapuj5n5vehcy

On a σ-ideal of compact sets

Tamás Mátrai
2010 Topology and its Applications  
Soc. 137 (3) (2009) 1115-1125] a G δ σ -ideal of compact subsets of 2 ω and prove that it is not Tukey reducible to the ideal I 1/n = {H ⊆ ω: h∈H 1/h < ∞}. This result answers a question of S.  ...  Solecki and S. Todorčević in the negative.  ...  Thus analytic P -ideals are rich in cofinal types. -We will show M ∩ K(2 ω ) < T I in Section 3. This gives a negative answer to [8, Question 3, p. 194 ].  ... 
doi:10.1016/j.topol.2009.06.014 fatcat:s5gr24iejzbv7b6fnj2uzp6sr4

Page 3632 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
A class of types, called low types, is identified. These are the types which can be realized by a random element of some ultrapower.  ...  One concern is the cofinality of the Z-prod / and the coinitiality of the nonstandard part. These cardinals are called the cofinality and coinitiality of 2. (CH implies these cardinals must be ¥;).  ... 

Maximal Tukey types, P-ideals and the weak Rudin-Keisler order [article]

Konstantinos A. Beros, Paul B. Larson
2019 arXiv   pre-print
This discussion segues into an examination of a refinement of the Tukey order -- known as the "weak Rudin-Keisler order" -- and its structure when restricted to these ideals of maximal Tukey type.  ...  Mirroring a result of Fremlin on the Tukey order, we also show that there is an analytic P-ideal above all other analytic P-ideals in the weak Rudin-Keisler.  ...  Two partial orders have the same Tukey type (or cofinal type) iff each is Tukey-reducible to the other. Tukey himself examined cofinal types in the context of convergence in topological spaces.  ... 
arXiv:1902.00968v2 fatcat:siird2dzfje6dpxwnqkicycihq

Analytic gaps

Stevo Todorčević
1996 Fundamenta Mathematicae  
For every nonatomic analytic P-ideal A on N there is a Borel monotonic map from A onto a cofinal subset of NN . P r o o f.  ...  Suppose A is an analytic ideal whose orthogonal A⊥ is not a P -ideal. Then there is a Borel monotonic map which transfers A to a cofinal subset of NN . P r o o f.  ... 
doi:10.4064/fm-150-1-55-66 fatcat:eqlmkjb2evbinpgxspp6trr3h4

Page 3160 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
(e) cf(Ky) > Ky. 03 MATHEMATICAL LOGIC AND FOUNDATIONS 3160 Let Y and # denote the ideal of null sets and the ideal of meager sets, respectively.  ...  Silver proved that every analytic set is Ramsey, while A. Kechris showed that every analytic set is K,-regular.  ... 

Cofinal families of Borel equivalence relations and quasiorders

Christian Rosendal
2005 Journal of Symbolic Logic (JSL)  
There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal.  ...  Families of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤ B , are constructed.  ...  We construct an analytic ideal I max generating a complete analytic equivalence relation.  ... 
doi:10.2178/jsl/1129642127 fatcat:dlb4kye6ubaspkiqqeo2qqazky

Page 3994 of Mathematical Reviews Vol. , Issue 83j [page]

1983 Mathematical Reviews  
3:03.054 thesis [“Model theory of fields (decidability and bounds for polynomial ideals)”, see Chapter III, Ph.D. Thesis, Rijksuniv. Utrecht, Utrecht, 1978].  ...  From this it follows that for the least a of cofinality w, such that R, is KM-expandable there is exactly one expansion of R, to a model of KM + ‘all classes are ramified analytical’.  ... 

Model Theory Methods for Topological Groups

Tomás Ibarlucía
2018 Bulletin of Symbolic Logic  
This correspondence can be inverted and we extend it to types and saturation.  ...  the second is the existence of a precipitous ideal on 1 .  ...  The first key result has applications on forcing projective or J 1+ (R) well-orderings of the reals, depending on the order type of the strong cardinals of K below V 1 , and on 2-step stationary forcing  ... 
doi:10.1017/bsl.2018.32 fatcat:wgj65guzljh6xazhthsokheyv4

Page 1204 of Mathematical Reviews Vol. 34, Issue 5 [page]

1967 Mathematical Reviews  
set of integers is cofinal, and (iii) if na20, where n is a positive integer, then az0.  ...  A partition # of X into closed subsets is regenerate if feO(X), f|\K ¢ A\|K for all K ¢ X imply fe A, and if for every maximal ideal J of A there exists a unique K ¢ # such that (/|K)~ is a maximal ideal  ... 
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