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University of North Texas

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Bozeman, Alan Kyle.
*Weakly**Dense* Subsets of Homogeneous Complete Boolean Algebras.

Degree: 1990, University of North Texas

URL: https://digital.library.unt.edu/ark:/67531/metadc330803/

The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^{c}(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B.
Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity.
In chapter III, we prove that for any integer n > 1, wd_{2}(B) = wd_{n}(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B).
In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras.
In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), i.e., the weak density of a homogeneous complete Boolean algebra B is at least as big as the cellularity. Also in this chapter, we introduce discernible sets. We prove that a discernible set of cardinality no greater than c(B) cannot be weakly dense.
In chapter VI, we prove the main result of this dissertation, i.e., d(B) ≤ min(2^< wd(B),sup{λ^{c}(B): λ < wd(B)}). In chapter VII, we list some unsolved problems concerning this dissertation.
*Advisors/Committee Members: Mauldin, R. Daniel, Jackson, Steve, 1957-, Brand, Neal E., Lewis, Paul Weldon.*

Subjects/Keywords: homogeneous complete Boolean algebras; weak density; cellularity; weakly dense sets; cardinal functions; Algebra, Boolean.

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APA (6^{th} Edition):

Bozeman, A. K. (1990). Weakly Dense Subsets of Homogeneous Complete Boolean Algebras. (Thesis). University of North Texas. Retrieved from https://digital.library.unt.edu/ark:/67531/metadc330803/

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

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

Bozeman, Alan Kyle. “Weakly Dense Subsets of Homogeneous Complete Boolean Algebras.” 1990. Thesis, University of North Texas. Accessed August 05, 2020. https://digital.library.unt.edu/ark:/67531/metadc330803/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Bozeman, Alan Kyle. “Weakly Dense Subsets of Homogeneous Complete Boolean Algebras.” 1990. Web. 05 Aug 2020.

Vancouver:

Bozeman AK. Weakly Dense Subsets of Homogeneous Complete Boolean Algebras. [Internet] [Thesis]. University of North Texas; 1990. [cited 2020 Aug 05]. Available from: https://digital.library.unt.edu/ark:/67531/metadc330803/.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Bozeman AK. Weakly Dense Subsets of Homogeneous Complete Boolean Algebras. [Thesis]. University of North Texas; 1990. Available from: https://digital.library.unt.edu/ark:/67531/metadc330803/

Not specified: Masters Thesis or Doctoral Dissertation