Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Donald Monk

Second Advisor

Keith Kearnes

Third Advisor

Natasha Dobrinen

Fourth Advisor

Agnes Szendrei

Fifth Advisor

Peter Mayr

Abstract

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.

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