Hausdorff gaps and towers in P(ω)/ Fin

We investigate which filters on ω can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size $\ge {\omega _2}$), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$-Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$.

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“…We show that the third possibility, namely, that all towers are nonmeager, is also consistent (this partially answers [14,Problem 63]).…”

Section: Fix α < 1 and Letsupporting

confidence: 70%

Towers in Filters, Cardinal Invariants, and Luzin Type Families

Brendle¹,

Farkas²,

Verner³

2018

J. symb. log.

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“…Since (A ∪ B) B = A/B, we can alternatively write 16 As usual, we adopt the standard convention to define 0 of P Comp /fin as [∅]∼ f in . This solution allows us to establish a homomorphism: P Comp (ω) → P(ω)/fin.…”

Section: A P Com P (ω)/Fin For Relative Sets and Its Algebraic Foundationmentioning

confidence: 99%

“…It practically assigned this notion to the 'static,' set-theoretic paradigm of thinking in the foundation of formal sciences. In addition, a natural conceptual hull of the concept determined by set-theoretic axioms (e.g., GCH), the so-called Haussdorf gaps or Cech-Stone's compactification -as described, e.g., in [16], [17] -makes this algebraic structure hardly elusive from the operational perspective and a dynamic paradigm of category theory.…”

Section: Introductionmentioning

confidence: 99%

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The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory

Jobczyk

2022

IEEE Access

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A category theory constitutes a convenient conceptual apparatus to organize the worlds of mathematical entities. The concept of fuzzy natural transformation as an abstract mapping on functors is one of the essential concepts of this theory. If we admit a piece of non-commutativity in its definition diagrams, then the 'upward' and the 'downward' diagram parts generate different result sets. In this way, we can introduce the concept of fuzzy natural transformation. We deal with the multi-fuzzy natural transformation if such a transformation is based on a multi-diagram. This paper aims to describe the multi-fuzzy natural transformation situation when the symmetric difference of the result is finite. It allows us to organize the whole spectrum of the result sets in a unique quotient algebra P comp (ω)/fin. Different algebraic properties of this structure will be explored, and a piece of classical encoding theory will be reconstructed in the environment determined by this algebra. In particular, the concepts of the k-multi similarity, the abstract k-multi similarity, the k-similarity balls, and the abstract k-similarity balls are introduced as some generalization of the idea of Hamming's distances and Hamming's balls. Finally, it is shown how to automate some verification processes in this context using an R-based programming environment.

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“…First we see (3) ⇒ (2) ⇒ (1). Let P be as in (2). Notice that P forces (a α , b α ) α<ω 1 to split by forcing Γ ⊂ ω 1 without the property of Proposition 2.6.…”

Section: Gaps In [ω]mentioning

confidence: 99%

Trees and gaps from a construction scheme

Lopez

Todorčević

2016

Proc. Amer. Math. Soc.

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Abstract. We present simple constructions of trees and gaps using a general construction scheme that can be useful in constructing many other structures. As a result, we solve a natural problem about Hausdorff gaps in the quotient algebra P(ω)/Fin found in the literature. As it is well known Hausdorff gaps can sometimes be filled in ω 1 -preserving forcing extensions. There are two natural conditions on Hausdorff gaps, dubbed S and T in the literature, that guarantee the existence of such forcing extensions. In part, these conditions are motivated by analogies between fillable Hausdorff gaps and Souslin trees. While the condition S is equivalent to the existence of ω 1 -preserving forcing extensions that fill the gap, we show here that its natural strengthening T is in fact strictly stronger.

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Hausdorff gaps and towers in P(ω)/ Fin

Cited by 5 publications

References 25 publications

Towers in Filters, Cardinal Invariants, and Luzin Type Families

Towers in Filters, Cardinal Invariants, and Luzin Type Families

The Fuzzy Natural Transformations, the Algebra P(ω)/fin and Generalized Encoding Theory

Trees and gaps from a construction scheme

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