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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

URL: https://scholar.colorado.edu/math_gradetds/43

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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

University of Colorado

2. Selker, Kevin. On Some Min-Max Cardinals on Boolean Algebras.

Degree: PhD, Mathematics, 2015, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/34

This thesis is concerned with cardinal functions on Boolean Algebras (BAs) in general, and especially with min-max type functions on atomless BAs. The thesis is in two parts:
(1) We make use of a forcing technique for extending Boolean algebras.
elsewhere. Using and modifying a lemma of Koszmider, and using CH, we prove some general extension lemmas, and in particular obtain an atomless BA, *A* such that *f*(A) = s_{mm}(A) = *w* < *u*(A) = *w*_{1}.
(2) We investigate cardinal functions of min-max and max type and also spectrum functions on moderate products of Boolean algebras. We prove several theorems determining the value of a function on a moderate product in terms of the values of that function on the factors.
*Advisors/Committee Members: James D. Monk, Keith Kearnes, Agnes Szendrei, Carol Cleland, Natasha Dobrinen.*

Subjects/Keywords: Boolean algebras; Logic; Set theory; Algebra; Logic and Foundations; Set Theory

Record Details Similar Records

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

APA (6^{th} Edition):

Selker, K. (2015). On Some Min-Max Cardinals on Boolean Algebras. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/34

Chicago Manual of Style (16^{th} Edition):

Selker, Kevin. “On Some Min-Max Cardinals on Boolean Algebras.” 2015. Doctoral Dissertation, University of Colorado. Accessed November 30, 2020. https://scholar.colorado.edu/math_gradetds/34.

MLA Handbook (7^{th} Edition):

Selker, Kevin. “On Some Min-Max Cardinals on Boolean Algebras.” 2015. Web. 30 Nov 2020.

Vancouver:

Selker K. On Some Min-Max Cardinals on Boolean Algebras. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2020 Nov 30]. Available from: https://scholar.colorado.edu/math_gradetds/34.

Council of Science Editors:

Selker K. On Some Min-Max Cardinals on Boolean Algebras. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/math_gradetds/34

University of Colorado

3. Wiscons, Joshua. Moufang sets of finite Morley rank.

Degree: PhD, Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/math_gradetds/2

We study proper Moufang sets of finite Morley rank for which either the root groups are abelian or the roots groups have no involutions and the Hua subgroup is nilpotent. We give conditions ensuring that the little projective group of such a Moufang set is isomorphic to PSL2(*F*) for *F* an algebraically closed field. In particular, we show that any infinite quasisimple L*-group of finite Morley rank of odd type for which (*B;N;U*) is a split *BN*-pair of Tits rank 1 is isomorphic to SL2(*F*) or PSL2(*F*) provided that *U* is abelian. Additionally, we show that same conclusion can reached by replacing the hypothesis that *U* be abelian with the hypotheses that the intersection of *B* and *N* is nilpotent and *U* is definable and without involutions. As such, we make progress on the open problem of determining the simple groups of finite Morley rank with a split *BN*-pair of Tits rank 1, a problem tied to the current attempt to classify all simple groups of finite Morley rank.
*Advisors/Committee Members: Keith Kearnes, Nathaniel Thiem, Natasha Dobrinen.*

Subjects/Keywords: BN-pair; group of finite Morley rank; Moufang set; Mathematics

Record Details Similar Records

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

APA (6^{th} Edition):

Wiscons, J. (2011). Moufang sets of finite Morley rank. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/2

Chicago Manual of Style (16^{th} Edition):

Wiscons, Joshua. “Moufang sets of finite Morley rank.” 2011. Doctoral Dissertation, University of Colorado. Accessed November 30, 2020. https://scholar.colorado.edu/math_gradetds/2.

MLA Handbook (7^{th} Edition):

Wiscons, Joshua. “Moufang sets of finite Morley rank.” 2011. Web. 30 Nov 2020.

Vancouver:

Wiscons J. Moufang sets of finite Morley rank. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2020 Nov 30]. Available from: https://scholar.colorado.edu/math_gradetds/2.

Council of Science Editors:

Wiscons J. Moufang sets of finite Morley rank. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/math_gradetds/2