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You searched for subject:(Cardinal Invariants). Showing records 1 – 2 of 2 total matches.

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University of Colorado

1. Scherer, Charles Frederich. Maximal Comparable and Incomparable Sets in Boolean Algebras.

Degree: PhD, Mathematics, 2016, University of Colorado

We consider the minimal possible sizes of both maximal comparable and maximal incomparable subsets of Boolean algebras. Comparability is given upper and lower bounds for familiar quotients of powerset algebras. The main upper bound is proved using a construction reminiscent of the construction of the reals from Dedekind cuts. Incomparability is placed in relation to the types of dense sets occurring, resulting in several upper bounds. Specifically, the existence of a countable dense set implies the existence of a countable maximal incomparable set, the latter being constructed using a game. A weaker result is proved for uncountable density with the aid of the diamond principle leaving open the question of whether the bound holds in ZFC. Advisors/Committee Members: Donald Monk, Keith Kearnes, Natasha Dobrinen, Agnes Szendrei, Peter Mayr.

Subjects/Keywords: Boolean Algebras; Cardinal Invariants; Logic; Set Theory; Logic and Foundations; Mathematics

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APA (6th Edition):

Scherer, C. F. (2016). Maximal Comparable and Incomparable Sets in Boolean Algebras. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/43

Chicago Manual of Style (16th Edition):

Scherer, Charles Frederich. “Maximal Comparable and Incomparable Sets in Boolean Algebras.” 2016. Doctoral Dissertation, University of Colorado. Accessed November 30, 2020. https://scholar.colorado.edu/math_gradetds/43.

MLA Handbook (7th Edition):

Scherer, Charles Frederich. “Maximal Comparable and Incomparable Sets in Boolean Algebras.” 2016. Web. 30 Nov 2020.

Vancouver:

Scherer CF. Maximal Comparable and Incomparable Sets in Boolean Algebras. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2020 Nov 30]. Available from: https://scholar.colorado.edu/math_gradetds/43.

Council of Science Editors:

Scherer CF. Maximal Comparable and Incomparable Sets in Boolean Algebras. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/math_gradetds/43

2. Montoya, Amaya Diana Carolina. Some cardinal invariants of the generalized Baire spaces.

Degree: 2017, University of Vienna

Diese Dissertation befasst sich mit den bekannten Kardinalzahlinvarianten des Kontinuums: Sie besteht aus zwei Hauptbestandteilen, in der Resulate vorgestellt werden, die in gemeinsamer Arbeit mit (in alphabetischer Reihenfolge) Jörg Brendle, Andrew Brooke-Taylor, Vera Fischer, Sy-David Friedman und Diego Mejía erzielt wurden. Der erste Teil dieser Arbeit beschäftigt sich mit der Verallgemeinerung der klassischen Kardinalzahlinvarianten des Kontinuums zu den verallgemeinerten Baire-Räumen κ^κ , wobei κ eine überabzählbare reguläre Kardinalzahl ist. Zuerst präsentieren wir eine Verallgemeinerung einiger Kardinalzahlen im Cichoń-Diagramm in diesen Kontext und einige der ZFC-Beziehungen, die zwischen ihnen gelten. Darüber hinaus untersuchen wir ihre Werte in einigen generischen Erweiterungen mittels < κ-support- und κ-support-Iterationen von verallgemeinerten klassischen Forcings. Wir weisen auf die Ähnlichkeiten und Unterschiede zu dem klassischen Fall hin und gehen auch auf die Einschränkungen der klassischen Methoden im verallgemeinerten Fall ein. Außerdem studieren wir ein bestimmtes Modell, bei dem die Ultrafilterzahl für κ klein ist, während gleichzeitig 2^κ groß ist und in der auch einige andere Kardinalzahlinvarianten diesen Wert annehmen. Im zweiten Teil konzentrieren wir uns ausschließlich auf den abzählbaren Fall: Wir stellen eine Verallgemeinerung der Methode der Matrix-Iterationen dar um Modelle zu finden, bei denen verschiedene Konstellationen der Kardinalzahlen im Cichoń-Diagramm zusammen mit der almost disjointness number erhalten werden können. Die Methode erlaubt uns auch, eine generische Erweiterung zu finden, in der sieben Kardinalzahlen im Cichoń-Diagramm unterschiedliche Werte annehmen.

The central theme of the research in this dissertation is the well-known Cardinal invariants of the continuum. This thesis consists of two main parts which present the results obtained in joint work with (alphabetically): Jörg Brendle, Andrew Brooke-Taylor, Vera Fischer, Sy-David Friedman and Diego Mejía. The first part focuses on the generalization of the classical cardinal invariants of the continuum to the generalized Baire spaces κ^κ , when κ is a regular uncountable cardinal. First, we present a generalization of some of the cardinals in Cichoń’s diagram to this context and some of the ZFC relationships that are provable between them. Further, we study their values in some generic extensions corresponding to < κ-support and κ-support iterations of generalized classical forcing notions. We point out the similarities and differences with the classical case and explain the limitations of the classical methods when aiming for such generalizations. Second, we study a specific model where the ultrafilter number at κ is small, 2^κ is large and in which a larger family of cardinal invariants can be decided and proven to be < 2^κ. The second part focuses exclusively on the countable case: We present a generalization of the method of matrix iterations to find models where various constellations in…

Subjects/Keywords: 31.10 Mathematische Logik, Mengenlehre; Mengenlehre / Forcing / Kardinalzahlinvarianten des Kontinuums; Set theory / forcing / cardinal invariants of the continuum

…notions . . . . 0.2 Cardinal invariants of the continuum . . . . . . . . . 0.3 Preservation… …Cardinal invariants on the uncountable 1 Cichoń’s Diagram on the uncountable 1.1 ZFC results… …1.1.1 Cardinal invariants of the generalized meager ideal 1.1.2 Slaloms and the invariants… …cardinal invariants in the model . . . . . . . . . . . 2.3.1 κ-maximal almost disjoint families… …scenario, however, one can isolate uncountable cardinals < c. Cardinal invariants (or… 

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Montoya, A. D. C. (2017). Some cardinal invariants of the generalized Baire spaces. (Thesis). University of Vienna. Retrieved from http://othes.univie.ac.at/47024/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Montoya, Amaya Diana Carolina. “Some cardinal invariants of the generalized Baire spaces.” 2017. Thesis, University of Vienna. Accessed November 30, 2020. http://othes.univie.ac.at/47024/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Montoya, Amaya Diana Carolina. “Some cardinal invariants of the generalized Baire spaces.” 2017. Web. 30 Nov 2020.

Vancouver:

Montoya ADC. Some cardinal invariants of the generalized Baire spaces. [Internet] [Thesis]. University of Vienna; 2017. [cited 2020 Nov 30]. Available from: http://othes.univie.ac.at/47024/.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Montoya ADC. Some cardinal invariants of the generalized Baire spaces. [Thesis]. University of Vienna; 2017. Available from: http://othes.univie.ac.at/47024/

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.